Journal of Experimental Psychology: General A Tri-Reference Point Theory of Decision Making Under Risk X T Wang and Joseph G Johnson Online First Publication, March 5, 2012 doi: 10.1037/a0027415 CITATION Wang, X T., & Johnson, J G (2012, March 5) A Tri-Reference Point Theory of Decision Making Under Risk Journal of Experimental Psychology: General Advance online publication doi: 10.1037/a0027415 Journal of Experimental Psychology: General 2012, Vol ●●, No ●, 000 – 000 © 2012 American Psychological Association 0096-3445/12/$12.00 DOI: 10.1037/a0027415 A Tri-Reference Point Theory of Decision Making Under Risk X T Wang Joseph G Johnson University of South Dakota Miami University The tri-reference point (TRP) theory takes into account minimum requirements (MR), the status quo (SQ), and goals (G) in decision making under risk The reference points demarcate risky outcomes and risk perception into functional regions: success (expected value of x Ն G), gain (SQ Ͻ ϫ Ͻ G), loss (MR Յ x Ͻ SQ), and failure (x Ͻ MR) The psychological impact of achieving or failing to achieve these reference points is rank ordered as MR Ͼ G Ͼ SQ We present TRP assumptions and value functions and a mathematical formalization of the theory We conducted empirical tests of crucial TRP predictions using both explicit and implicit reference points We show that decision makers consider both G and MR and give greater weight to MR than G, indicating failure aversion (i.e., the disutility of a failure is greater than the utility of a success in the same task) in addition to loss aversion (i.e., the disutility of a loss is greater than the utility of the same amount of gain) Captured by a double-S shaped value function with inflection points, risk preferences switched between risk seeking and risk aversion when the distribution of a gamble straddled a different reference point The existence of MR (not G) significantly shifted choice preference toward risk aversion even when the outcome distribution of a gamble was well above the MR Single reference point based models such as prospect theory cannot consistently account for these findings The TRP theory provides simple guidelines for evaluating risky choices for individuals and organizational management Keywords: expected utility, value functions, risky choice, reference points, risk preference Supplemental materials: http://dx.doi.org/10.1037/a0027415.supp ments (MR) Second, we introduce a tri-reference point (TRP) theory, which takes into consideration these three reference points and their interaction with outcome distributions in determining risk perception and risky choice (see Wang, 2008, for an earlier brief introduction of this framework) We also developed a basic mathematical formalization of the TRP value function (see the supplemental material) Third, we report empirical tests of specific predictions derived from key assumptions of the TRP theory in comparison to predictions of other decision theories, particularly prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992) In closing, we briefly analyze the potential significance of the TRP theory in risk analysis and risk management and limitations that call for future research A life without adventure is likely to be unsatisfying, but a life in which adventure is allowed to take whatever form it will is sure to be short —Bertrand Russell Risky decision making as an essential and integral part of individual and organizational behaviors has been a central topic for psychologists, economists, and other scholars At the heart of most decision theories is the formulation of the subjective value function of the decision maker We propose in this article a subjective value function that is demarcated by three decision reference points We integrate research findings from the behavioral decision-making literature on status quo (SQ)-related choices, the management science literature on goal (G) setting, and risksensitive foraging theory on survival-related minimum require- Reference Points and Payoff Distributions in Risky Choice X T Wang, Psychology Department, University of South Dakota; Joseph G Johnson, Department of Psychology, Miami University The studies reported in this article were in part supported by National Science Foundation Grant SBR-9876527 to X T Wang We thank Peng Wang and Yi Shi for their assistance in conducting Experiments 1, 2, and This article was prepared in part while X T Wang and Joseph G Johnson were doing research at the Max Planck Institute for Human Development in Berlin, Germany; while X T Wang was a visiting professor at the Guanghua School of Management, Peking University; and while Joseph G Johnson was supported by National Institute of Mental Health National Research Service Award MH14257 to the University of Illinois and National Science Foundation Grant 0851990 Correspondence concerning this article should be addressed to X T Wang, Psychology Department, University of South Dakota, 414 East Clark Street, Vermillion, SD 57069 E-mail: xtwang@usd.edu Need for Considering Payoff Distributions Most theories of risky decision making have at least a nominal relation to expected utility theory (Bernoulli, 1738/1954; von Neumann & Morgenstern, 1944) At the heart of expected utility (EU) theory and many contemporary models of decision making has been the idea of utility maximization The classic work by von Neumann and Morgenstern (1944) showed that the idea of EU maximization is derivable from a small set of axioms of behavioral consistencies in risky choice behavior