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Incorporating complexities into the explanation of decision making

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Title:
Incorporating complexities into the explanation of decision making strategy simulations and an empirical test
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Decker, Nathaniel
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University of South Florida
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Risk
Context
Dynamic
Aspiration
Trajectory
Dissertations, Academic -- Psychology -- Masters -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Summary:
ABSTRACT: This investigation of risky decision making models the standard forced-choice two outcome lottery task by incorporating elements of complexity present in real-world decision making. Potential decision criteria such as current wealth and quality of life information were made available to examine the influence of time-dependent contextual cues on decision strategy selection, since previous investigations of decision making have not included specific contextual cues that would allow for people to use complex or "dynamic" decision strategies. Two studies explored simulated decision strategies requiring more or less complexity. Results suggest that strategies using dynamic, time-dependent criteria provide important advantages over simpler strategies.Also, as 'aspirations' become closer to the most likely outcome and as trajectories include a larger margin of previous experiences, there is more control over improvements to the likelihood of ending up in an extremely good place over an extremely bad place. Certain changes to the decision environment seem to affect the accuracy of dynamic decision strategies, which in turn can help or hinder their effectiveness. As a test of convergence, an empirical test was conducted to compare actual decision strategy use with simulated decision strategies. Two distinctly different decision tasks were used: one required only passive choices between two lotteries and the other required active changes to a given lottery situation. Information about lottery outcomes, current wealth, and quality of life were provided to participants to provide additional context to the decision environment.Participants seemed to be using a variety of different strategies, including strategies that focus on dynamic information. Simple risk policies were often very good at describing risk preferences, though a subset of participants were relying on a complex decision strategies. There were also systematic differences in dynamic decision strategy usage. The combination of simulations and the empirical investigation elucidate the advantages to exploring risk preferences with attention to different decision strategies in specific environments, especially including more complex or "dynamic" decision strategies.
Thesis:
Thesis (M.A.)--University of South Florida, 2008.
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Includes bibliographical references.
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by Nathaniel Decker.
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Title from PDF of title page.
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Document formatted into pages; contains 103 pages.

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oclc - 405639416
usfldc doi - E14-SFE0002407
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ABSTRACT: This investigation of risky decision making models the standard forced-choice two outcome lottery task by incorporating elements of complexity present in real-world decision making. Potential decision criteria such as current wealth and quality of life information were made available to examine the influence of time-dependent contextual cues on decision strategy selection, since previous investigations of decision making have not included specific contextual cues that would allow for people to use complex or "dynamic" decision strategies. Two studies explored simulated decision strategies requiring more or less complexity. Results suggest that strategies using dynamic, time-dependent criteria provide important advantages over simpler strategies.Also, as 'aspirations' become closer to the most likely outcome and as trajectories include a larger margin of previous experiences, there is more control over improvements to the likelihood of ending up in an extremely good place over an extremely bad place. Certain changes to the decision environment seem to affect the accuracy of dynamic decision strategies, which in turn can help or hinder their effectiveness. As a test of convergence, an empirical test was conducted to compare actual decision strategy use with simulated decision strategies. Two distinctly different decision tasks were used: one required only passive choices between two lotteries and the other required active changes to a given lottery situation. Information about lottery outcomes, current wealth, and quality of life were provided to participants to provide additional context to the decision environment.Participants seemed to be using a variety of different strategies, including strategies that focus on dynamic information. Simple risk policies were often very good at describing risk preferences, though a subset of participants were relying on a complex decision strategies. There were also systematic differences in dynamic decision strategy usage. The combination of simulations and the empirical investigation elucidate the advantages to exploring risk preferences with attention to different decision strategies in specific environments, especially including more complex or "dynamic" decision strategies.
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Incorporating Complexities into th e Explanation of Decision Making: Strategy Simulations and an Empirical Test by Nathaniel Decker A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Psychology College of Arts and Sciences University of South Florida Major Professor: Sand ra L. Schneider, Ph.D. Doug Nelson, Ph.D. Edward Levine, Ph.D. Date of Approval: March 20, 2008 Keywords: Risk, Context, Dyna mic, Aspiration, Trajectory Copyright 2008, Nathaniel Decker

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i Table of Contents List of Tables iii List of Figures iv Abstract vi Introduction 1 Vantage Point Dependency 2 Goal Dependency 5 Aspirations about the Future 9 Modeling Decision Complexity: Dynamic Systems 12 Study One: Investigate Decision Strategy Simulations Pilot Simulations 17 Decision Policies 18 Lottery-Based Strategies 19 Contextual or Aspiration-Based Strategies 20 Procedure 21 Preliminary Results 21 Current Simulation Study One Methods 30 Materials 33 Design Strategies. 34 Orders. 35 Measurements. 35 Participants 35 Procedure 36 Study One Analysis of Simulations The Aspiration Criterion: A Thorough Investigation 37 New Phase of Strategies: Outcome and Cumulative Trajectory 41 Why there is Skew and the In sensitivity to Expected Value. 43 The Direction of Skew for a Trajectory Strategy. 43 Why Cumulative Trajecto ries differ from Outcome Trajectories. 44 The Number of Previous Outcomes Being Evaluated. 45 Study One Results Summary 47

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ii Study Two: Test for Parameters of the Decision Environment Study Two Methods 49 Study Two Analysis of Simulations An Examination of Trial Length 50 The Implications of Going Broke 53 Riskless versus Very Risky Lotteries 54 Study Two Results Summary 56 Study Three: Empirical Investigation of In teraction between Situation and Context Pilot Empirical Work 58 Methods for Empirical Study 64 Participants 64 Stimulus 64 Design 65 The Micro-World 66 Procedure 67 Results of Empirical Study 69 Participants Who “Went Broke” 69 Decision Strategy Comparison 70 Simple Differences in the Amount of Risk 72 Risk Preferences as a Function of Lottery-Dependent Characteristics 73 Risk Preferences as a Function of Current Wealth 75 Risk Preferences as a Function of Wealth X Valence 79 Risk Preferences as a Functi on of Trajectory Information 81 Discussion of Empirical Study 84 General Discussion 90 References 97 Appendices 100 Appendix A: Quiz Sample 101 Appendix B: Social Status Example 103

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iii List of Tables Table 1: List of Strategy Type s used in Pilot Simulations. 34Table 2: Summary of Soci al Status Indicators. 68Table 3: Percentage of Participants Predicted by each Decision Strategy Type. 71Table 4: Number of Participants for Point Biserial Correlation 79

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iv List of Figures Figure 1: An example of Bernoulli’s (1738/1954) simple utility curve. 3 Figure 2: S-shaped Utility Curve (Tversky & Kahneman, 1981). 4 Figure 3: The final outcome distributions for the baseline strategies. 22 Figure 4: Final outcome distributions for strategies based on models of decision-making. 23 Figure 5: The percentage of the virtual pa rticipants for a give n strategy found in the extremes of the final outcome distribution. 25 Figure 6: Final outcome distributions fo r the strategies ba sed on aspiration level criteria. 27 Figure 7: The percentage of the virtual pa rticipants for a give n strategy found in the extremes of the final outcome distribution. 29 Figure 8: Theoretical Example of how distri butional variability is manipulated with respect to current progress. 37 Figure 9: Rendering of the “optim al” skewed distribution. 38 Figure 10: Comparison of different aspi ration level strategy final outcome distributions with the optimal skew ed distribution under pilot (36 trials) conditions. 39 Figure 11: Comparison of different outco me and cumulative trajectory final outcome distributions with the “o ptimal” skewed distribution under pilot conditions. 42 Figure 12: Outcome Trajectories using diffe rent number of previous outcomes as determinants for risk preference. 45 Figure 13: Comparison of 36 trial lottery distributions and 72 trial lottery distributions for aspira tion level strategies. 51

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v Figure 14: Comparison of 36 trial lottery distributions and 72 trial lottery distributions for trajectory strategies. 52 Figure 15: Percentage of pa rticipants who dropped out of the study (via reaching zero) for each static strategy (lef t) and dynamic strategy (right). 53 Figure 16: Aspiration level st rategies for riskless v. ve ry risky lottery set. 55 Figure 17: Ticket Transfer Para digm-Move Example Diagram. 59 Figure 18: Ticket Transfer Para digm-Choice Example Diagram. 60 Figure 19: Risk preference by Valence for the Choice task in Previous Research. 61 Figure 20: Risk preference by Valence comparison of Move vs. Choice tasks from Previous Research. 62 Figure 21: Comparison of previous and current Move v. Choice data. 73 Figure 22: Individual Risk Attitudes at each Wealth Level. 76 Figure 23a: Wealth Level by Va lence for Choice Task. 80 Figure 23b: Wealth Level by Valence for Move Task. 75

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vi Incorporating Complexities into th e Explanation of Decision Making: Strategy Simulations and an Empirical Test Nathaniel Decker ABSTRACT This investigation of risky decision ma king models the standard forced-choice two outcome lottery task by incorporating elem ents of complexity present in real-world decision making. Potential decision criteria su ch as current wealth and quality of life information were made available to examine the influence of time-dependent contextual cues on decision strategy selec tion, since previous investigatio ns of decision making have not included specific contextual cues that would allow for people to use complex or “dynamic” decision strategies. Two studies explored simu lated decision strategies requiring more or less complexity. Results suggest that stra tegies using dynamic, time-dependent criteria provide important advantages over simpler st rategies. Also, as ‘aspirations’ become closer to the most likely outcome and as trajectories include a larger margin of previous experiences, there is more control over impr ovements to the likelihood of ending up in an extremely good place over an extremely bad pl ace. Certain changes to the decision environment seem to affect the accuracy of dynamic decision strategies, which in turn can help or hinder their effectiveness.

PAGE 8

vii As a test of convergence, an empirical test was conducted to compare actual decision strategy use with simulated decision st rategies. Two distinctly different decision tasks were used: one required only passive choices between two lotteries and the other required active changes to a given lottery s ituation. Information a bout lottery outcomes, current wealth, and quality of life were provi ded to participants to provide additional context to the decision environment. Participants seemed to be using a vari ety of different strategies, including strategies that focus on dynamic information. Simple risk policies were often very good at describing risk preferences, though a subset of participants were relying on a complex decision strategies. There were also system atic differences in dynamic decision strategy usage. The combination of simulations a nd the empirical investigation elucidate the advantages to exploring risk pr eferences with attention to diffe rent decision strategies in specific environments, especially including more complex or “dynamic” decision strategies.

PAGE 9

1 Introduction Over the past several years, explan ations of decision making have been predominantly utility-based. Some advan cement has been made by Prospect Theory, which incorporates several ps ychological phenomena such as reference dependence and loss aversion, though still ultimately using a utility-type curve to explain many risky choice phenomena. Other recent advances in the study of decision making have provided for deviations from the prediction of a utili ty-based theory through a compartmentalized application of various heuristic s or biases. Research from this “heuristics and biases” perspective has become particularly narrow, generally focusing on singular deviations from the standard utility-based models. This study takes a broader approach to the understanding of decision making. Instead of relying predominantly on utility-bas ed explanations, it may be worthwhile to also take into account the more complex elemen ts from the real world. There are several fundamental complexities to the study of d ecision making in the real world. After decades of research, three of the most obvi ous or important fundamental complexities that arise repeatedly are vantage point depe ndencies, goal dependencies, and aspirations. Several models have made attempts to account for some or all of these complexities, including the Prospect Theory model (K ahneman & Tversky, 1979), Aspiration Level and Goal Trajectory models (Schneider & Lopes, 1986; Lopes, 1987, 1996), and models

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2 using a dynamic systems approach (Busemeyer 2004). These models have various ways of accounting for the three fundamental complexities. The purpose of this thesis is to anal yze these fundamental complexities in contextually rich environments by synthesi zing several different elements of existing models. Two different approaches were taken to better understanding the role of complexities in decision making. First, simula tions were created to look at the idealized or statistical aspects of h ypothesized decision strategies. Using a Monte Carlo approach, the simulations yield outcome distributions with properties that are helpful in understanding the potential impact of different strategies. A second approach used an experimental decision environment enhanced with contextual information, including time-dependent or “dynamic” context. In addition, a convergent methods approach comparing the simulations with empirical test s is used to look for common patterns and explanations in the behavior of actual participants. In order to better understand the fundament al complexities mentioned previously and how they relate to th e larger question of decisi on-making, a brief overview of vantage point dependency, goal dependency, an d aspiration levels will be provided, as well as a discussion of how they relate to current approaches to understanding risky decision making. A thorough explanation of the current study will then be introduced, along with the findings and implications reached. Vantage Point Dependency In the context of this project, “vantage point” dependency refers to reference dependence as it relates to a reference point The term “reference dependence” is the

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3 notion that a participant’s res ponse to a given problem varies due to the pers pective of the participant. The simplest utility models (Bernoulli, 1738/1954) pr edict that people will evaluate risks differently de pending on how far away the outcomes are from the reference point of zero on a utility curve (shown in Figure 1). Figure 1: An example of Bernoulli’ s (1738/1954) simple utility curve. These models work by assuming a systematic relationship between the actual value of a commodity/lottery a nd the subjective experience that lottery elicits. Typically, the presumed relationship, as shown in Figur e 1, is that a given increase in objective value is experienced as smaller the farther aw ay the values are from zero. For instance, when presented a 50/50 chance of $0 or $100, the difference in objective value between the two options ($100) elicits a larger ‘subjective value’ th an the difference between a 50/50 chance of $500 or $600. This notion of a marginally decreasing relationship

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4 between objective and subjectiv e value is important in that it introduced a way for Bernoulli to explain why peopl e are not neutral to things with the same change in expected value. This helps to illustrate th e importance of a reference point for explaining behavior. However, with the Prospect Theory model and the creation of the S-shaped utility function (Kahneman & Tversky, 1979) shown in Figure 2, particip ants can evaluate choices differently depending on the valence of the decision, or dire ction (positive of negative) from the reference point, in addition to the distance from the reference point. In Prospect Theory, the indivi dual first converts th e values in the decision problem into valences: either gains, which are choices that result in a “good” or better state than the current reference point; or losses, which are ch oices that result in a “bad” or worse state than the current reference point. These ga ins or losses are then weighted by their “subjective value.” This allows the valence of the decision problem to have an impact on the outcome of the decision. Figure 2: S-shaped Utility Curv e (Tversky & Kahneman, 1981).