These axioms appeared so reasonable and parsimonious that they have been used widely to define rational decision making However, one common limitation of these normative models of decision making is their lack of WANG AND JOHNSON consideration of the variance and distribution of expected outcomes The use of a single value of expected utility for each choice option is done at the cost of valuable information about payoff distributions in each of the choice options As a result of this focus on EU, each choice option is represented by a single value: n EU ϭ pi ⅐ u͑xi͒ iϭ1 where pi is the probability of outcome i, xi is its objective value, u is a utility function, and n is the number of possible outcomes Note that mathematically, this expression is an expectation; the variance in this expectation is not incorporated, despite the role it may play Outcome distributions and variance in payoffs are important for decision making under risk In the field of behavioral finance, following the pioneering work by Markowitz (1952, 1959), risk is primarily measured by variance in expected monetary returns Similarly, in the management literature, risk is also commonly conceived as reflecting variation in possible outcomes and their subjective values (e.g., March, 1988; March & Shapira, 1992).1 Despite this ubiquity, few models of behavioral decision making explicitly incorporate variance (however, see Busemeyer & Townsend, 1993; Pollatsek & Tversky, 1970; and Wang, 2002, 2008, among others, for notable exceptions) Need for Considering Multiple Reference Points In the context of judgment and decision making, we define a reference point as any value subjectively selected and used by an agent for the purposes of comparison, classification, and evaluation of possible outcomes associated with a decision EU theory was first formed on the basis of final assets (Bernoulli, 1738/1954) without considering any reference point Later developments started to focus on changes from SQ (e.g., Markowitz, 1952; Edwards, 1954; Helson, 1964; Kahneman & Tversky, 1979), resulting in value functions with a kink at the reference level (e.g., Tversky & Kahneman, 1992) and EU models that measure targetoriented utility (e.g., Bordley & Kirkwood, 2004; Castagnoli & Calzi, 1996; Fishburn, 1977) More recently, an increasing number of studies suggest that people make decisions using multiple reference points and show some advantages of using a theory of multiple reference points to interpret and predict choice behavior The emphasis of the TRP theory on additional reference points G and MR has been inspired by many scholarly works The idea that people use multiple reference points has been suggested by Parducci (1965) in the context of range- and frequency-based judgment; by Neale and Bazerman (1991) in the context of negotiation; by Yates and Stone (1992) in categorizing various reference points; by Mellers, Schwartz, Ho, and Ritov (1997) in terms of comparisons between an obtained outcome and various real and counterfactual outcomes; by Ordo´n˜ez, Connolly, and Coughlan (2000) regarding assessment of fairness and satisfaction; and by Higgins (1997) in his self-regulation theory, where he makes a distinction between a promotion regulatory focus and a prevention regulatory focus Multiple reference point models have also been developed to account for foraging choices of bees and birds (e.g., Hurly, 2003) The existence of G, SQ, and MR is evident in situations from a student who received a B in his first organic chemistry exam (his SQ) and aims to get an A in the second exam and would definitely drop the class if he gets a C on the exam; to an NCAA basketball coach with an MR of being selected for the tournament and a G of reaching the Final Four; to a job candidate who evaluates salary offers using her desired objective and bottom line for salary in reference to her current salary; and to a CEO of a large corporation who has a G to exceed the profit expectation for the next quarter, a SQ of current profits, and an MR of maintaining the necessary cash flow for the quarterly operations Because no theory can formulate all possible reference points, it becomes essential to capture key features of reference points A theory that relies on many reference points to reach its descriptive accuracy would at the same time lose its generality and normative strength There has been no compelling argument in the literature about the ideal number of reference points that should be included in a model of risky choice To strive for a balance between accuracy and generality, we propose the following four criteria for reference point selection These reference points should (a) be theoretically and practically present in many choice tasks; (b) affect choice preference, as supported by abundant empirical evidence; and (c) reflect specific standards rather than general desires, ambitions, fears, or wishes Also, (d) decision outcomes that cross over these reference points should have a greater psychological impact than the same amount of change between two reference points Works by Kahneman and Tversky (1979; Kahneman, 2002) have demonstrated that the carrier of subjective value is not the total wealth but changes from the SQ We take this recognition further in conjunction with the studies of G and MR We propose that the changes in value and the value of changes are also assessed using a G reference point for positive changes and an MR reference point for negative changes To make adaptive decisions at risk and under task