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5 Reference dependence is commonly de monstrated in phenomenon known as “framing effects.” Tversky and Kahneman ( 1981) proposed that decision makers use the valence of information as a critical reference point. Framing effects are predicted when a decision maker is presented with a choice be tween a riskless opti on and a risky option under circumstances that could be consid ered a loss (e.g., people dying, losing money) versus a gain (e.g., people being saved, gain ing money). Tversky and Kahneman (1981) found that participants are likely to choos e the riskier option when they adopt a loss perspective and are likely to choose the ri skless option when facing what seems like a gain. Reference dependence suggests that th e subjective experience elicited by a lottery depends on the perception of the individual, and as a result, a model built on only these simple utility-type curves can gauge subjectiv e experience as it relates to the reference point of the participant. Reference depe ndence is important in the explanation of decision-making because it requires that the re ference point being taken into account be used to make distinctions depending on th e direction of the choice outcomes from a neutral point. Goal Dependency Vantage point dependencies of distance a nd direction from zero represent static versions of reference dependence. “Goal trajectories” provide an additional dynamic dimension of the reliance on reference poi nts when making decisions. The reference point can be similar to those previously stat ed, but in addition to evaluating risky choices based on distance from zero (as with Utility theory) and direction from zero (as with Prospect theory), decision makers may also take into account the e xperience of previous

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6 outcomes over time as they relate to the e xpectancy of future outcomes. People are influenced by previous successes and failures, such that previous successes create a tendency to view the current goal as continuing their successes by approaching a more positive state, whereas previous failures cr eate a tendency to view the current goal as preventing their future failures by avoiding a more negative state. From this perspective, goals are formed based on info rmation about previous outcomes as well as information about the outcome trajectory fr om the previous decisions to the current decision. This explanation would operate cons istent with the promotion and prevention focus described by Regulatory Focus theo ry (Higgins, 1998). The goal trajectory approach takes into account the decision make r’s previous experiences as well as their future expectations. Goals are set with respect to whether the decision maker has recently experienced improvements or decrements in outcomes. In short, when people make decisions, goal trajectories are using ‘where they are’ w ith respect to ‘where they were’ to make inferences about ‘where they want to be.’ The construct of goal trajectories is in spired by a clever study developed by McKenzie and Nelson (2003). McKenzie and Nelson investigated framing effects using a glass of water at half capacity that was described as having just previously been either empty or full. They found that a majority of the participants interp reted a filling glass as “half full” and an emptying glass as “half em pty.” McKenzie and Nelson conclude that frames and framing effects are useful in providing additional contextual information about the situation. Participants evaluate glasses differently by using the inferences about the trajectory of the water in the glass. A pparently, this information led participants to

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7 believe the glass was either filling or em ptying until it was at half capacity. Hence people interpret events based on outcome trajectories. Schneider and colleagues (2005) elaborated on this idea by incorporating outcome trajectories into a risky decision making tas k. They hypothesized that people’s goals would be highly sensitive to outcome trajecto ry information. She devised a task that makes outcome trajectory and goal related in formation more obvious. Participants start off in a risky situation and are given the opportu nity to try to improve the situation, and in doing so either decrease or increase the amount of risk they are facing. Schneider argued that this is a more ecologically valid paradi gm than the standard risky choice paradigm, wherein participants behave counter intuitiv ely by taking risks when they are in loss situations but playing it safe for gains. Because the new paradigm places the decision within an ongoing dynamic context, preference s should be more intu itive as a result (e.g., not taking big risks when facing losses). Indeed, Schneider and colleagues found that preferences were very different from the standard risky choice paradigm. Instead, Schneider and colleagues found th at people tend to play it safe for both gains and losses, but especially for losses. Schneider and colleagues (2006) then later used her new paradigm to determine if and how outcome trajectories affect the experi ence of decision-making. She argued that people’s affective reactions would differ predictably given the overall valence of experiences across a series of decisions. Sh e argues that basic adaptive mechanisms may lead to negative affect when losing and pos itive affect when winning as a functional way to assess and respond to longe r-term consequences. Specifi cally, when one is losing over

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8 a period of time, protective measures are in order; however, when one is winning over a period of time, one can afford to be more open to a variety of options. People have often argued that framing e ffects should be ignored, but Schneider argues that it is reasonable for people to re spond differently to starting with nothing and winning up to a given value versus starting off with a lot and losing down to the same value. This is exactly what Schneider and colleagues (2006) found; affect changed markedly as participants c ontinued through a series of lotteries depending on whether they experienced a positive or negative tre nd across the entire se ries (e.g., how much money they received at the beginning of the task + how much they earned [positive trend] or lost [negative trend] during the task). Th ese affective differences in this context seem to operate in a fashion consis tent with Damasio’s (1995) somatic marker hypothesis. The somatic marker hypothesis focuses on somatosens ory reactions to stimuli over a period of time as a means for gauging a choice option’s us efulness. All else equal, changes in affective response are likely to signal improveme nts or declines in one’s position and are thus adaptive in more natural scenarios. Goal trajectories pose a problem for decision researchers because trajectory informati on varies depending on the participants’ evaluation of the lottery scenario, previous lotteries in a series, etc. As a result, individual participants’ decision pattern s may vary depending on their different interpretations of the current lottery state relative to previous lotteries and, perhaps more importantly, previous outcomes

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9 Aspirations about the Future In the same vein as the previously men tioned goal trajectories, Lopes (1981) and others have argued that maki ng a series of decisions can have different meanings and consequences than making a single decision with the same expected value. Lopes addressed a comment by Samuelson (1963) in wh ich he argued with a colleague that if one play of a lottery was not desirable, then mu ltiple plays of that lott ery also couldn’t be rationally considered preferable. Lopes (1981) argued that lotteries could be interpreted differently depending on the number of play s given. When given a large number of chances to play, people are much more likely to end up close to the expected value of outcomes. For example, if given a 50/50 chance between $0 and $5, after a single play, there is no chance of ending up with the aver age value $2.50, only either $0 or $5; after ten plays however, there is a relatively good ch ance of ending up at or near the expected value of $25. The central limit theorem would di ctate that the greater the number in the series of games, the higher the likelihood th at the experienced cumulative outcome will be close to the expected value. Hence, a single play does not capture the change in outcome distributions associated with a series of events. In short, a large number of chances to play a particular gamble yield a higher likelihood to arrive at some multiple of the expected value such that the standard de viation of outcomes is smaller compared to fewer plays of the same gamble. Lopes (1981) goes on to argue that it is reasonable for people to base their decisions on something other than expecta tions, given this diffe rence in experienced outcomes between long run and short run gamb les. While vantage point dependencies

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10 and goal dependencies reflect the importance of the reference point for encoding the size of values, their valence, and their impact with respect to recent outcomes and current goals, there may be individual differences that cannot be explai ned relative to the reference point, but further depend on the im plicit goals of participants. Lopes and Schneider have long argued that decisi on strategies are based less on momentary considerations such as the expected va lues of gambles and more on the future implications of an increasingly time-dependent decision environment. To that end, Lopes (1987) proposed a dual criteri on strategy to address how aspi rations serve as threshold values. SP/A theory (Lopes, 1987) involves two decision criteria in formulating a decision: the first is the security/potential component, wh ich suggests that motivations correspond with how influential the good or bad outcomes are. Specifically, the ‘security’ modality instantiates that the worse outcome is more influe ntial than the better outcome, while the ‘potential’ modality instan tiates that the better outcome is more influential than the worse outcome. This component functions in a dispositional way, dissimilar to the s-shaped utility functi on suggested by the Prospect Theory model (Tversky & Kahneman, 1981). Instead of each valence representing a different evaluation of objective value, the particular motivation to remain secure to avoid large losses (as with the “negative” realm) or achie ve some large potential gain (as with the “positive” realm) exist with relative simulta neity, wherein the most prevalent motivation is adopted.

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11 The second component of the SP/A theory is the aspiration level component. This is a situational variable, such that the envi ronment introduces opportuni ties or constraints, which in turn influence choices. Lopes ( 1987) presents three potential sources of aspiration levels; direct asse ssment of what is reasonable to look forward to, direct contextual influence of othe r alternatives in the choice set, and outside contextual influences, such as the rules of a particular decision task. Lopes (1987) suggests that as people make decisions in the real world, of ten relative to some persistent goal, (e.g., making enough to pay the rent; other financia l, often noncompensatory goals) they may change their fiscal strategy depending on how they perceive themselves to be doing relative to that goal. This should be an important aspect of decision modeling because it suggests that an effective model must accoun t for strategy changes as a result of longerterm goals and aspirations. The aspiration level phenomenon is im portant because it extends the evaluation of risky decision-maki ng beyond a comparison of expected values. For the purpose of this investigation, we are interested in the noncompensatory processing which can result from the aspi ration level phenomenon. Noncompensatory decision criteria are non-additi ve, all-or-nothing determinants of choice that relate to distinctions people make in their envir onment (Paine, 1975). While compensatory criteria allow for other criter ia to tradeoff such that bei ng good with one characteristic (great miles per gallon in a car) can compensa te for being bad with another characteristic (a smaller interior), noncompensatory decisi on criterion do not consid er other criteria if the conditions of the noncompensatory criteria are not met. Noncompensatory criteria work like a gateway, determining whether one ne eds to consider the other criteria; if the

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12 conditions for a noncompensatory criterion are met for a given subset of choices, then those choices are evaluated using the remaini ng criteria. The choice subset that does not meet the conditions of the noncompensatory crit erion is ignored. Nonlinearity can often arise from situations involving noncompensat ory processing, which can create problems for linear models of decision-making. For example, if the decision pr oblem involved buying a car, one noncompensatory decision strategy would be to not buy a car if it were red. Regardless of the other criteria that can be used in the act of choosing a ca r, if it is red, it will not be purchased. When a participant changes th eir strategy by ignori ng all of the other characteristics of a choice for a given lo ttery, that is consid ered a noncompensatory decision strategy, because the “other criteria” (e.g., the valence of the lottery, previous outcomes, etc.) cannot affect the participant’s choice when they are in the given situation. When decision strategies are attached to aspiration levels, the aspiration threshold influences the strategy by creating a disconti nuity or nonlinearity. As a result, additive linear models can not account for changes in st rategy for situations th at “do” versus “do not” meet the aspiration, because these m odels would require unwieldy “heuristics,” addendums, and exceptions to explain why one chose from a limited subset of choices given one’s aspirations as they relate to the particular environmental information. Modeling Decision Complexity: Dynamic systems Partly in an attempt to account for n onlinear and noncompensatory criteria and partly to explain the problem associated with how simple processes and entities can operate dynamically, (e.g., depending on their in itial criterion values and current criterion

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13 values) dynamic systems approach to deci sion-making was created. Aspects of these systems have been used in research on j udgment and decision-making in a number of different ways including the Decision Fi eld Theory model (Busemeyer & Townsend, 1993), explanations of capital investment (R apoport, 1975), and models of managerial behavior (Sternman, 1989). The important aspects that make a particular system “dynamic,” as laid out by Busemeyer (2001), are as follows: 1. Actions occur in a series over ti me in order to achieve a goal; 2. Said actions are “interdependent,” e.g., later actions depend on previous actions; 3. And the system environment must change spontaneously, as a consequence of earlier actions, or both. Busemeyer (2001) and later Gonzalez ( 2005) point out several characteristics found in dynamic systems tasks that are pa rticularly useful when studying decisionmaking. ‘Dynamic systems’ models offer advant ages in testing for differences related to the fundamental complexities discussed earlie r. Dynamic systems modeling address goal dependencies due to the existence of a goa l-oriented time series, allowing for timedependent stimuli in the short term and the l ong term to be introduced and analyzed. In addition, the dynamic systems methodology incl udes a contextual environment with specific implicit (e.g., avoidi ng zero) and explicit (e.g., ma king money) aspirations. However, by necessity, these dynamic decision making tasks are generally oversimplifications of real-world situations and as a result require some additional realworld knowledge of the decision environment. In addition, the exact stimulus presented

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14 to each participant is not under the complete control of the experimenter, due to the interaction of previous information with current information. As a result, making comparisons by averaging across individual de cisions or participan ts no longer suffices as an analytical tool. Therefore, dynamic decision making tasks have to be carefully planned and executed, and different analytical tools are required. There have been previous forays into extending dynamic systems into decisionmaking. Dynamic systems models are often us ed with respect to complex multi-attribute decision-making (e.g., Decision Field Theory (Busemeyer & Townsend, 1993)). Dynamic decision-making has also been used to model decision behavior with respect to task rules. This is accomplished by using im plicit task-specific goals/aspirations along with explicit task instructions. An exam ple is a firefighting task given by Brehmer (1992) and later adapted by Omodei and Wearing (1995). In this task, participants were given a hypothetical situation of managing resources to fight a fire. Participants were put through a computer simulation that included thin gs like feedback dela ys and implicit goal contingencies (e.g., don’t let the firehouse bur n down). These dynamic systems models often use this type of “micro-world,” whic h is broadly defined as a complex computer task which, though relatively simple, allows re searchers to evaluate and control a larger set of decision-making characteristics essent ial in real world dynamic decision-making environments. ‘Dynamic systems modeling’ often relies on mathematical simulation as a type of analysis. First, simulations allow for easy and controlled additi on of dynamic system variables. For example, in a simulation, the addition or removal of certain decision

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15 strategy constraints (compe nsatory or noncompensatory) from the micro-world can emphasize the impact of different decision criteria under different circumstances. Second, simulations can extract the distributional characteris tics of combining participant experiences with given stra tegy criteria (and criteria interactions) by isolating those specific criteria. Third, simu lations remove the individual differences and individual error normally found between participants by determining the actions taken by each “participant” a priori As a result, a simulation pr ovides a distribution of possible outcomes and the relative likelihood for what mi ght occur if participan ts were to operate using only a particular decision criterion or set of criteria with in a specified environment. While many previous investig ations of risk preferences have taken into account vantage point criteria, relatively few have attempted to account for the more complex elements of goal dependency and aspirations. Of those previous investigations which included complex elements, none have include d a thorough analysis of the statistical properties associated with time-dependent decision criteria, nor have there been systematic attempts to determine whether these more complex decision criteria are evident in the decision making of study part icipants. This investigation entails two methodologies: a series of decision strategy simulations to compare and contrast the distributional and statistical aspects of vantage point de pendency, goal dependency, and aspiration levels; and an empirical manipul ation to compare what we learn using simulations to actual human decision-maki ng behavior in a dynamic decision-making environment. This study will then attempt to provide a richer understanding of the implications associated with vantage point dependency, goal depende ncy and aspirations

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16 by comparing the effectiveness of each leve l of complexity under simulated ‘optimal’ conditions, as well as the effectiveness of each at prediction of how real people make decisions.

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17 Study One: Investigate D ecision Strategy Simulations The purpose of using simulations in th is investigation was to address the differences in outcomes between a variety of decision strategies that depend to a greater or lesser extent on vantage point dependency, goal dependency, and as piration levels. In previous work, Schneider and colleagues (2006) developed a series of simulations to compare the longer-term implications of se veral different models of decision-making. This previous work will be described, followe d by an explanation of the extensions to the simulation methodology which is currently ut ilized, followed by an evaluation of the results from the current simulation. Pilot Simulations Our approach in this project is to apply Monte Carlo simulation methodology addressed as an analysis t ool by dynamic systems modeling to investigate the relationship between different kinds of d ecision-making models (risk poli cies, valence-dependent, and aspirations-based). In order to lay the groundwork for the investigation into decisionmaking in a dynamic decision making environment, mathematical simulations were performed for the purposes of this experiment in three phases. Th e first phase was to simply capture the baseline distributional in formation about risk pr eferences. The second phase was to compare and evaluate the di stributional aspects of several decision strategies that embody the models of late. The third phase was to compare and evaluate the distributional aspects of deci sion strategies that use contextual information to form an

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18 aspiration criterion in determining choice. For this mathematical simulation, “virtual participants” were created to use a specifi c decision strategy that resembled either baseline conditions, decision-making mode ls that only use local intra-lottery characteristics as criteria for determining risk preference (Prospect Theory: Kahneman & Tversky, 1979; Risk as threat and op portunity: Highhouse & Yuce, 1996, Lopes, 1987, Schneider & Lopes, 1986; Risk as variance: Savage, 1954) or aspiration level models which use extra-lottery contextual information as criteria for determining risk preference; that is, criteria which used the “cumulative total” value (the concatenation of their starting value and their outcomes so far) as a m eans for deciding when to be risky or safe. In order to look at vantage point dependency and larger contextual effects, several sets of virtual participants were created to make deci sions across a series of 36 lotteries. Each set of virtual participants was ascribed with a particular strategy that fit one of three major groups. Decision Policies The Decision Policy strategies include the most basic strategies that represent a constant policy that operate independent of lottery characteristics or environmental constraints. These strategies differ exclusively with respect to the presumed shape of their utility curve. The risk-a verse strategy can be said to have a concave utility function, similar to the utility curve supported by Bernoulli (1738/1954), and always leads to selecting the less risky option. The risk-s eeking strategy presumes a convex utility function, and opposite to the risk-averse strategy always leads to the selection of riskier options. The random strategy can be said to imply a straight utility relationship between

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19 objective and subjective value, which produ ces indifference in the choice between options of equal expected value. As a resu lt, option selection is randomly determined. Lottery-based Strategies The lottery-based strategies are dependent on static characteristics of the lottery options for strategy selection. This set emul ates the risk preference patterns of several popular models of decision-making. The first strategy presented represents the Prospect Theory Model (Tversky and Kahneman, 1981). In this strategy, lotteries that have positive outcomes are evaluated as gains and therefore are met with risk aversion and lotteries that have negative outcomes are eval uated as losses and th erefore are met with risk-seeking behavior, similar to the predic tions of the Prospect Theory S-shaped subjective value function (Kahneman & Tversk y, 1979). When the choice is between mixed lotteries (containing both positive and ne gative outcomes), they are met with risk aversion, based on the loss aversion phenomen on suggested by the Prospect Theory model. The next strategy is meant to represent the Security/Potentia l component of the SP/A model (Lopes, 1987; Schneider & Lopes, 1986), or is otherwis e often described as Cautious Optimism or “Risk as th reat and opportunity” (Highhouse & Y ce, 1996). In this strategy, lotteries that have all negative outcomes lead to risk being interpreted as a threat and therefore the safer option is chos en, while lotteries that have all positive outcomes lead to risk being interpreted as an opportunity and theref ore the riskier option is chosen. When lotteries have mixed outco mes, they are met with loss aversion, due to the typical tendency to prioritize security ov er potential, or threat over opportunity.