constraints, people strive to reach a goal and avoid falling below a minimum requirement at the same time Moreover, to measure the performance against G and MR, an additional reference point, the SQ, is required Thus, a parsimonious model of risky choice would include G and MR in addition to the SQ There is ample empirical evidence for the effects of each of the three reference points on risk preference, although they are typically studied independently First, behavioral studies of human decision making highlight the importance of the current state, or SQ, especially in the context of prospect theory (see Kahneman & There are, however, several issues regarding the use of variance as an index of risk worth mentioning The same amount of variance in payoffs may belong to different distributions and thus different degrees of risk Psychologists have shown that the use of variance as a measure of risk is adequate only if gambles have normal distributions (see Sarin & Weber, 1993, for a review) Some other researchers define risk as probabilistic deviation from target returns (e.g., Fishburn, 1977; Mao, 1970) Similarly, managers often perceive risk in terms of downside outcomes only (e.g., March & Shapira, 1987) Meta-analyses conducted by Shafir (2000) and Weber, Shafir, and Blais (2004) have shown that for both humans and other animals, the coefficient of variation in outcomes, a relative measure of risk per unit of return, predicts choices across a broad range of situations better than absolute measures of risk (e.g., outcome variance) TRI-REFERENCE POINT THEORY Tversky, 1979, 2000) Second, many studies in management science focus on how goal settings (G) affect risky choice and task performance (see Heath, Larrick, & Wu, 1999; Locke, 2002; Payne, Laughhunn, & Crum, 1980, 1981) Goals can also serve as the inflection point or reference standard for satisfaction versus dissatisfaction (Mento, Locke, & Klein, 1992) The importance of MR in risky decision making is also well illustrated in the extant literature A well-known example is Tversky’s (1972) elimination-by-aspects model, which assumes that people make choices by gradually eliminating less attractive alternatives on the basis of an MR for each of the attributes of the alternatives In the negotiation literature, MRs are frequently used to guide negotiation, such as Raiffa’s (1982) analysis, which provides a lower bound (the MR) for a negotiated agreement Animal behavior also appreciates the MR, in that birds and bees avoid high-variance foraging options when they have an energy surplus to minimize the chance of crossing the energy MR for survival, just as they seek high-variance foraging options when they have an energy deficit to maximize their chance of exceeding their energy MR (Kacelnik & Bateson, 1997; Real & Caraco, 1986; Stephens & Krebs, 1986) In real life, goals and bottom lines can be either determined endogenously as in the case of salary negotiation or set exogenously as task requirements by others However, in either case, goals and bottom lines are subjective reference points used in the process of decision making Even for the Gs and MRs imposed by others, they have to be accepted and transformed by the decision maker into subjective reference points By the same token, the three reference points for any individual are sufficient for doing self-performance evaluation even when the evaluation standards involve more than three categories (e.g., grade categories for students or the ranks used for promotion) In sum, the three reference points can be elicited in at least four different ways First, task situations (e.g., a student taking an exam) naturally determine the three reference points Second, the three reference points can be imposed as task requirements by others Third, the reference points can be distinct as a result of social comparisons in different social groups Fourth, the three reference points can be learned and adjusted through personal and organizational experience two reference points However, options below the two reference points did not result in predominantly risk-taking behavior, as would be expected if the G was the only reference point for these managers When options were between the two reference points, managers exhibited a mixture of risk-taking and risk-avoiding choices, suggesting that the managers considered both the SQ and the G in performing the task In a recent study, Xie, Xie, Ren, and Yu (2009) examined the effects of G and MR settings on risky choice in an experiment of dynamic stock investment involving multiple trials with feedback In a computer-simulated investment task, participants had the opportunity to invest in one of five preselected stocks for each of four trials (quarters in a fiscal year) Participants could invest all or part of their money each time, and whatever was not invested would be placed in a savings account with a fixed return rate The five stocks had the same expected return but different variances The G and MR were explicitly stated for each participant, as were several investment-related indices including SQ, SQ–G distance, and SQ–MR distance The results of a regression analysis of the data revealed separate main effects of MR and G in investment choices, supporting the prediction that decision makers consider multiple reference points The participants were more likely to choose stocks of higher variance when they were below their MR or G Koop and Johnson (2012) conducted a study using gambles specially designed to test the assumption that people use multiple reference points They presented participants with a series of pairwise choices among two-outcome gambles with actual monetary consequences The three reference points were established simultaneously by endowing participants with an initial amount that they could gamble (SQ), presenting participants with the possibility of earning bonus entries into a raffle for one of many gift certificates (G), and retaining the possibility of failing to gain entry (MR) The study showed that individuals can use the MR, SQ, and G within a single risky decision task Participants prioritized the attainment (or maintenance) of these three reference points, even when doing so produced decisions that ran counter to the predictions of other candidate theories such as simple risk aversion, expected value, or prospect theory Empirical Evidence of Using Multiple Reference Points When Making Risky Choices As pointed out by Ordo´n˜ez et al (2000), decision makers are often confronted simultaneously with more than one referent; however, little is known about the simultaneous impact of such multiple reference points In an empirical study, Ordo´n˜ez et al demonstrated a simultaneous impact of multiple reference points on ratings of salary satisfaction and fairness The reference points in this study were salary values of compatible others A more direct test of simultaneous effects of G and MR on risky choice comes from a study in which Sullivan and Kida (1995) investigated investment decisions by imposing two reference points for performance evaluation: a level of current return (an SQ) as a result of investments from the preceding year and a target level of performance (a G) that the company had imposed for the current year In general, the participants (corporate managers) were largely risk averse when investment options were above the TRP Theory: An Introduction Assumptions of TRP-Dependent Evaluation Assumption Guided by the four selection criteria discussed above, three distinct reference points are considered People strive to obtain a G while simultaneously avoid falling below an MR in comparison to their SQ.2 Furthermore, because Gs and MRs are measured relative to the SQ, we expect decision makers to use the SQ as a zero point when evaluating distributions of choice options This assumption does not exclude the possibility that one may take a sequential strategy to attend to MR first and then only G if distributions of choice outcomes are all above the MR That is, if all options are above MR, only G is relevant 4 WANG AND JOHNSON Assumption The magnitude of the three reference points follows the order3 of MR Ͻ SQ Ͻ G and effectively divides the value scale of choice outcomes into four functional regions: success (expected value, x, is above G, thus x Ն G), gain (improvement from the SQ, thus SQ Ͻ x Ͻ G), loss (deterioration from the SQ, thus MR Յ x Ͻ SQ), and failure (falling below the MR, thus x Ͻ MR).4 Note that in terms of the TRP outcome regions, “mere” gain and “mere” loss are no longer the same as defined in a single reference point (SQ) theory, where a gain may or may not reach a G and a loss may or may not fall below an MR Assumption Consistent with a basic assumption of other reference-point-dependent decision theories (e.g., Kahneman & Tversky, 1979; Tversky & Kahneman, 1992; March & Shapira, 1992), psychological value is reference-point dependent, such that a small change (increase or decrease) in objective value is subjectively greater when it passes across a reference point, en route to a different region, than when it remains in the same region Assumption The four regions can be further classified in a hierarchy that conveys their relative importance Specifically, avoiding failure is most important, followed by achieving success, followed by the SQ This assumption is consistent with a longstanding security-first principle in financial investment and business management (e.g., Roy, 1952) as well as a Darwinian order of natural selection from survival of the fittest to reproduction of the survived This reference point priority order implies failure aversion (i.e., the disutility of a failure is greater than the utility of a success in the same task) in addition to the notion of loss aversion (i.e., the disutility of a loss is greater than the utility of the same amount of gain; e.g., Kahneman & Tversky, 1979) One way to formalize Assumptions 1– is a double-S value function separated by the SQ The S above the SQ consists of a concave segment in the success region and a relatively convex segment in the gain region, whereas the S below the SQ consists of a relatively concave segment in the loss region in comparison to a more convex segment in the failure region The S below the SQ is expected to be steeper than the S above the SQ, due to the assumption that the psychological impact of the reference points is in an order of MR Ͼ G Ͼ SQ The supplemental material shows one possible way such a double-S-shaped function could be formed Assumption The settings of reference points are mainly determined by situational and social factors in task environments (e.g., economical, ecological, social, relational, organizational, and cultural variables) and fine-tuned by dispositional factors (e.g., risk attitude, regulatory focus, subjective life expectancy, and selfefficacy) and communicational factors (e.g., decision frames and anchors) To justify and demonstrate the above assumptions, first consider the case illustrated in Figure 1A, where, along the linear value dimension