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20 The next strategy is meant to represent the Risk as variance or modest risk tolerance model, which is similar to a m odel presented by Savage (1954). This model favors modest amounts of risk on low variance gambles and differentia tes lotteries with riskless or “sure thing” options by selecting the riskier option, while in all other cases, the riskier option is rejected. This model still involves a reference poi nt that incorporates both distance from zero and direction. However, instead of direction from zero in terms of objective values, the risk as variance model adjusts risk preference relative to the amount of variability (size di fferences) between outcomes. Contextual or Aspirati on-based Strategies The aspiration-based set of strategies co mbines contextual information and goals as criteria for decision strategies. Specifica lly, they define goal-d irection as a decision criterion, namely the cumulative value across the series of simulation outcomes is the focus for changing risk preference. This in corporates the Aspira tion component of the SP/A model with the dispositional tendencies. Specifically, when a virtual participant is above a certain desired wealth threshold or ‘aspirat ion level,’ they focus on potential and are willing to take risks, while when a vi rtual participant is below a certain wealth threshold, they focus on security and are not willing to take risks. Security-focused individuals will set their aspiration levels highe r so that they only take a risk when they are substantially well off. Potential-focused individuals will set their aspiration level lower in order to take advantage of the opportunities possible with the riskier options.

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21 Procedure These 3 sets of strategies were evaluate d using Monte Carlo style simulations. 36 lottery pairs were presented in one of f our different counterbalanced orders. 10,000 virtual participants were used for each or der, for a total of 40,000 per strategy. These virtual participants were a ssigned to pick the more ris ky or less risky option based on their given strategy. The results of the simulations were evaluated by examining characteristics for each strategy comparing fi nal outcome distributions, central tendency, variability, and skew within the final outcome distributi ons, as well as showing the percentage of participants that are consolidat ed into either the high (denoted “richer”) and low (denoted “poorer”) extremes of the fina l outcome distribution. The expected value for the series of 36 lotteries was $1000 in al l cases, and hence all distributions had the same mean. Preliminary Results of th e Initial Investigation For the Risk Policy strategies, we test ed the baseline conditions for always choosing the more risky of two options (de noted “Risk-seeking”), always choosing the less risky of two options (denoted “Riskaverse”), and a condition in which risk preference was determined completely randomly (denoted “Random”).

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22 Figure 3: The final outcome distribu tions for the baseline strategies. “SimLo” designates an entirely risk-averse strategy, “SimHi” designates an entirely risk-seeking strategy, and “Random” designates an entirely random strategy. As shown in Figure 3, we found that d ecision strategies involving only riskseeking choices have a final outcome dist ribution with a large amount of variance, meaning there are fewer participants at or around the mean and more participants who ended up at the extremes (e.g., very “rich” or ended up very “poor”). Conversely, a decision strategy that involves only risk-ave rse moves has a final outcome distribution with a small amount of variance, meaning th ere are more participants at or around the mean and fewer participants at the extremes. Also, a decision strategy that involves only random choices has a final outcome distribution th at is intermediate to the risk-averse and risk-seeking strategy.

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23 For the lottery-based strategies, we test ed the strategies based on the Prospect theory model (Tversky and Kahneman, 1981), th e Risk as Threat and Opportunity or Security/Potential component of the SP/A model (Lopes, 1987; Schneider & Lopes, 1986), and the Risk as variance model (Savage, 1954). We found that decision strategies that are based on these models of decision-ma king which happen to only use local, static characteristics of individual lotteries (shown in Figure 4) are similar to Risk Policy strategies in that they differ systema tically in distributional variability. Figure 4: The final outcome distribu tions for the strategies based on models of decision-making. Risk policies from the previous Figure are included. These local, lottery-based strategies were symmetrical just like the baseline strategies, but differed in the amount of variance. The amount of variance in a particular

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24 final outcome distribution was completely dete rmined by the number of risky moves that a particular strategy makes for a given series of lotteries. The more risky choices that a strategy demands throughout the se ries of lotteries, the higher the vari ance or the more the final outcome distribution looks like th e “always risk-seeking” strategy. The more risk-averse actions, the lower the variance or the more the final outcome distribution looks like the “always risk-averse” strategy. The percentages of participants in the ex tremes for the final outcome distribution are shown in Figure 5. This illustrates the di fference that strategies make in determining the likelihood of ending up at the extremes. The likelihood of ending up in the extremes is very low for strategies th at include more risk-averse choices, while the likelihood of ending up in the extremes increases as a numbe r of riskier choices in creases. In addition, there is relative symmetry found in both lott ery-based and baseline strategy conditions. For the most part, there is the same percen tage of participants in the higher outcome groups as the lower outcome groups.

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25 0% 2% 4% 6% 8% 10% 12% Risk AverseRisk as Variance Prospect Theory Model Risk as ThreatRandomRisk Seeking Strategy% of Participa n < $400 > $1600 Figure 5: The percentage of the virt ual participants for a given strate gy found in the extremes of the final outcome distribution. Th e names of the strategies follo w the same naming conventions discussed earlier. From comparing these strate gies, we found that there we re systematic differences in strategies based on linear models, and those differences were limited to the amount of variance in the final outco me distribution. The mean s of these final outcome distributions were all identical, as was th e likelihood of ending up in a good place as a bad place within a given strategy. Some of the strategies shared a near identical amount variance, namely the Prospect Theory model, the Risk as Threat model, and the Random baseline condition. However, the similaritie s between the Risk as Threat model, the Prospect Theory model, and the Random ba seline condition existed because the stimulus set had approximately the same number of negati ve lotteries as positive lotteries. If there were more negative lotteries than positive lotteries, then the Risk as Threat model would

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26 have exhibited more risk-averse behavior a nd would hence have had a distribution that looked more like the risk-averse strategy’s fi nal outcome distribution, while the Prospect theory model would have exhibited more ri sk-seeking behavior and would hence have a distribution that looked more like the risk-seek ing strategy. For the aspiration-based set, we tested strategies that used an aspiration criterion in determining choice by compari ng the current wealth with a specific aspiration. Unlike other strategi es, we demonstrated that de cision strategies using this, noncompensatory, longer-term aspiration base d strategy (shown in Figure 6) created skew in the final outcome di stribution, increasing the like lihood of ending up in one tail of the distribution while simultaneously d ecreasing the likelihood of ending up in the other tail of the distribution.

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27 Figure 6: The final outcome distribu tions for the strategies based on aspiration level criteria. The names of the strategies follow the same naming conventions discussed earlier. Specifically, for the single aspiration leve l strategies (denoted ‘Cumulative total: 800’ and ‘Cumulative total: 1200’), when a virt ual participant has le ss than the specific aspiration level (800 or 1200 resp ectively) as their current total, the decision strategy dictated to act risk-averse until the specific aspiration level was reached, then for as long as that specific aspiration level was exceeded, the strategy dictated to act risk seeking. This causes final distributions to look like th e risk-averse strategy di stribution on the left side but like the risk-seeking strategy on the right side. Final outcome distributions for these strategies are skewed to the right, su ch that there were fewer people who ended up very poor and more people who ended up very wealthy. Of course, the positive skew also offsets the median and the mode to th e left, meaning that these strategies also

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28 increase the likelihood that you wi ll end up slightly below average. In other words, for a small increase in the likelihood of being slightly less than average, one can substantially decrease chances of being extremely poor and increase chances of being extremely rich. The dual criterion strategy operated such that as long as a virtual participant was between the two specific aspiration values (400 a nd 1600), the decision st rategy dictated risk aversion, and if those values were exceeded, th e decision strategy dictated risk seeking. The comparison of these groups can be seen in Figure 7, which depicts the percentage of virtual participants who e nded up at the tails of the final outcome distribution.

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29 Figure 7: The percentage of the virt ual participants for a given strate gy found in the extremes of the final outcome distribution. Th e names of the strategies follo w the same naming conventions discussed earlier. From comparing these strategies, we lear n that asymmetries can be created in a final outcome distribution if a long-term aspi ration level is used as a criterion in the decision strategy. Of the stra tegies tested, only those th at rely on aspirations can differentially manipulate th e long run probabilities of ending up rich or poor. From the pilot work, we learned that th e final outcome distributions for Risk Policy and Lottery-based strategi es that use static criteria can only be symmetrical, such that users of those strategies were equally likely over time to end up very rich or very poor, though their risk preferences provided some control over the combined likelihood of landing in the extremes. However, given the use of the long-term aspirations as a

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30 decision criterion, there is the capacity to create an asymmetry in the final outcome distribution. Decision strategies that use decision policies or local, lottery-b ased criteria as the determining factors for risky choice are limited in their ability to change the likelihood of ending up in a good place over a bad place. However, decision strategies that use contextual or aspiration-based cr iteria have the benefit of differentially controlling the likelihood of ending up in a relatively good place over a relatively bad place. This pilot work lends itself to the im portance of time-dependent criteria, as well as the dynamic systems modeling, which is used to test for time-d ependent criteria. Figuring out precisely how and why an aspi ration level decision strategy works is important because finding a point at which th ere is the “most” skew or the “most” long run benefit may help us figure out when and how actual people might use these strategies. In addition, we expect trajecto ry-based decision stra tegies to make an impact in the long run beyond simple lottery-based strategies be cause trajectory base d decision strategies satisfy the conditions of dyna mic criteria: They are timedependent and preferences change dynamically. Current Simulation Study One Methods For the current project, the previously outlined simulation methodology has been revamped in a number of ways so as to further investigate the influence of timedependent criteria that take into account the contextual, historical, and goal-related information of the decision maker. These changes are as follows: (1) Aspiration levels were more thor oughly manipulated, so as to identify the “optimal” levels of skew for a given simu lation environment. In addition, aspiration

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31 levels that occur between two possible cu rrent total values were used to reduce distributional “lumps” that are artifacts of our stimuli. (2) A new phase of strategies that use outcome trajectories as their basis were created and compared. These strategies allo w us to evaluate how the dynamic reference points mentioned earlier operate in our simu lation. These strategies use information about the previous outcomes to determin e their current choice, similar to goal dependency. These outcome trajectory strategies come in two forms. The first form looks at the previous seve n lotteries and makes a distinction with respect to risk preference depending on th e number of the outcomes that are good vs. bad.1 For example, if of the previous seve n lotteries, four are the “worse” outcome and three are the “better” outcome, a trajector y strategy would make a choice based on the fact that there is a ‘majority’ of worse outco mes. This strategy operates without using distance from zero or direction from zero in de termining risk preference, and is important because it gives a crude measure for individual effect of outcome trajectory on the final outcome distribution separate from the other influences of reference points. The outcome trajectory strategies are named based on the virtual participant’s inferred interpretation of the future directions for the trajectory. The virtual participants who are using the Continuation strategy are inferring from thei r trajectory that if they are currently doing “well” (the trajectory has more “wins” than “losses) that they will continue to do well and they attempt to take fu ll advantage by exhi biting risk seeking 1 Because our simulation uses two-outcome lotteries, on e outcome is usually better than the other. Hence “better” outcomes are ‘good’ relatively speaking inso far as they are better than “worse” outcomes, even though in the absolute sense they may be positive, negative, or zero.

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32 behavior, while if they are currently doing “poorly” (the tr ajectory has more “losses”) they infer that they will con tinue to do poorly and they at tempt to diminish potential future losses by exhibiting risk aversion. The Discontinuation strategy exhibits the opposite inference from the trajectory information. For example, if a virtual part icipant using the Disc ontinuation strategy is currently doing “well,” they infer that they will not continue to do well and they attempt to diminish potential future losses by exhibi ting risk aversion, while if the same virtual participant is currently doing “ poorly,” they infer that they will not continue to do poorly and they hence attempt to take advantage of the future gains by e xhibiting risk seeking behavior. The second form of goal-dependent strategi es make a distinction with respect to risk preference depending on whether a par ticipant has won or lost money over the previous seven lotteries. Specifically, since each lotter y has a specific outcome, the outcomes for the previous seven lotteries are summed, and then risk preference is determined by whether that sum is positive or negative. This is important because it includes the additional reference point inform ation that is left out of the previous incarnation of goal trajectories, so as to al low a direct comparison of the interaction between static and dynamic reference point information. Using the new decision environments additional final-outcome likelihood distributions, measures were created and compared for centr al tendency, variability, and skew, and the percentage of pa rticipants that dropped out or “died,” or ended up in the high and low extremes of the final outcome distribution.

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33 Materials/Stimuli Each virtual participant ‘saw’ a set of 36 monetary two-outcome lotteries. Each virtual participant began the expe riment with 200, 1000, or 1800, and the final adjusted expected value for th e series of lotteries is 1000. Outcomes of the lotteries accumulate as the virtual participant conti nues through the study. The lotteries can be positive or negative, and each will have an e xpected value that ranges from +400 to -400. Design This study involves testing stra tegies using a series of two-outcome lotteries in a Monte-Carlo-style simulation. Each virtual pa rticipant is created to follow an assigned algorithm derived from the strategies in Ta ble 1. The measures of interest are the outcomes that virtual particip ants received based on each trial, the cumulated earnings from trial to trial, and th e distribution of fi nal outcomes across a given strategy. In addition, to reduce order effects, four lotter y orders were used across all strategies. Strategies. All the strategies addressed in the preliminary investigation were redone. In addition, the strate gies regarding outcome trajec tories described in Table 1 were performed and compared.