x, the distance between adjacent points is identical Thus, the difference in linear utility between Points A and B is the same as that between Points B and C or between Points C and D However, once G, SQ, and MR are specified, as shown in Figure 1B, the equal distances would entail different subjective values, according to Assumption This is because the segments of equal distance on the value dimension now straddle different reference points and different functional regions Finally, in Figure 1C, we can contrast the effect of crossing reference points to remaining in a single region In this case, increasing value from A to B (or from Figure Effects of reference points on outcome evaluation Each neighboring pair of points are equidistant, producing an equal change in psychological value between points in A When reference points—minimum requirements (MR), status quo (SQ), and goals (G)—are introduced in B and C, the same distance results in differential perception of change in psychological distance C to D) produces a larger subjective value difference than an increase from B to C In Figure 1B, the importance ordering established in Assumption suggests the difference between A and B should yield the highest subjective value increment because it crosses the MR, which differentiates between failure and nonfailure, or between death and survival Moving from C to D (i.e., across the G) should generate the next highest increment because it involves the difference between achieving success and lack thereof The difference in subjective value between B and C is expected to be lowest because the change can be viewed as fluctuations around the SQ Note that failure aversion is a special type of loss aversion, where a loss (movement from B to A) has a greater impact on psychological well-being than does the same amount of gain (movement from C to D) Applying TRP Theory to Decisions Between Risky Options In this section, we offer some examples of how TRP theory can be applied to choices between risky prospects We use these examples to extract some implications of the theory that can be empirically tested Assume that the four points shown in Figure 1B are used as expected values in deriving four pairs of independent gambles (see Figure 2) In particular, assume that for each of the four expected values (e.g., Point A in Figure 1), one high-variance There are other possible orders of reference points besides the most common order of MR Ͻ SQ Ͻ G For example, one possible order is MR Ͻ G Ͻ SQ after successfully completing a task and before goal updating, whereas another possible order is SQ Ͻ MR Ͻ G after failing a task and before starting over again These two orders of reference points are more transient but still subject to the choice rules of the TRP theory described in a following section (i.e., risk averse when SQ Ͼ G, and risk seeking when SQ Ͻ MR) Note that these four regions not explicitly include the value x ϭ SQ This could technically be defined as a fifth region, such as mere preservation or stagnation, although we refrain from doing so; this does not affect subsequent interpretations or analyses TRI-REFERENCE POINT THEORY Figure Pairs of choice options with identical mean but different variances High-variance options (A, B, C, D) are paired with low-variance options (AЈ, BЈ, CЈ, DЈ, respectively) of equal expected value MR ϭ minimum requirements; SQ ϭ status quo; G ϭ goals gamble (A in Figure 2) and one low-variance gamble (AЈ) are created, producing four pairs of gambles Finally, assume that the MR, SQ, and G for a particular decision maker are those given in Figure When encountering a choice between the risky option A and the safer option AЈ, we predict a strong risk (variance)-seeking preference for A because it offers the only chance of staying above the MR In contrast, when making a choice between options B and BЈ, we predict a strong risk-averse preference for the safe option BЈ to avoid a disastrous potential outcome of B Although the small upper tail of B offers a potential gain that BЈ does not, the possibility of failure outweighs this potential gain (Assumption 4) Similarly, a weak risk-seeking preference for C over CЈ is expected because the small upper tail of the distribution above G would be valued higher than the small lower tail of the distribution below the SQ A risk-averse preference for DЈ is expected because the safer option DЈ is well above the goal, whereas for the risky option D, the cumulative probability of getting better than DЈ would be offset by the risk of falling below the goal These predictions contrast with those of prospect theory, which predicts risk seeking for losses (choice of A and B in Figure 2) and risk aversion for gains (choice of CЈ and DЈ in Figure 2), particularly when a loss or gain is almost certain (with a medium to high probability) In general, for gambles with equal expected value and normally distributed outcomes, the shorthand rule we call the mean-variance principle is to be risk or variance seeking when the expected (mean) value of choice outcomes is below MR (or G) but be risk or variance averse when the expected value is above MR (or G) This principle is straightforward for gambles with their values distributed across either MR or G but not both (see Figure 2) In the latter case, a TRP-dependent trade-off has to be made between expected value of success and expected value of failure We tested these situations in Experiments and In addition to the qualitative predictions made using the TRPdependent assumptions