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34 Table 1: List of Strategy Ty pes used in Pilot Simulations. Strategy Explanation Risk-averse Virtual participants exhi bit risk-averse behavior across all lotteries by choosing the less risky option, which reduces variance. Risk-seeking Virtual participants exhi bit risk-seeking behavior across all lotteries by choosing the more risky option, which increases variance. Non-systematic, or “Random” Virtual participants used a nonsystematic method to determine risk preference. Randomly chose between riskier and less risky options. Modest Variance Virtual participants are w illing to accept some but not all of risk. When faced with lotteries that st art with a high initial variance, they choose the less risky option, and when faced with lotteries that start with a low or moderate initial variance, they choose the riskier option. Prospect Theory Virtual participants act in accord with the S-shaped value function described in Prospect Theory. When facing a ‘loss’ lottery, participants chose the ri skier option. When facing a ‘gain’ lottery, participants choose the less risky option. When facing a lottery with mixed outcomes, pa rticipants exhibit loss aversion and thus choose the less risky option. Risk As Threat Virtual participants avoi d risks when they are a threat, and only exhibit risk-averse behavior when faced with a potential loss. When facing a ‘gain’ lottery, pa rticipants choose the riskier option. When facing a ‘loss’ lotte ry, participants choose the less risky option. When facing a lottery with mixed outcomes, participants exhibit loss aversi on and thus choose the less risky option. Cumulative Total: Aspiration Virtual participants act based on the amount of money they have accumulated so far. If the virtua l participant's cumulative total is below the aspiration value, they choose the less risky option, whereas if the virtual participan t's cumulative total is above the value, they choose the riskier op tion. Proposed as piration values: 775, 975, 1025, 1225. In addition, we will include one asymmetric dual aspiration, where if a participant has less than 375, they exhibit risk seeking. The reverse, in which a participant’s cumulative total being below a given aspiration value elicits risk seeking behavior and a participant’s cumulative total being above a given aspirati on level elicits risk aversion, will also be evaluated at all of the proposed aspiration values.

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35 Strategy Explanation Goal Trajectories, Number of Outcomes Based on the number of better or worse outcomes for the previous seven lotteries, if there are more of the better than worse outcomes received, one of the risk policies is exhibited, while if there are more of the worse than better outcomes received, the other risk policy is exhibited. “Sure thing” outcomes, in which both outcomes of a given lottery are the same, were randomly attributed as either better or worse. Continuation infers that if things are going well (a majority of better outcomes), they will continue to go well and so ri sk-seeking should be employed, while Discontinuation infers that if things are going well, they will not continue to go well, a nd so risk-aversion should be employed. Goal Trajectories, Cumulative Values of Outcomes If the overall change in current total for the previous seven lotteries is positive, the higher option is improved simulating risk-seeking behavior while if it is negative, the lower option is improved, simulating risk-averse behavior. If equal to zero, the participant exhibits risk aversion. Note. Strategies were named based on their similarity to current theories and ideas in human decision making literature. Prospect theory was developed by Kahneman and Tversky (1979). Orders. Two randomized orders of 36 two-outco me lotteries were created. Then these two orders were reversed (the first lott ery in the initial order became the last lottery in the new order, the second lottery in the in itial order became the se cond to last in the new order, etc.) to create two completely new orders, making a total of 4 orders. Measurements. Using Excel, two databases were created for each simulation strategy: The first database logs the outcome value of each virtual participant at each lottery and the second database shows the accumulation of outcome values across the lotteries completed thus far. Participants Forty thousand virtual participants were created to simulate each strategy condition. Ten thousand participants were r un through each of the 4 orders for each of the 12 strategies for a total of 480,000 virtual participants.

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36 Procedure The procedure applied to the current si mulation methodology is the same as the procedure used in the pilot simulation discussed earlier. Study One Analysis of Simulations The simulations were analyzed by comparing the outcome distributions after 36 trials, including: m easures of central tendenc y, variability, and skew; the percentage of participants th at end up in the high and low extremes of the final outcome distribution; and, where applicable, the percen tage of participants who “went broke.” First, additional findings pert aining to aspiration level stra tegies are addressed. Then, any novelty that can be attributed to the outcome trajectory stra tegies is addressed, followed by a step-by-step explanation of how specific changes to the decision environment affect different types of stra tegies (risk policy, static lottery-based, aspiration level-based, and trajectory-based). The Aspiration level Criterion: A Thorough Investigation While investigating aspiration-based or ‘cum ulative total’ strategies, we looked to answer two distinctly different questions with the current simulations: (1) Why does skew occur when using a cumulative total strategy? and (2) How can we ‘set’ and aspiration level so as to achie ve the maximal level of skew? The answer to question one may be most easily answered by r ecognizing that the cumulative total functions over time as the curre nt measure of success. In this sense, the cumulative total can be seen as a running tally of a virtual participant’s progress thus far in the task. As a result, making decisions ba sed on cumulative wealth allows participants

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37 to manipulate variability differently base d on where they consider themselves doing comparatively good or bad with respect to the “average” or expectation. However, participants do not necessarily have to use what they expect as their aspiration level, they may just as well use some external constr aint (e.g., “my rent is $500, so regardless of what I expect, I need at least that much.”) Figure 8: Theoretical Example of how distributio nal variability is manipulated with respect to current progress. For instance, if thus-far in the task, y our “current total” is comparatively quite low, you might consider that because you want the highest opportunity of getting at least the mean, you would decide to taking fewer ri sks, or exhibiting the risk-aversion strategy (represented by the blue line in figure 8). When doing comparatively well with a high current total, you might try to take advantage of the poten tial to achieve a very high amount by taking more risks, or exhibiting the risk seeking strategy (shown in red). Hence, a sort of “skew” is created as a re sult of manipulating dist ributional variability depending on the rough estimate of current pos ition on the final outco me distribution.

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38 N u m b e r o f P a r t i c i p a n t s Final OutcomeOptimal Distribution Figure 9: Rendering of the “optimal” skewed di stribution. Variability is minimized when you consider yourself to have less than average and maximized when yo u consider yourself as having more than average based on the goals asserted by Sc hneider and colleagues (2007) to be the focus of real world decision making. Schneider and colleagues (2007) suggests th at if people knew they were going to do worse than the average, they would form a goal that attempts to increase the likelihood of getting as close to the aver age as possible. In addition, if people knew they were going to do better than the average, Schneider assert s that they would form a goal that attempts to maximize the potential to get as far away fr om the average as possible. As a result, a pattern emerges such that one minimizes risk when one would interpret themselves as doing worse than the average (risk aversi on) and maximizes risk when one would interpret themselves as doing better than th e average (risk seeking). Because the risk averse and risk seeking strategies each repr esent a particular risk manipulation technique (always safe choices or always risky choices respectively) within the constraints of the

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39 decision environment2, combining half of each together across the mean should create a representation of the optimal skewed distribution that woul d satisfy both of these goals. 0 2000 4000 6000 8000 10000 12000 14000 4 0 0 2 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0 2 2 0 0 2 4 0 0 N u m b e r o f P a r t i c i p a n t s Final Outcome Optimal Distribution Cumulative total: 775 Cumulative Total: 975 Cumulative total: 1025 Cumulative total: 1225 Figure 10: Comparison of different aspiration level strategy final outcome distributions with the optimal skewed distribution under pilot (36 trials) conditions. The aspiration level strategies are named after the cutoff po int used to switch from risk aversio n (when current total is less than the given value) to risk seeking (when cur rent total is more than the given value). As shown in Figure 10, the aspiration leve l strategies are found to look similar to the optimal distribution (also shown). In addition, if one uses the mean or overall expected value of the lottery series (in our simulations, 1000) as one’s aspiration level, a distribution of final outcomes is formed that is comparatively the most similar to the 2 Total lottery distributional variability is determined by a multitude of factors, such as the range of expected values for individual lotteries, the range of differences in the size of ticket values, the total number of lotteries, etc.

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40 optimal skewed distribution. This does not s uggest that using the e xpected value as the aspiration level is necessarily optimal in its own right, since individual preference may still dictate overall tendencies toward risk, in additional to external goals playing an integral role in aspiration formation. For instance, even though usi ng the expected value as th e aspiration level “cutoff” creates the most skew for th e distribution, an individual mi ght set their as piration level “cutoff” as a small margin more than what’s expected so as to s till tend towards risk aversion, since the larger the as piration level is compared to the expected value, the more the distribution of outcomes tends toward the ri sk averse strategy. As an aspiration level becomes unattainably large, a participant would always have a current total that is less than their aspiration level, and hence the final outcome distribution would become the risk aversion final outcome distribution, whereas if the aspiration level is set as always attained (or otherwise unreas onably low), the distribution of final outcomes becomes the risk seeking stra tegy distribution. As a result, having information about “what is expected” in the decision environment is helpful in achieving the most ef fective use of the aspi ration level strategy. Without a reasonable expecta tion, an aspiration level may end up having been set too high, which would result in a predominance of risk averse behavior or too low, which would result in a predominance of risk seeki ng behavior. If your goals coincide with those posited by Schneider and colleagues (200 7), then you should use an aspiration level as we have described it, as it will create distributional asymmetry in a manner that is consistent with those goals.

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41 New Phase of Strategies: Outcome and Cumulative Trajectories Even though aspiration levels can genera te asymmetry and achieve an outcome distribution that looks similar to the optimal skewed distribution, they are limited in their scope when it comes to the real world. A fixed aspiration level cannot account for changes in the overall direction of experiences. For example, if one sets an aspiration level, and then one’s environment systema tically fluctuates between doing well for a while (better than average), then doing poorly for a while (worse than average), then doing well for a while again, one’s aspiration level strategy would only suggest risks be taken with respect to how much one has not to the fluctuations in the environment. While this is still perfectly reasonable, it is no t necessarily optimal, as the main goal is to take risks when one’s situation is evaluated as better than average, and there is obviously something systematic going on in a ti me-dependent manner not accounted for by aspiration levels. The problem is that aspiration levels are “fixed”; they aren’t sensitive to changes in environmental trajectory only changes in cumulative outcomes In part to account for the fixed nature of aspiration levels, outcome trajectory strategies, which do not have a fixed valu e for risk preference distinctions, and cumulative trajectory strategies, which focu s on changes to the cumulative total instead of the actual value, were investigated. A trajectory focuses on directly experienced information from the recent past to make choices about the immediate future (Schneider & Barnes, 2003). This differs from aspiration level strategies because in most cases there is no fixed cumulative value being used by the participants; only recent salient information about the direct past few events. The criterion for action with respect to risk

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42 in an outcome trajectory relies only on outcome s evaluated as “good”, “bad”, “better,” or “worse” over a particular interval. As a re sult, information about all of the previous performance is not neces sarily required. Refer to p. 26 and 27 to see the specifics of how outcome trajectory strategies and cumulative trajectories work. 0 2000 4000 6000 8000 10000 12000 14000-400 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 Optimal Distribution Outcome: Continuation Outcome: Discontinuation Cumulative Trajectory, Cont. Figure 11: Comparison of different outcome and cu mulative trajectory fina l outcome distributions with the “optimal” skewed distributi on under pilot conditions. The nu mber of previous trials used is seven unless otherwise stated. As can be seen in figure 11, using an outco me trajectory strategy or a cumulative trajectory strategy can generate skew in the final outcome distribution such that one is more likely to end up at one end of the fina l outcome distribution over the other. The shape of these trajectory stra tegies is mesokurtic (similar sh ape as a normal distribution) while the shape of the aspiration levels is leptokurtic (higher probability of ending up at the mean and at extreme values compared to the normal distribution).

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43 Why there is Skew and the Insens itivity to Expected Value. Trajectory decision strategy outcome distributions are skewed b ecause, by evaluating a subset of previous outcomes as “better” than average or “wor se” than average and then looking for a majority, one is provided a rough estimate of progress thus far in the task, much like the current total provides a rough es timate of progress as it is us ed with the aspiration level criterion. Outcome trajectory strategies do not requi re that you be able to add up all the previous outcomes into a statistic that represent all the previous outcome information (current total), unlike aspirati on level decision strategies. As previously stated, the final outcome distributions for aspiration level decisi on strategies become more or less skewed as the aspiration level cutoff point becomes closer or further away from the overall expected value of the lottery series (1000). These trajectory strategies do not use the actual current total as the determinan t for risk preference, but rather changes or perceived changes in performance as the determinant for ri sk preference, so the expected value does not need to be known to effectively use a goal-trajectory base d decision strategy. The Direction of Skew for a Trajectory Strategy. The direction of skew (left or right) depends on the behavior exhibited by the decision maker when a majority of better/worse outcomes is acquired. When a ma jority of better outcomes is experienced and risk seeking is exhibited, skew follows the same direction as the aspiration level criterion. However, when risk aversion is exhibited during a majority of better outcomes being experienced, skew goes in the opposite direction. In addition, cumulative trajectories can be seen here as le ss skewed than outcome trajectories.

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44 The skew changes direction between th e two outcome trajectory strategies because behavior with respect to “how well you are doing so far” is different between the two strategies. For example, if when usi ng the Continuation strategy, a decision maker experienced a majority of “better” outcomes, the strategy suggests that since the decision maker would consider themselves as doing well and therefore suggest risk-seeking behavior, which follows similarly to the goals laid out by Schneider and colleagues (2007); the skew for the Continuation strate gy follows in the same direction as the aspiration level strategies, sin ce risks are being taken at sim ilar times relative to one’s evaluation of current progress (e.g., “when I eval uate my progress as above average, I am risk seeking”). However, if you were using the Discontinuation strategy and you experienced a majority of “better” outcom es, you would then exhi bit risk aversion, and hence skew goes in th e opposite direction. Why Cumulative Trajectories differ from Outcome Trajectories. For cumulative trajectory strategies, a similar evaluation of s ituated progress in the task is taking place, however the element being evaluated (change to cumulative total) is heavily influenced by the fluctuation in the values of the individual lotteries. Fo r instance, if one of the five previous lotteries had an extremely low expected value (for example, if the previous five outcomes were -50, -450,150, 50, 0) or if the expected values for the previous five lotteries were all of a similar direction from zero (for example, if the previous five outcomes were 100, 50, 150, 0, 50), then the changes in cumulative total are not necessarily representative of changes in rela tive success in the task. Skew seems to only present itself when a decision strategy uses an accurate measure of performance, not

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45 simply overall changes in cumulative value due to changes in the static task environment. Cumulative trajectory strategies are problem atic because they are overly sensitive to order effects and extreme expected values in certain specific in stances like the above mentioned examples. However, that is not to say there won’t be instances where cumulative trajectory accurately gauges relati ve progress in the task, and hence skew is still present, however it is reduced compared to outcome trajectories. 0 2000 4000 6000 8000 10000 12000 14000-400 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 Optimal Distribution Outcome: 5 Previous Trials, Cont. Outcome: 7 Previous Trials, Cont. Outcome: All Previous Trials, Cont. Figure 12: Outcome Trajectories us ing different number of previous outcomes as determinants for risk preference. The number of Previous Outcomes being Evaluated. Figure 12 compares different versions of the Con tinuation outcome trajectory st rategy. Each version of the Continuation strategy uses a different-sized subset of previous outcomes from which it attempts to determine a majority. For the strategy labeled “5 previous trials,” the

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46 decision strategy only evaluated the five trials that immediately preceded the current trial to search for a majority (so in this case, if th ere were 3 or more of the “better” outcomes). “7 previous trials” only used the immediatel y-preceding 7 trials, and all previous trials used all the previous trials. As can be seen in figure 12, the skew in the final outcome distribution is greater for the decision strategies that use more of the previous outcome information. For example, if instead of only looking at the majority of the previous seven outcomes, I were looking at the majority of the previous nine outcomes, the final outcome distribution would be further skewed. If one were to use fewer of the previous outcomes to establish a majority of “bette r” over “worse” outcomes, one would see a reduction in the amount of distributional skew. In addition, by looking at all the previ ous outcomes (shown in figure 12 above), the distribution becomes maximally skewed, b ecause all the previous outcomes are being taken into account. As with the aspira tion level decision strategies, the more accurate the information about previous and current perfor mance (e.g., closer approximation to what’s expected3), the more effectively a decision maker can choose to be risk-seeking or risk averse consistent with their goals about wher e they end up relative to what they expect (e.g., Schneider, 2007). 3 The accuracy of previous inform ation for outcome trajectories is relative to the “expectation” of receiving the “better” of the two outcomes equa lly often as receiving the “worse” of the two outcomes.

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47 Study One Results Summary After further investigation of the as piration level decision strategies, we established an aspiration level’s sensitivity to th e expected value of the series of lotteries. Aspiration levels require information about “w hat is expected” at th e end of the lottery series (overall expected value). The more accurate the information about “what is expected” is to the overall expected value fo r the lottery series, the more skew can be generated in the out come distribution. In addition, we have created decision stra tegies that use the previous outcomes and changes in cumulative total as determinan ts for risk preference, which are insensitive to expected value, but are also sensitive to one’s ability to remember previous events. Outcome trajectory strategies, which establis h a majority from a subset of previous outcomes evaluated as “better” or “worse” compared to the alternative outcome, can skew the distribution of outcomes, though less-so compared to aspiration level strategies. Cumulative trajectory strategies, which use the changes in cumulative total across a subset of previous lotteries, can also skew the distribu tion of outcomes, though less-so compared to outcome trajectory strategi es and aspiration le vel strategies. As the subset of previous outcomes being used in a trajectory strategy is made larger, the amount of skew that can be generated in the distribution becomes larger as well. Aspiration level decision strategies and trajectory decision strategies are in essence using a similar means to impact the outcome distribution; some measure of performance relative to “what is expected” is used to switch between risk-s eeking and risk-averse

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48 behavior when current performance is evalua ted as either “bette r than expected” or “worse than expected.”

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49 Study Two: Test for Parameters of the Decision Environment It is presumable that by changing the envi ronment in which decisions take place, the effectiveness of certain decision strategies may change. For example, how will different decision strategies be affected by incr easing the number of lotteries in the series, removing certain kinds of lotteries, or introducing a real-world noncompensatory component? In an attempt to address these co ncerns, we present several variations on the previously presented methodology to invest igate how the effectiveness of decision strategies may change unde r certain circumstances. Study Two Methods We suggested three specific adaptations to the decision environment used in the previous simulation study to investigate how subtle changes to the decision environment might affect the performance of di fferent decision strategy types. (1) 72 trials were used instead of 36 in an attempt to eliminate anomalies in the final outcome distributions. Also, the effectiveness of tr ajectory-based strategies is investigated. (2) In an attempt to introduce a non-com pensatory real-world component, all of the decision strategies were re-tested under the constraints that if a participant fell below a certain wealth threshold (went broke), they no longer accumulate any more wealth (equivalent to dying).

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50 (3) In order to examine the effects of each strategy in an environment similar to the experimental conditions in which Prospect theory model and other models have been tested, a version of the decision environmen t was simulated using decisions that were either entirely riskless or very risky. Unless otherwise explicitly stated, all materi als, orders, valences, series lengths, simulations types, and procedures are the same as in Study 1. Study Two Analysis of Simulations An Examination of Trial Series Length When the total length of the trial series was increased from 36 trials to 72 trials, there was little change in any of the strategies. This parameter for decision simulations can be thought of similarly to increasing the ‘n’ for a sampling distribution in inferential statistics. By increasing the number of tria ls each participant goes through and holding all else equal, it is a transition from one sampling distribution to another more representative sampling distribution. The more representative sampling distribution makes decision strategies more normally distribut ed. As a result, risk policies and static lottery-based strategies remained relativ ely unchanged, aside from appearing more normally distributed in shape. However, should a decision strategy use the number of previous outcomes (as with the trajectory strategies) or make some other calculation based on overall distributional variability (as with aspiration leve l strategies), there are likely to be other changes in distributional shape.

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51 0 2000 4000 6000 8000 10000 12000 14000-400 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400Number of ParticipantsFinal Outcome Optimal Distribution, 36 trials Cumulative total: 775, 36 trials Cumulative total: 1225, 36 trials Optimal Distribution, 72 trials Cumulative Total: 775, 72 trials Cumulative Total: 1225, 72 trials Figure 13: Comparison of 36 trial lottery distri butions and 72 trial lottery distributions for aspiration level strategies. The median and mode values for the aspiration level strategy final outcome distributions were further skewed roughly $150 mo re to the right compared to 36 trials. Aspiration-based strategies with 72 trials are also much more leptokurtotic, resulting in a higher likelihood at the mean and in the ex tremes. Even though there were more opportunities for virtual partic ipants to reach intermediate final outcomes, the distributions for the 72 lottery series do not a ppear to be much smoother. Because there are more trials with which to vary distributional variability based on cumulative total, the skew is likely to increase just as a result of the afforded opportunities. However, the skew that is added based on in creasing this parameter is limited. If trial length was extended infinitely (as with a continuous pr obability distribution), the aspiration level

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52 strategies would look like some variant of the optimal final outcome distribution, with different amounts of skew depending on their “cutoff” values. 0 2000 4000 6000 8000 10000 12000 14000-400 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400Number of ParticipantsFinal Outcome Optimal Distribution, 36 trials Outcome: Continuation, 36 trials Optimal Distribution, 72 trials Outcome: Continuation, 72 trials Figure 14: Comparison of 36 trial lottery distribution s and 72 trial lottery distributions for trajectory strategies. However, goal-dependent/trajectory st rategies, shown in Figure 14, lost a substantial portion of skew, ma inly because the subset of seven previous lotteries which were used to evaluate progre ss now represented a smaller per centage of the total trials, and hence were less reflective of total progr ess in the task. The functionality of a trajectory strategy with limited information about previous outcomes (e.g., limited to only the previous seven outcomes) is such th at the more previous information you can evaluate (your limit) compared to the total amount of previous information (the lotteries series length) the more accurate your derived evaluation of your current progress relative to the “mean” or average participant.

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53 The Implications of “Going Broke” The percentage of participants who “wen t broke” is shown in Figure 15. The risk aversion policy had the fewest number of pa rticipants drop out and the risk seeking policy had the largest number of participants drop out. Other static strategies had participants drop out relative to their overall riskiness. Because the random, Prospect Theory, and Risk as Threat strategies all ha ve roughly the same amount of final outcome distributional variability in this instance, it only makes sense that roughly the same percentage of participants would drop out of the study for those three strategies. There are, however, slight differences between the ra ndom, Prospect Theory, and Risk as Threat as the result of simple order effects. 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00% 50.00% 55.00% 136Percentage of Participants that Dropped Out Risk Seeking Cumulative: Continuation Outcome: Continuation Outcome: Discontinuation Aspiration Level: 975 Risk Aversion 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00% 50.00% 55.00% 136Percentage of Participants that Dropped Out Risk Seeking Risk as Threat Random Prospect Theory Modest Variance Risk Aversion Figure 15: Percentage of participants who dropped out of the study (via reaching zero) for each static strategy (left) and dynamic strategy (right). However, aspiration level strategies s how a strong benefit in this decision environment, with a large reduction in the pe rcentage of participants who drop out, as shown in figure 15. It is important to note th at the aspiration level criterion as a measure of progress is sensitive to the distance of the current total to zero, while outcome

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54 trajectory strategies are insens itive to one’s current total. Hence, outcome trajectory strategies showed no benef it over static strategies unde r these conditions. Also, cumulative trajectory strategies use changes in cumulative total as the criterion, and hence cannot take into account information about what the actual cumulative total is but only how it has changed As a result, it is understandable that the cumulative trajectory strategy shows no improvement compared to other trajectory strategies or static strategies. Riskless versus Very Risky Lotteries The next change in the decision envir onment involved removing from the choice set all the lottery pair s that did not include a “sure th ing” option, in addition to doubling the difference in the ticket separation values. This was done in an attempt to approximate the “riskless” choice options used when for the initial proposition of Prospect Theory (Tversky & Kahneman, 1981). For example, if the lottery pair in cluded a 50/50 chance between $100 and $200 or a 50/50 chance between $50 and $250, it wa s excluded from this manipulation. However, if the lottery pair included a 50/50 chance between $100 and $200 or a sure thing of $150, all values were doubled, making the new lottery choi ce between either a 50/50 chance of $200 and $400 or a sure thing of $300. Because there were both positive and negative lotteries being doubled, the expected value remained the same. When the lottery choice set was limited to a choice between lotteries that were either riskless or very risky as explained prev iously, risk policies showed a vast change. For the risk averse strategy, all virtual partic ipants ended with a final outcome that was

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55 equal to the mean of the dist ribution (since they always sele cted the riskless choice) and for the risk seeking strategy, vi rtual participants’ final outcom es were more spread out, due to the increase in total variability to the decision environment. Lottery-based strategies based on valence showed relatively no change, while the decision strategy that uses the amount of ticket separations (modest variance) became akin to a risk policy, since there was only one type of inter-lottery variation. 0 2000 4000 6000 8000 10000 12000 14000 200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Aspiration Level:775 Aspiration Level: 975 Aspiration Level: 1025 Aspiration Level: 1225 Figure 16: Aspiration level strategies for riskless v. very risky lottery set. Note that there are only 40,000 virtual participants per strategy, one-thir d the participants compa red to previous outcome distributions. Aspiration-based strategies showed an increase in skew, since an increase in inter-lottery variability (lotteries were made to be more risky) meant that changes in to the cumulative total were more meaningf ul as a gauge of progress in the task. Outcome trajectories, however, are ineff ective in the face of riskless outcomes, because there is no way to extrapolate meani ngful information. For instance, if I am

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56 offering you a choice between two sets of two outcomes with equal probability, the first set consists of 5 dollars or 15 dollars while the second set consists of 10 dollars or 10 dollars, if you choose the second set, there is no meaningful way to differentiate getting one of the 10 dollar outcomes as better or wo rse than the other 10 dollar outcome. As soon as a majority of riskless options has been chosen, the decision strategy has no way of distinguishing a majority of better-worse outcomes, unless there is a stochastic method (flipping a coin) which would lead one to tend back towards risk seeking behavior. Study Two Results Summary Under decision environments with different characteristics or conditions, such as longer trial lengths, non-compensatory dr opping out conditions, and changes in the availability of intermediate risk (as in the riskless v. very risky environment), there are advantages and disadvantages for each type of decision strategy, but especially for dynamic decision strategies like the aspirati on level strategies or the goal trajectory strategies. When a lottery series of 72 trials is used instead of 36 trials, static decision strategy outcome distributions are found on the whole to appear more normally distributed, while aspiration level decision strategy outcome distributions are more leptokurtotic. When a non-compensatory drop-out compone nt (dropping out of the study) is introduced, the advantage of one static strate gy over another is directly related to the amount of overall final outcome distributional variability (less variab ility leads to fewer virtual participants “going broke”). Howeve r, by skewing the final outcome distribution via a dynamic decision strategy, it is possible to capitalize on the a dvantages of static

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57 strategies under different non-compensatory co nstraints. Dynamic strategies such as aspiration level strategies show advantag es for this non-compensatory component without simple differences in final outcome distributional variability; the same number of participants drop out of the study for the aspiration level decision strategy as the risk averse strategy. Under conditions similar to the initial ev aluation of Prospect Theory (Tversky & Kahneman, 1981), where no intermediate risks are present but all risks are markedly larger, risk-averse behavior leads to certai nty with respect to the final outcome, riskseeking behavior leads to a spread out fi nal outcome distributi on, and other static strategies still appear entire ly symmetrical. Aspiration leve l strategies capitalize on the advantages of having markedly larger risks. Outcome trajectory strategies are limited in their effectiveness without intermediate risks. In the real world, decision-making takes place under a myriad of different contexts; it becomes increasingly difficult to understand which aspects of the decision environment are most important in determining strategy formation. Based on our observations of simulated environments, aspi ration levels show a clear advantage in many of the contexts where static strategies show no influence on final outcomes. Hence we used an empirical test to determine if one aspect in particular (cumulative wealth) would be used as a determinant for risky choice by actual participants.

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58 Study Three: Empirical Investigation of Interaction between Situation and Context In addition to the simulations, this project includes an empirical investigation to elucidate which strategies might come closes t to what people actually do. First, I will address previous empirical work pertinent to the current approach. Then, the current empirical test is outlined, followed by a discussion of the various analyses and comparisons used for the empirical test. Pilot Empirical Work In addition to the simulations on risky choice, Schneider and colleagues have looked at two different tasks to examine how people’s risk preferences differ. Schneider (2002) designed a decision paradigm mentione d earlier that may be preferable to the standard risky choice paradigm because it makes goal trajectory information more available to the participant. To see how act ual strategies may differ, the standard risky choice paradigm and the new goa l-trajectory paradigm (ticket transfer task) are directly compared. In the passive “Choice” task, participants are shown two static lotteries and are asked to simply choose which of the two lotterie s they prefer to play. In the active goaltrajectory paradigm (called the “Move” task from here on), participants are given a single lottery and are instructed that they are about to play the lottery. Before playing, their task is to improve the lottery in one of two ways, both of which ultimately lead to the same choices in the Choice task. In this paradigm, participants are either instructed to actively

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59 increase or decrease the amount of risk they are exposed to (F igure 12) if they are in the move condition, or they are subjected to th e standard passive choice task in which participants choose between two lotteries that vary in the amount of risk (Figure 13). The active goal seeking paradigm was inve nted by Schneider to “situate” decision makers in a given decision environment. With passive choice, people are just looking at two different options, and there is no context with respect to how these lotteries relate to their goals or activities. However using th e Move task, participants are given context about the choice, and are asked to make a cha nge in their current situation to influence future outcomes. Schneider’s Move task combines aspects of situated cognition by attributing a decision context and then allowing participants to exert some control over the potential outcomes. Figure 17: Ticket Transfer Paradigm-Move Exam ple Diagram. In Ticket Transfer paradigm, participants are asked to actively change the amount of risk in a given lottery. The arrows are meant to represent the possible ticket moves.

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60 Figure 18: Ticket Transfer Paradigm-Choice Example Diagram. This is an example of the Ticket Transfer paradigm task that calls on participants to passively choose one of the two possible lotteries for a given lottery pair. In her first investigation of the Ticket Transfer paradigm phenomenon, Schneider and colleagues (2005) compared risk preferen ces for lotteries that were positive and negative using the Choice and Move tasks. Using the Choice task, Schneider replicated the preference patterns predicted by the S-sh aped value function of Prospect Theory. Specifically, when both possible outcomes were monetary gains, par ticipants were more likely to choose the less risky choice opti on, and when both possible outcomes were monetary losses, participants were more likely to choose the more risky choice option (shown in Figure 14).

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61 0.10 0.30 0.50 0.70 0.90Lg NEGSm NEGN w/ 0N w/ PP w/ 0Sm POSLg POS% Risky ChoicesValenceStandard Passive Choice Figure 19: Risk preference by Valence for the Choice task in Previous Research. The higher on the Y-axis, the more participants there were who exhi bited risk-seeking behavior in that valence. However, when participants were give n the Move task, their responding was not consistent with the S-shaped value func tion. Specifically, participants were predominantly risk averse throughout (shown in Figure 15). Schneider and colleagues argue that by providing the decision maker with a situated de cision task involving outcome trajectory information, the decision ma kers are in a better position to understand the meaning of risk and to deal with it in a manner characteristic of other goal-related activities. This enhanced understanding lead s to preference pattern s that are markedly different from Prospect Theory predictions and more consiste nt with standard intuitions of how to deal with risk.

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62 0.10 0.30 0.50 0.70 0.90Lg NEGSm NEGN w/ 0N w/ PP w/ 0Sm POSLg POS% Risky ChoicesValenceMove vs. Choice Move Data Choice Data Figure 20: Risk preference by Valence comparison of Move vs. Choice tasks from Previous Research. The higher on the Y-axis, the more participants there were who exhi bited risk-seeking behavior in that valence. In an extension of this study to non-mone tary outcomes, Schneider and colleagues (2005b) used “health units” as lo ttery outcomes, and told partic ipants that these units are meant to represent decisions about improvement s or detriments to current health. This adaptation using “health units” was meant to contextualize the Ticket Transfer task beyond the use of simple monetary outcome s. Schneider and colleagues (2005b) found the same pattern of preference reversal fo r the Choice task, and the same lack of preference reversal for the Move task. In a third manipulation, Schneider and colleagues (2006) added a component to the Ticket Transfer tasks (both Move and Choi ce) such that the lotteries actually played and participants were paid some multiple of their winnings. Given this new capability, Schneider and colleagues (2006) developed what is essentiall y a “trajectory”

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63 manipulation, such that half of the participants would start o ff with less in their initial “capital,” but would have an overall positive experience up to the status quo, while the other half of the par ticipants would start off with mo re, but would have an overall negative experience down to the status quo. From this goal trajectory manipulatio n, Schneider and colleagues (2006) found that there were no substantive differences in risk preference given the different trajectory conditions (though the standard choice versus move differences were replicated); however, participants were asked affect ques tions about how they were feeling at the beginning of the study and about how they we re feeling at the end of the study. The responses to these affect que stions showed an overall improvement in mood for the positive trajectory condition, and an overall deterioration in mood for the negative trajectory condition. This shows the importan ce of long-term considerations such as “trajectories” are important because they im pact individual participant experiences. Because these long-term considerations are often overlooked in standard risky choice tasks, we plan to extend the usage of accumulation and experiential outcomes into the more fully contextualized microworld described previously. In summary, participants seem to have different patterns of risk preference depending on whether they are shown a task which requires only passive choices between two static options or are show n a task which engages them in self-initiated action to improve the value of the outcomes in a given situation. This suggests participants are sensitive to being ‘situated’ in risky environments as opposed to simply being provided with static choices.

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64 Methods for Empirical Study The purpose of our current empirical st udy is to examine the importance of various decision criteria across these two different decision tasks (the passive Choice task and the active Move task) with respect to risk preferences. Participants completed either the Move task or the Choice task. In a ddition, to incorporate the inclusion of timedependent contextual information, several dynam ic contextual cues were made available, specifically (a) results of each of the plays of the selected lottery, (b) curr ent actualized wealth and (c) wealth reflected as “social status” at various intervals. These dynamic contextual cues exist to provi de additional information for pa rticipants to potentially use in guiding their decisions, as th is type of dynamic informati on is often available in the real world. Also, participan ts were provided with noncompensatory elements, namely the requirement of keeping one’s current w ealth above zero to avoid “going broke.” Participants 202 undergraduate psychology students receive d extra credit toward their course grade for participating in this laboratory session; 111 were shown the choice version, while 91 were shown the move version. Due to our participant pool, the majority were female and between the ages of 18 and 22. Of those, several participants were dropped from the analysis (16 from the Choice versi on and 17 from the Move version), as they failed to pass a short quiz, designed to test if they understood the task. Stimulus Participants were provided with lotteries that look the similar to the lotteries shown in figure 17 (excluding the arrows and the “OR”) and 18. As discussed

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65 previously, for the “Choice” ta sk, participants passively choose between 2 two-outcome lotteries (figure 18), while for the “Move” ta sk, participants start with a two-outcome lottery and improve one of the two outcome s (figure 17). Each two-outcome lottery consists of outcomes which were either both positive (e.g., 50 & 100), both negative (e.g., -50 & -100), or mixed (some combinati on of positive values, negative values, or zero). The two outcomes were separated by di screte intervals of ‘50’. The largest disparity between any two outco mes was 3 intervals (‘150’). Design There were 4 orders in each condition, a nd all participants began with the same initial score. This score served as a stand in for currency. A higher score represent more currency. Also, the expected value for the seri es of lotteries was ke pt consistent between all orders and with the s imulation study. There were 38 unique lotteries (from an expected value of ‘-250’ to ‘325’) and each lo ttery type was presented twice with a few exceptions4 for a total of 75 lotteries in each order. Each order consisted of 3 practice lotte ry pairs, 8 quiz questions, and 75 twoticket lotteries or lottery pairs, followed by severa l questions about the participant’s strategies regarding th e lotteries and their status in the micro-world (to be discussed later). In addition several que stionnaires were used to gauge individual differences in risk preferences, namely Higgins’s Regulat ory focus survey and Elke Weber’s risk tolerance survey. 4 So as to achieve a slight positive trajectory from the initial score of ‘600’ to the overall expected value of ‘1000’, two of the most positive lotteries were not d uplicated, and one of the negative lotteries was seen three times.

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66 The Micro-world The task consists of a micro-world desi gned as an oversimplification of realworld decision-making with respect to risk pr eferences. Participants were provided with a series of forced-choice two-outcome lotte ries, and the outcomes of those lotteries accumulated into a displayed value referred to as their “current wealth.” This current wealth value or “score” repres ents the participant’s current socio-economic status in the micro-world. Based on the simulation results5, we set an initial sc ore of 600. Current wealth was adjusted after each lottery was played based on the randomly determined outcome, similarly to the simulation study. In order to further contextu alize the lottery playing tas k, a variety of information about quality of life relative to a participant’s current wealth was provided to participants at occasional intervals. This information in cluded a brief summary of the participant’s current living conditions as proportional their amount of current wealth (This information is outlined in Table 2). The larger the participant’s current wealth, the richer that participant was considered to be. As menti oned previously, particip ants receive feedback about their standing in the micro-world incr ementally after every 5 lotteries throughout the experiment. The characteristics for the different levels of social status in the simulated microworld are designed to represent important real -life human concerns. These characteristics include availability of nourishment, the qual ity of housing, social influence, level of monetary wealth and capital, and access to medical care. Ba sed on a participant’s current 5 We decided on 600 as an initial score in order to have some participants who would “go broke” in the study, but no so many as to negatively impact our results.

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67 wealth, these characteristics changed at specif ic “milestones” or cutoff points. Also, if one of these milestones was reached and a par ticipant had zero or le ss than zero in their cumulative total, the participant was notified that they “died” in the simulated microworld, and they were directed to answer a ny final questions and leave the room early. Procedure A desktop computer was used to run th e program, which cons isted of either a series of pairs of lotteries as in the Choice ta sk or a series of singular lotteries as in the Move task. Participants were in one of two conditions; eith er they were asked to choose one of the lotteries to play (f or the Choice task) or they were asked to choose one of the two ‘tickets’ (outcomes) to improve before the lottery played (for the Move task). Once a selection had occurred, a random drawing took place, and the ‘winning’ outcome was displayed on screen before continuing to the next lottery. After every 5 lotteries, participants were provided with the contextual feedback disc ussed earlier (an example is provided in Appendix B), or if their current wealth was at or below zero, they were informed of their demise and no more lotterie s were provided. Once participants finished all the lotteries, either by completing them or by “going broke,” they were asked several questions regarding their overall decision stra tegy, as well are their reasoning for why they would choose a particular type of lottery (for Choice) or ticket to improve (for Move).

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68 Table 2: Summary of Social Status Indicators. Residence Availability of Nourishment Method of transportation Monetary capital Medical coverage Destitute Subsidized in an unsafe neighborhood Severely limited Walk or public transit Limited to survival No medical coverage Lower Forced to share an apartment with others in mostly unsafe neighborhood Can maintain healthy eating but cannot afford restaurants Bike or moped, cannot afford car Most spent on living expenses, but some money can be saved Limited emergency medical coverage Lower middle Can afford own apartment, or share nicer house with a roommate in moderately safe neighborhood. Maintain healthy eating and restaurants a couple of nights a week Can afford Economy Car About half of money earned is spending money Partial medical coverage /HMO Upper middle Can afford own house or apartment in safe neighborhood Can eat at restaurants any time Can afford a nice car A large portion of money can be saved or invested in the future Full medical coverage Very Rich Can afford any house in completely safe neighborhood Personal chef and nutritionist Luxury vehicle Almost all of money can be saved or invested in the future Top notch health coverage and personal trainer Note: These were provided to participants in the empirical portion of the experiment.

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69 Results of Empirical Study For the empirical investigation, the go al was to understand what types of contextual information people might be us ing when forming their personal risk preferences. To do this, we present an analys is of the data starting from the simplest of possible risk strategies (Risk Policies) and investigate progr essively more complex risk strategies. After simple risk policy strategies, the next le vel of complexity focuses on static lottery characteristics, such as valen ce and variability. The mo st complex strategies include aspiration levels and dynamic goal tr ajectory strategies. We also explore differences between the Choice task and the Move task conditions at each level of complexity, as well as the implications that arise from participants “going broke.” Participants Who “Went Broke” For participants who were ri sk-seeking more than half of the time, the rate at which they “dropped out” of the study was 33% while for participants who were riskaverse more than half of the time, the rate at which they dropped out of the study was only 28%, which demonstrates that being ri sk-averse helps your ch ances to avoid going broke (confirming what was already shown defi nitely using the statistical simulations). Also, for participants who went broke usi ng the risk-averse policy, the minimum number of trials or “life experien ces” was 15 and the maximum number of trials was 65, whereas for the participants who were reliably risk seeking, the minimum was 10 and the maximum was 45. Thus, participants who we re predominantly risk averse generally outlived participants who were predominantly risk seeking.

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70 Decision Strategy Comparison In addition to a comparison across different lottery characteri stics (e.g., valence) and interpersonal characteristics (e.g., current wealth), tests were conducted to compare each participant’s risk preferen ces with simulated risk strate gies. This project uses the decision strategies simulated previously to work backwards and see if “actual” participants are making the same choices as an idealized “virtual” participant might respond under the same conditions. This was done to get a better understanding of whether participants are reacting to specific contextual information from the decision environment at each of the different levels of complexity (basic risk policy differences, static lottery-dependent, wealth-focused/aspiration-based, goal trajectories). Strategy comparisons were conducted as fo llows: For basic risk policy decision strategies, we simply had to look at the numbe r of times a participant was risk-averse or risk seeking. Static, lottery-dependent deci sion strategies focused on the valence of each lottery and the amount of risk in each lottery to determine their choices. We just look at how the characteristics of the lotteries affect preferences. For dynamic decision strategies such as the aspirati on level strategies or the trajec tory strategies, which require more robust contextual information, an anal ysis was done for each individual based that participant’s complete “decision environment. ” This “decision environment” included information specific to that individual (inc luding which lotteries were presented, what previous outcomes they received, their current total at each trial, etc.); aspiration level decision strategies used the information about current wealth at each trial, and the goal

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71 trajectory decision strategies used a running tally of inform ation about previous outcomes at each trial. This analytical technique takes the same basic principle behind creating virtual participants from decision stra tegies. However, instead of putting the virtual participants into their own unique environment (as with a randomly generated outcome), “virtual” participants are placed in the environmen t experienced by “actual” participants. Therefore, if an actual participant is using a particular decision strategy, then the risk preferences of the “vir tual” participant would be the same as the risk preferences of the accompanying “actual” participant. The number of participants who were be st predicted by a pa rticular decision strategy type is presented at each level of complexity, and comparisons between difference strategies are discussed with referen ce to Table 3. In the case of a tie (two or more strategies predicting a participant with the same accuracy), we used parsimony and credit was attributed to the simpler of the two decision strategies. The samples for each task type only reflect participants who had complete decision environments (i.e., participants who “went br oke” were not included). Table 3: Percentage of Participants Pred icted by each Decision Strategy Type. Risk Aversion Risk Seeking Prospect Theory Risk As Threat Modest Variance Asp. Levels Goal Trajectories % Best Predicted Choice (n = 63) 32% 18% 11% 10% 3% 14% 13% Move (n = 53) 43% 9% 2% 21% 11% 4% 9% Avg. Best Prediction Choice (n = 63) 79% 80% 69% 61% 67% 66% 69% Move (n = 53) 82% 73% 65% 69% 72% 66% 76% Note: % Best Predicted represents the proportion of participants (excluding those who ‘went broke’) whose preferences were most accurately predicted by the given decision strategy. The average best prediction represents the propor tion of actual responses that were accurately predicted for participants who were best predicted for the given decision strategy.

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72 Simple Differences in the Amount of Risk There was an overall tendency toward s risk aversion for both the Move and Choice task. For the Choice task, participants took risks an averag e of 45% of the time and for the Move task, participants took risks an average of 36% of the time. As shown in Table 3 (p. 62), when the risk preference s of actual participants were compared to simulated risk policies for the Choice task, 32% of participants were best predicted by the risk-averse strategy while 18% were best predicted by the risk-s eeking strategy. For Move, 43% of participants were best predicted by the risk-a verse strategy, while 9% were best predicted by the risk-seeking strategy. In addition, on average for both Move and Choice, risk policy strategies predicted roughly 80 % of actual responses, compared to the other decision strategies whic h all predicted roughly 65-70% of actual responses, with the exception of risk seeking in the Move task. Th is shows that simple risk policy strategies have the largest margin of prediction for act ual participant responding, likely due to their simplistic nature. In addition, risk aversion tends to be a more predominant strategy for the situated Move task than for the passive Choice task.

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73 Risk Preferences as a Function of Lottery-Dependent Characteristics 0.1 0.3 0.5 0.7 0.9Lg NEGSm NEGN w/ 0N w/ PP w/ 0Sm POSLg POS% Risky ChoicesValenceMove vs. Choice Previous Choice Data Previous Move Data Current Choice Data Current Move Data Figure 21: Comparison of previous and current Move v. Choice data. To look for differences with respect to valence, and to see if our results are consistent with previous findings, Figure 21 shows the comparison of the current data from the Move task and Choice task with prev ious versions of the Choice task and Move task. Note that for both Choice and Move, ther e appears to be a substantial reduction in the number of risks taken for the negative lotterie s, as well as a general ‘middling’ of risk preferences effect across all th e valences (fewer risks taken/ avoided where there used to be a large majority). The substantial reducti on in the number of risks taken may have had something to do with the potential for particip ants to “go broke,” as avoidance of this noncompensatory element would suggest an incr ease in risk-averse be havior and was not included in previous manipulations.

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74 The simulation comparison for decision st rategies that use static lottery characteristics (specifically, de cision strategies similar to the Prospect Theory model or Risk as Threat model) allowed us to see if different individuals we re using a consistent strategy that focused on the vale nces of the lotteries or the amount of variability between the outcomes in the lotteries. As discu ssed in Study 1, the Prospect Theory decision strategy produces risk-seeking behavior for lotteries with all negative outcomes and riskaverse behavior for lotteries with either all positive outcomes or a mix of positive and negative outcomes. The Risk as Threat deci sion strategy exhibits risk-seeking behavior for lotteries with all positive outcomes and risk-averse behavi or for lotteries with either all negative outcomes or a mix of positive and negative outcomes. The Modest Variance decision strategy exhibits risk-s eeking behavior when there is the potential for a riskless or “sure thing” outcome, otherwise exhibi ting risk-aversion. The percentage of participants who were best predicted by these strategies is shown in Table 3 (p. 62). For Choice, the effectiveness of the Pros pect Theory-based decision strategy at predicting actual participant responding fo r choice is the same as the predictive effectiveness as the Risk As Threat-based deci sion strategies (10%). For Move, Prospect Theory only predicts 2% of the participants while the Risk as Th reat decision strategy predicts 19% of the pa rticipants. This suggests the diffe rences articulated in Schneider (2002, 2005) are indeed correct; Prospect Th eory does not adequately account for decision making with respect to risk when under Choice task conditions and Prospect Theory does even worse at attempting to accoun t for decision-making with respect to risk when dealing with the active Move task. In addition, the Modest Variance decision

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75 strategy was a worse predictor for Choice than for Move. This suggests that passively choosing between two options when one of th em is riskless (e.g., Choice) does not have the same impact on avoiding riskless options as being forced to actively configure a riskless option (e.g., Move). Risk Preferences as a Function of Current Wealth. Because participants may be using information related to their individual current wealth, we investigated indivi dual risk preferences at each wealth level in both Move and Choice. Instead of simply averaging across each wealth level for all participants, we wanted to reflect patterns of be havior at each wealth level, and so first made sure that a participant spent a substantial amount of time at a particular wealth le vel before their risk preferences contributed to the analysis. This was to accoun t for the fact that not all individuals saw all the wealth levels for a prolonged period of time; it wouldn’t make sense to compare people’s risk preferences abou t particular wealth levels if they weren’t spending a significant amount of time there. First, the amount of time a pa rticipant spent at each wealth level was tabulated. Then for all the participants who spent a large enough amount of time at each particular wealth level (more than 5 trials), their indivi dual ratios of risk pref erences (percentage of risk-seeking behavior) were averaged within each participant and th en across each wealth level. The amount of risk seeking behavior at each wealth level is shown in Figure 22.

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76 n = 52 n = 40 n = 89 n = 72 n = 79 n = 67 n = 60 n = 50 n = 25 n = 16 0.1 0.3 0.5 0.7 0.9 ChoiceMoveProportion of Risk SeekingProportion of Risk Seeking at each Current Wealth Level Poor Lower Middle Middle Upper Middle Rich Figure 22: Individual Risk Attitudes at each Wealth Level. ‘N’ is the number of participants who had more than 5 responses at the given wealth level. Figure 22 shows that when individual risk preferences are evaluated across wealth levels, the pattern of risk seeking for Choi ce steadily increases as participants have a higher current wealth level, whereas the pattern of risk seeking for Move steadily decreases as participants have a higher curren t wealth level. Because there are a different number of individuals at each wealth leve l and different individuals spent different amounts of time at each wealth level, the va riances are not homogenous, so a statistical comparison of these groups would not be prudent. To better capture the poten tial individual usage of current wealth levels, a comparison of actual participan ts’ risk preferences and virt ual participants imbued with decision strategies focused on current wealth was analyzed. As di scussed in Studies 1

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77 and 2, an aspiration level strategy predicts that a participant exhibits risk-averse behavior until their current wealth exceeds a particul ar goal or “cutoff” point, whereupon that participant exhibits risk-seeking behavior until the current wealth falls below their goal. To see if participants were focused on speci fic current wealth leve ls, several different “cutoff” point values were utilized. The pe rcentage of actual pa rticipants who were found to be best predicted by the aspiration le vel decision strategies across all of these various cutoffs are shown in Table 3 (p. 62). There were several participants for C hoice (and many fewer for Move) who were best predicted by aspira tion level strategies, which suggest s that something is going on in Choice that makes current wealth more salien t as a decision criterion than for Move. There were other participants whose prefer ences resemble with both aspiration level decision strategies and risk policies. As an aspiration level decision strategy had a progressively lower cutoff value (e.g., 775), the participants who were consistent with it were also consistent with the risk seeking risk policy. The same held true for higher cutoff values (e.g., 1225) and the risk-avers e risk policy. This makes establishing whether a participant was using current wealth as a criterion or just a simple risk policy quantitatively difficult to differentiate. To provide for the notion that participants formed their individual risk preferences using current wealth and not just a simple risk policy, the self -report style questions asked directly to participants about their overall strategy we re analyzed. Participants were asked direct confidentia l questions at the end of th e lottery series about their individual decision strategy. When participants were asked, “What was your overall

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78 strategy?” 19% of participan ts for the Move task responde d with some mention of a cumulative total or current status as a pr edominant factor, while 52% responded with a similar strategy answer for the Choice task. Despite the overlap between risk polic ies and more complex dynamic decision strategies, aspiration level deci sion strategies that use specifi c “cutoff” values to switch between risk policies may not be the way participants are using current wealth. Participants are not told about the total rang e of current wealth st ates or the range of possible scores, so it would not be possible to effectively se t cutoffs based on particular expectations. However, for our simulations accurate expectation information for the lotteries was known a priori to the design of the decision st rategy. The stochastic nature of the simulation procedure allowed us to optim ize an aspiration level decision strategy in the general sense, since we knew what the overall expected value was; however, each participant is limited in the actual outcomes th ey will experience, and the range of values they have previously experienced is changing dynamically throughout the task. Therefore aspiration levels may be changing over time or they may be set with other factors in mind. Trying to match their perf ormance with fixed aspirations having cutoff points relative to information not explicitly given to participants isn’t a fair ‘litmus test’ of whether participants might have been usi ng their current total in their decision-making. Instead of using a “cutoff poi nt” to utilize current wealt h, participants may, in the general sense, be taking more or less risks as their cumulative total increases. To account for this potential utilization of current wea lth, a point biserial co rrelation was performed for each participant to investigate potential within-subject relationships between current

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79 wealth and risk preferences. Pa rticipants who were entirely ri sk averse or entirely risk seeking were not included for obvious reasons. Table 4 shows the pattern of results for these biserial correlations with positive corr elations suggesting greater risk seeking at higher wealth levels. Table 4: Number of Participants for Point Biserial Correlation Positive Significant Positive Non-significant Negative Non-significant Negative Significant Choice (n = 89) 12* 37 35 5* Move (n = 72) 1* 22 36 13* While the vast majority of participants have non-significant correlations, there are many more positive significant correlations for Choice and many more negative significant correlations for Move. The relatio nship between risk se eking behavior and current wealth explained here are consistent with the patterns shown in Figure 22. This suggests that for Choice, when people are usi ng current wealth in their decision-making, they do so by taking more risks as their cu rrent wealth increases, while for Move, when people using current wealth in their decisionmaking, they do so by taking fewer risks as their current wealth increases. Risk Preferences as a Function of Wealth X Valence Although we have shown there is evidence to suggest that curre nt wealth plays a role in forming risk preferences, most of the research over the last several decades regarding risk preference patterns focuses on th e importance of lottery valence. As such, the interaction between wealth levels and valence for Choice is shown in Figure 23a, while the interaction between w ealth levels and valence for Move is shown in Figure 23b. Each point represents the tota l proportion of risky choices ac ross all the participants at

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80 each particular wealth level for each valence. No limiting procedure was used because all participants saw the same number of each ty pe of lottery valence; so, in this context, wealth level is a byproduct of experience and homogeneity of va riance across wealth levels cannot be assumed. 0.1 0.3 0.5 0.7 0.9 Lg NEGSm NEGN w/ 0N w/ PP w/ 0Sm POSLg POS ValenceWealth level by Valence for Choice Task Choice--Rich Choice--Upper Middle Choice--Middle Choice--Lower Middle Choice--Poor Figure 23a: Wealth Level by Valence for Choice Task. The X axis indicates the proportion of riskseeking behavior for each weal th level at each valence. There are three noted points of interest for the Choice by Wealth level by Valence interaction (Figure 23a); First, risk pr eferences for negative lotteries appear inconsistent/pattern-less, in that they fluc tuate between risk aversion and risk seeking across all wealth levels; Sec ond, when participants are ‘poo r’, they exhibit strong risk aversion when faced with a mixed lottery ( N w/ P ) or a choice between a sure gain and a risk that includes zero ( P w/ 0 ); Third, when participants ar e at the “Rich” wealth level, they are compelled to take more risks when facing any of the lotteries which contain positive outcomes ( N w/ P P w/ 0 Sm POS Lg POS ) than another other wealth levels.

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81 0.1 0.3 0.5 0.7 0.9 Lg NEGSm NEGN w/ 0N w/ PP w/ 0Sm POSLg POS ValenceWealth Level by Valence for Move Task Move--Rich Move--Upper Middle Move--Middle Move--Lower Middle Move--Poor Figure 23b: Wealth Level by Valence for Move Task. The X axis indicates the proportion of riskseeking behavior for each weal th level at each valence. Notable points of interest for the Move by Wealth level by Valence interaction (Figure 23b) are that preferences between va lences seem relatively consistent across different wealth levels, with the potential ex ception of the “Rich” wealth level. When comparing Figures 23a and 23b, it should be not ed that the differences between wealth levels across valences are not the same for the Move task as the Choice task. Risk Preferences as a Functi on of Trajectory Information Another potential explanation for how participants may still be using dynamic information in a way that isn’t captured by the aforementioned comparisons of aspiration level strategies is that participants ar e sensitive to the overall changes in the trajectory of previous outcomes but not necessarily in their current wealth value As shown in Study

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82 1, trajectory decision strategies provide an analysis of “relative progress” based on sensitivity to subtle changes in outcomes over the short term and/or long term. The trajectory decision strategies compared with “actual” participants work the same way they were described in Study 1 ( p. 26 and 27), inferring a risk preference from a majority of previous outcomes evaluate d as the “better” outcome or the “worse” outcome. “Continuation” type trajectory stra tegies infer that a majority of “better” outcomes leads to the conclusion that outcomes will continue to be good, and hence exhibits risk-seeking behavior, while “Discontinuation” type trajectory strategies infer that a majority of “better” outcomes leads to the conclusion that outcomes will no longer continue to be good, and hence exhibits risk-a verse behavior. Each participant had their own “trajectory” depending on their individual previous ou tcomes. The information for the comparison of these trajectory decision strate gies as predictors of actual participant responding can be found in Table 3. Only th e “Continuation” version of the trajectory decision strategies was compared to actua l participants because the “Continuation” version is fundamentally consistent with the goal-directed behavior of the aspiration level decision strategies. Comparing only this ve rsion of trajectory strategies makes the predictability of aspiration levels and goa l trajectories directly comparable. Also, trajectory strategies that required a ‘super’ ma jority of previous outcomes (e.g., 5 out of 7) as “better” to insight ri sk-seeking behavior were incl uded in addition to the ‘weak’ majority (e.g., 4 out of 7) addressed in study 1 and 2, because actual participants may require a larger margin of previous outcomes be “better” before they are willing to take risks.

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83 The effectiveness of trajectory decision stra tegies at predicting actual participant responding is comparatively high for both Move and Choice given the underlying assumptions that participants were using ei ther the previous 7 outcomes or all of the previous outcomes. Consequently, there ar e also more participants who are best predicted by trajectory strate gies that use all previous outcomes than the trajectory strategies that use a limite d number of previous outcomes, which suggests that participants may not overtly se nsitive to subtle changes in trajectory over time, but may be sensitive to changes in l ong-run trajectory information. Also, the comparatively high prediction of actual responses for the Move ta sk (76%) suggests that participants who are using goal trajectory strategies do so more in a consistent manner. This may occur because there is less outcome information be ing displayed throughout the Move task, as there are only at most two valu es being shown in the task at any one given point in time.

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84 Discussion of Empirical Study For the purpose of this investigation, we compared the results of the Move vs. Choice tasks with respect to risk preferen ces in the dynamic task environment. We further compared those participant resu lts to the patterns of responding found in simulations, taking into consideration the Move v. Choice task distinction. When comparing Move v. Choice tasks, we investig ated differences in responding depending on several levels of complexi ty in the decision environmen t: basic overall risk policy differences, risk preferences with respect to valence, risk preferences with respect to when the lotteries were presented in time, risk preferences with respect to current wealth, and risk preferences as they related to goal trajectories. Because the Choice task has a slightly hi gher tendency for risk seeking behavior than the Move task, it could be argued that something about the Move task and situating participants in the decision e nvironment has an overall effect on their actions with respect to risk. However there may very well be more complex factors mitigating those differences. Risk preferences with respect to valence in large part replicated the findings of Schneider and colleagues (2005). Group differe nces indicated that the passive Choice task approximated the preference patter n outlined by Tversky and Kahneman (1981), while the active ‘Move’ task did not follow the pa ttern of preference. The strength of the Prospect Theory pattern was less pro nounced in our study than in previous manipulations, and may have been influen ced by the addition of the noncompensatory “going broke” condition, which was not present in previous manipulations. In addition,

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85 when actual participants were compared to static, lottery dependent decision strategies, the Prospect Theory model was underwhelming in its ability to predict actual participant responding for both Move and Choice, while the Risk as Threat model performed relatively well especially for Move. Risk preferences at each wealth level had different patterns depending on task type; the Move task showed a weak tendency to decrease in risk-seeking behavior as wealth increased while the Choice task showed an overall tendency to increase in risk seeking behavior as wealth increased. One interpretation is that participants under the Move task, who are actively improving the values of the outcomes, have feelings of more ‘control’ over their situa tion, and attribute the increas e in cumulative total to ‘dispositional’ factors (e.g., th ey acquired a good score base d on their skill at improving outcomes). As a result, when they achieve a high current wealth state, they attempt to skillfully “hold on” to that high curre nt wealth state by reducing risk. Choice, on the other hand, is the act of passively choosing between two lotteries, and may not necessarily lead participants to attribute their high score to dispositional factors, but rather to ‘s ituational’ factors (e.g., they acquired a good score based on random luck, a rigged game, etc.). As a resu lt, when they achieve a high current wealth state, they presume it must be some manner of random chance that got them to the higher current wealth state, or that they are lucky, and that continuing to take risks will supply them with even more wealth. This is all speculative however, and more investigation is required to understand particip ants’ explicit reasoning, par ticularly with respect to

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86 whether there are differences in situational/dispositional attribution for the Choice/Move tasks. In addition, there are problems associated with assuming participants know more about what to “expect” from the lotteries than they actually do. Participants have no contextual cues to give them a sense of the overall range of possible values in the short term, nor do they have accurate information a bout what they should expect in the longer term. For future investigati ons, this problem might be solv ed by providing participants with information about what they can expect in the short term and long term before they begin to make decisions, or alternatively gi ving them more contextual information about a particular goal, to see if they set an aspiration level as a re sult. This is because in the real world, people often have information about what to expect in the short term and long term (e.g., weekly salary) and have partic ular goals or noncompe nsatory requirements (e.g., cost of living expenses). For the purpose of this investigation, however, correlations were performed between individuals’ risk preferences and thei r current totals to see if participants generally change their risk preferences with respect to an overall increase or decrease in current wealth. We found that several par ticipants for the Choice task had a positive correlation between risk-seeking behavior a nd current total (e.g., as current wealth increased, participants were more riskseeking ) and several participants for the Move task had a negative correlation between risk-seek ing behavior and cu rrent total (e.g., as current wealth increased, participants were more riskaverse ). This suggests the differences between the Choice and Move task s may be meaningful within individuals

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87 and are not just a byproduct of overall curr ent wealth differences between individuals (e.g., people who were “doing well” the whol e time exhibiting risk-seeking behavior while people who were “doing poorly” the whol e time exhibiting risk-averse behavior). The self-report data suggests that several participants were identifying that their own decision strategies include a current wealth component. Both tasks (Move and Choice) had a substantial percen tage of participants respond us ing some reference to their “current state” in thei r explanation of thei r overall strategy. The choice version shows more participants offering up this self-report response, which is actu ally consistent with the results found when comparing the task by we alth level by valence interaction (Figures 23a/b). Specifically, when wealth level is aggreg ated within valences for both Move and Choice, differences between wealth levels s eem to persist most strongly for the Choice task and further differences between the hi gher (Rich) and lower (Poor) wealth levels seem to apply almost entirely when participants are facing positive lotteries. When participants are rich, they ma ke more risky choices, and when participants are poor, they make more risk-averse choices. This sugge sts participants are adopting a criterion similar to the Security-Potential/Aspira tion strategy of Lopes (1987). When poor, participants might be interp reting risks as security thre ats, and hence avoiding them, whereas when rich, participants might be inte rpreting risks as potenti al opportunities, and hence adopting them. However, in the Move version, wealth levels aggregated within valences show no strong consis tent differences. This suggests that for Move, the risk preference pattern across valen ce is consistent at each curren t wealth level, which would

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88 provide a reason for the overall effectiveness of lottery-depende nt strategies, particularly the Risk as Threat decision strategy. The current wealth level analyses as a whole suggest that the Choice and Move tasks present decision information to participants such that current wealth is interpreted to mean two different things. The Move ta sk, which has been hailed by Schneider and colleagues (2005, 2006) as a way to situate participants into improving outcomes in an active, goal-seeking manner, may invoke additi onal contextual information (e.g., situated risk) which influences risk preferences, pot entially to the detriment of other more important contextual factors (e.g., current wea lth as a measure of long-run progress). The passive Choice paradigm, on the other hand, is one in which wealth levels were able to make an impact on risk preferences. The data also suggests that without provo cation or manipulation, the participants in the Choice condition who did end up using the wealth levels as a criterion for risky choice, did so in a manner such that the l ong run outcomes would be skewed to the right, resulting in an increase to the likelihood of ending up in the extreme higher end of the spectrum, a decrease to the likelihood of ending up in the extreme lower end of the spectrum. The problem still exists, however, that lotteries in the negative valence were largely unaffected by the wealth level manipul ation, and hence the usage of cumulative total as a criterion is seemingly limited. When instead of current wealth, participan ts were analyzed with respect to goal trajectory information (which consequently does not use specific fixed cutoff points), a moderately sized portion of par ticipants were consistently pr edicted as using one of these

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89 strategies for both Move and Choice. The usefulness of trajectory strategies in circumstances where the overall range of valu es is not known was discussed in Studies 1 and 2, and it is surprising that participants we re found to be consistent with this type of decision strategy, given the number of assumptions required to assert it.

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90 General Discussion Vantage point dependencies, aspirations, and goal dependencies were investigated in a number of different ways to provide insight into how people make decisions in heavily contextual external world. Study 1 used statistical simulations to investigate how each of these contextual fact ors from the external world might affect long run outcome distributions. Overall, these simulations pr ovided us with a means to find out about idealized strategies for making risky choices We investigated if and how decision strategies that use dynamic and time-dependent criteria such as aspiration levels and goal trajectories are advantageous in the probabilistic sense. We found that aspiration levels and goal trajectories are advantageous because they provide some way of gauging one’s current position relative to the “average” or expected position. The ability to skew the long-run distribut ion (in either direction) is something “static” strategies simply cannot do. In many real-world contexts, using a dynamic strategy can provide an advantage over a static strategy. In fa ct, there is no case in which a dynamic strategy is suboptimal to a static st rategy if used appropr iately, because every dynamic strategy is at its core essentially the concatenation of two or more static strategies using some dynamically-acquired value that is a rough measure of progressthus-far in the decision environment. Study 2 placed the decision strategies fr om Study 1 and from the pilot simulation into several variations on c ontextualized decision environments. We showed that under

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91 different contexts, the effectiv eness of different decision stra tegies may change, but that the core reasons why dynamic decision strate gies can be advantageous over static decision strategies still persist. Study 3 was designed to find out which t ypes of decision strategies people are actually using. As a result of the overwhelming long-run benefits from using dynamic criteria found in studies 1 and 2, we hypot hesized there would be evidence that participants use the provided dynamic criterion (cumulative to tal) in strategic way. We found that while participants appear to have systematic differences in risk preferences with respect to dynamic and time-dependent cr iteria, there is not an overwhelming usage of dynamic decision strategies over other deci sion strategies. Neve rtheless, our results may underestimate the use of dynamic criteria, as it is much more difficult to specify exactly how these criteria might be used a nd how the criteria may be changing over time. Risk Policy strategies are the simplest possible decision strategy, where either risk-averse behavior or risk seeking behavior are predom inant throughout. Differences between simulated risk policy decision strategi es are limited to diffe rences in variability for the outcome distribution; risk-aversion has a narrow distribution and risk-seeking has a spread out distribution. Simple risk po licies were the most commonly used decision strategy for both Move and Choice, particul arly the risk-avers e decision strategy. Lottery-based strategies use lottery-dependent vantage point information, specifically the valence of the potential outcom es (as with the Prospect Theory and Risk as Threat decision strategies) or the resp ective difference between the potential outcomes (as with the Modest Variance decision stra tegy). For simulated lottery-based decision

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92 strategies, there were no differences in th e long run outcome distributions aside from different amounts of spread. In the empirical study, the efficac y of Prospect Theory as a predictor of actual behavior was extremel y poor. The Modest Variance strategy and the Risk as Threat strategy didn’t do much be tter, especially in the Choice task. Aspiration level strategies and goal trajecto ry strategies each use contextual cues from the decision environment that are ti me-dependent and constantly-updating. Simulations of these dynamic decision strategi es showed a specific change in the shape of the outcome distribution via the addition of skew. The skew in the outcome distribution articulates that th ese strategies have the capacity to influence one’s long-run results in ways static lottery-dependent d ecision strategies cannot. Also, the predictive strength of these dynamic decision strategies is somewhat surprising, considering the large number of assumptions required to asse rt them. Also, there are substantial task differences, with the Choice task having a la rger portion of partic ipants who used an aspiration level decision strategy as consistent with the goals addressed in Schneider and colleagues (2007) compared to the Move task. The usage of current wealth as a crit erion was more pronounced in the Choice version of the task possibly because the elem ents of the Move version that appear to require skill outweigh any othe r strategic planning. By as king participants to improve outcomes for each lottery, participants are presum ably led to believe that risky choice can be skillfully mastered, and current wealth isn’ t used as a measure of progress in the task, but as a measure of skillful mastery at th e task. The Choice version, however, does not

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93 include elements which might be interprete d as requiring ‘skill’, and is perhaps more open to allowing current total to be used as an inference about future goals. These task differences have been shown to influence which types of contextual information from the environment are being us ed to formulate decisi on strategies. This experiment studied these task differences in isolation, but the two ta sks described herein have strong implications for the real world. Decisions that are pres ented in the passive form of static choice have markedly different real world implications than decisions that are presented in the active form of initiating progress. So the pros associated with presenting a decision in its passive form is that participants are likely to be more receptive to the external context, with the downside that they are not as situated in the individual ri sky environments. Presenting a decision in its active form has the benefit of providing a rea listic context for each individual decision, even though some external cont ext may not be as salient. It has also been shown that people are in fact using dynamic information when making decisions, and by comparing different decision strategies, we provide evidence for which strategies make bett er predictors. Risk aversi on is the best predictor of decision making, which is consistent with th e most fundamental principles of utility theory from Bernoulli (1738/1954). Aside from that, lots of different decision strategies are being used for both Move and Choice; a lthough some are used more in the Choice task and others are more frequent in the Move task. Trying to es tablish a single best predictive decision strategy was not acco mplished, nor necessarily should it.

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94 Nevertheless, it appears as though the interpretation of “r isk” as something to be avoided is highly predictive among actual part icipant responding, especially when risk corresponds to everyday notions like “danger” or “threat.” Thinking of safety from risk is a commonality in this respect. One coul d consider that decision strategies which exhibit entirely risk-averse behavior throughout or include a component of risk-aversion when one’s current vantage point or future outlook are poor or negative as this type of “avoiding” risk. The risk-aversion policy, th e Risk as Threat strategy, Aspiration Level decision strategies and “Continuation” Goal Traj ectory strategies all ex hibit this notion of avoiding risk. When all those are taken into account, over 75% of participants’ strategies are accounted for by the notion of avoiding risk. Of those decision strategies, several in clude an additional component for taking risks when one’s vantage point or future out look are good. The Risk as Threat strategy, the aspiration level strategies, and the “Conti nuation” goal trajectory strategies all include this notion of seeking risk as well as avoi ding risk. Of the 75% of participants who predominantly avoid risk, about half were also riskier when their va ntage point or future outlook was good. By comparing different deci sion strategies, we have s hown that there are long-run statistical advantages and disa dvantages to using different ty pes of contextual information in the environment (static vs. dynamic). It may seem reasonable to base one’s decision making on the lottery environment (e.g., vale nce); however, strategically speaking, if one’s goal is to give oneself a long-run advantage, focusing on where one is and where one wants to go from there (e.g., dynamics) is the best way to go, as long as the options

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95 your are provided with have the same moneta ry expected value. How one comes to know their “current status” can be achieved either by cues provided about one’s current status or by observing outcomes to see one is doing we ll relatively speaking. In addition, the exhaustive investigation of aspiration level simulations and goal trajectory simulations provides evidence that having a meaningless or uneducated ‘goal’ will not provide one with the same type of long-run advantage as having a meaningful or accurate goal might. We found that as an ‘aspirati on’ becomes more appropriate to what can be best expected and as a trajectory includes a larger margin of previous experiences, there is better control over the long-run probability distributi on, specifically the amount and direction of skew in the long-run outcome distribution. Ho wever, if one’s implicit goals are to tend towards a specific risk policy (e.g., remain pr edominantly risk-averse), aspiration level decision strategies can use an adjusted cutoff point to reflect a general tendency towards a risk policy (e.g., increasing the cutoff point for one’s aspiration level strategy tends towards risk-aversion), and goal trajectory de cision strategies can require a stronger or weaker majority to reflect a general te ndency towards a risk policy (e.g., requiring a stronger majority of “better” outcomes before one is willi ng to take risks tends towards risk-aversion). More research is required to determine the driving force between the differences in the Move and Choice tasks. Though our study provided additional insight into differences with respect to Move and Choi ce, we have yet to provide a systematic explanation of why those differences exist. In addition, our st udy was limited in its investigation of the potential influence on decision-making by dynamic criteria because

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96 of the large number of assumptions regardi ng cutoff points and implicit goals. Future studies will do well to develop additional pote ntial methods for investigating the impact of current wealth levels on risky choice usi ng a more systematic manipulation. Also, the empirical results suggest that the dynamic decision making perspective might provide some additional insight in understanding risk preferences for choices that require actual skill or ability that can improve over time by using the Choice v. Move task manipulation. Most importantly, since the predictability of actual particip ant preferences is spread out across multiple decision strategies and different levels of decision complexity, attempting to evaluate all decision making with respect to risk thr ough a single contextual cue or piece of the decision environment is sh ortsighted. More research is required to determine what types of cues are commonly us ed by individuals, but not necessarily for the purpose of finding a single unifying decisi on strategy that pred icts all behavior.

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97 References Bernoulli, D. (1738/1954). Exposition of a new theory on the measurement of risk. Econometrica, 22 23. Brehmer, B. (1992). Dynamic decision making: human control of complex systems. Acta psychologica, 81(3), 211-41. Busemeyer, J. R. (2001). Dynamic decision making. In International encyclopedia of the social and behavioral sciences: Met hodology, mathematics, and computer science Amsterdam: Pergamon. Busemeyer, J. R. (2004). Dynamic decision making. In N. J. Smelser, & P. B. Baltes (Eds.), International encyclopedia of th e social and behavioral sciences (pp. 3903-3908). Oxford, UK: Elsevier Science Limited. Busemeyer, J. R., & Townsend, J. T. (1993). Decision field theory: A dynamic-cognitive approach to decision making in an uncertain environment. Psychological review, 100 (3), 432. Damasio, A. (1995). Descartes' error: Emoti on reason and the human brain New York: Putnam. Higgins, E. T. (1998). Promotion and prevention: Regulatory focus as a motivational principle New York: Academic Press. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47 (2), 263. Lopes, L.L. (1981). Decision Making in the Short Run. Journal of Experimental Psychology: Human Learning and Memory, 7(5), 377-85. Lopes, L. L. (1987). Between hope and fear: The psychology of risk. Advances in experimental social psychology, 20 (3), 255. Lopes, L. L. (1996). When Time Is of the Essence: Averaging, Aspi ration, and the Short Run. Organizational Behavior and Human Decision Processes, 65(3), 179-89.

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98 McKenzie, C. R. M., & Nelson, J. D. (2003). What a speaker’s choice of frame reveals: Reference points, frame selection, and framing effects. Psychonomic bulletin & review, 10 (3), 596. Omodei, M.M., Wearing, A.J. (1995). The Fi re Chief microworld generating program: An illustration of computer-simulated micr oworlds as an experimental paradigm for studying complex decision-making behavior. Behavior Research Methods, Instruments & Computers 27(3), 303-316. Rapoport, A. (1975). Research Paradigms for Studying Dynamic Decision Behavior. Utility, Probability, and Human Decisi on Making. Ed: Dirk Wendt & Charles Vlek. Samuelson, Paul A. (1963). Risk and Uncerta inty: The Fallacy of the Law Of Large Numbers. Scientia 98, 108-13. Savage, L. J. (1954). The Foundations of Statistics. New York, NY: John Weiley & Sons. Schneider, S. L. (2002). Reference points and risk preferences USA. Schneider, S. L., & Barnes, M. (2003). Emerging Perspectives on Judgment and Decision Research. Ed: Sandra Schneider & James Shanteau. Cambridge University Press. Schneider, S. L., & Hudspeth, C. (2004). Why reference dependence matters: Goal trajectories and risk exposure. Presented at The Annual Meeting of the Society for Judgment and Decision Making. Schneider, S. L., Hudspeth, C. (2005). On the Dynamics of Risky Decisions: Seeking Goals and Making Choices. Presented at The Annual Meeting of the Society for Judgment and Decision Making. Schneider, S. L., Hudspeth, C. & Decker, N.K. (2006). Good Reasons for Framing Effects: Goal Trajectories and Experience. Presented at The Annual Meeting of the Society for Judgment and Decision Making. Schneider, S.L., Decker, N.K., & Hudspeth, C. (2007). A Fundamental Advantage of Dynamic Experience-based Strategies for Risky Choice. Presented at The Annual Meeting of the Society for Judgment and Decision Making.

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99 Schneider, S. L., & Lopes, L. L. (1986). Re flection in preferences under risk: Who and when may suggest why. Journal of experimental psychology. human perception and performance, 12 (4), 535. Tversky, A., & Kahneman, D. (1981). The fram ing of decisions and the psychology of choice. Science, 211 (4481), 453.

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100 Appendices

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101 Appendix A: Quiz Sample ID Number:____________________ _________________ $50 $50 $50 $50 $50 $0 $50 $0 $50 $0 $50 $50 $0 $50 $10 0 $50 $0 $50 $10 0 $150 Do you have a better chan ce of drawing a ticket wort h $0 or $100 ? ____________ If you played this lottery over and over, on average, what ticket value would you draw only 3 out of every 20 times? _____ _____________________ If you were given the chance to randomly draw a ticket from this lottery, which ticket value would you be least likely to draw? _____ ___________________ In this lottery, do you have a better ch ance of drawing a winning ticket or a losing ticket? __ ____________ ___________

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102 -$100 $100 -$100 $100 -$100 $100 -$100 -$50 $50 $100 -$100 -$50 $0 $50 $100 -$100 -$50 $0 $50 $100 In this lottery, wh at value is the least likely outcome? ___ ___________________ In a single play of this lo ttery, do you have a better chan ce of drawing a ticket with the value of $100 or -$50? ____ _____________ What is the best possible outcome that you could expect in playing this lottery one time? __________________ How many tickets would result in yo ur losing money? __________________

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103 Appendix B: Social Status Example