above, we also developed a mathematical model to formalize the TRP theory This work is provided in the supplemental material available online TRP Theory and Other Multiple Reference Point Theories It is important to note how TRP theory relates to existing theories in a similar vein Lopes (1987) advocated the view that instead of maximizing EU, decision makers strive to maximize the probability of meeting a goal or aspiration level Her two-factor theory (also called the SP/A model) integrates a dispositional tendency to seek either security (S) or potential (P) with situationally driven aspiration (AP) levels (see also Diecidue & van de Ven, 2008, for a discontinuous utility model of success and failure defined by an aspiration level) TRP assumptions are consistent with the general arguments of the two-factor theory However, the two theories have different emphases The two-factor theory emphasizes the effects of dispositional motivations on risk preference but does not explicitly involve the notion of multiple reference points For instance, although a security-minded individual may pay more attention to low outcomes, he or she may not distinguish between low outcomes that are above versus below an MR threshold For a potential-minded individual, the two-factor model predicts risk seeking for losses with a higher aspiration level However, a TRP prediction would be different if the outcome of a gamble option may fall below the MR setting, as illustrated by the predicted risk-averse choice between B and BЈ in Figure In their managerial decision-making models, March and Shapira (March, 1988; March & Shapira, 1987, 1992) assumed that the risk preference of a decision maker is affected by differential attention to two reference points, one for success and one for survival The TRP theory and the variable risk preferences theory (March, 1988; March & Shapira, 1987, 1992) share several fundamental assumptions in common, but they are yet distinct in a number of important aspects The most obvious common ground between these two models is that both survival (minimum requirement) and success (goal) are critical parameters for determining risk preferences of decision agents Moreover, the variance of outcomes, which is regarded as a measure of risk, is also a central assumption for both models, which differentiate them from other models that emphasize expected value as a major determinant of risky choice The most significant difference between the models lies in the assumed mechanism that generates risky choices Although the TRP model is value-function based, the March and Shapira (1987, 1992) model is value-function free The TRP evaluation of choice options depends on the specific properties of its utility function to assign values to alternative options, which, in turn, serve as input into any choice strategy In contrast, in the variable risk preferences model, a random-walk process is used to determine the degree of risk or outcome variability that is acceptable on the basis of the relative position of current wealth relative to the survival point or aspiration level for success Therefore, the attention of a decision agent (as a free parameter of the model) shifts between the success and survival foci As a result, the decision agent actively selects an option with acceptable risk to fulfill his aspiration or try to survive The outcomes of such choices accumulate and change the current wealth of the agent over time Thus, the model is capable of simulating market dynamics by replacing failed agents with new startups However, in the TRP context, instead of creating choice options, the decision makers choose among available choice options by measuring outcome distributions against reference points Empirical Tests of TRP Theory Predictions In this section, we report five experiments designed to examine specific TRP predictions in comparison to those derived from WANG AND JOHNSON expected utility theory and prospect theory In the first three experiments, we examine the effects of the three reference points without imposing any external requirements In Experiment 4, we evaluated the double-S-shaped value function derived from our mathematical model by adding a constant to a sure option three times so that the expected value of the sure option and its gamble equivalent moves from below MR to between MR and SQ, to between SQ and G, to above G In Experiment 5, we further tested the three inflection points suggested by the TRP value function and examined the overlap between the TRP and prospect theory predictions In particular, we compared risk preferences in conditions of mere gains and mere losses with absence of MR and G with conditions in which the same gains and losses were placed under the constraints of MR and G These empirical tests differ from previous studies on the effects of multiple reference points (e.g., Koop & Johnson, 2012) in several important ways First, the current experiments are designed to test key predictions derived from TRP theory above and beyond the general assumption that people use multiple reference points Second, we examined, in the first three experiments, the effects of implicit, self-determined reference points on risk preference without imposing explicit reference points Third and in particular, we examined specific TRP predictions in contrasting situations where the mean expected value is either above or below a reference point and where the distribution of payoffs is either between MR and G or spreading over both of the points Experiment 1: Salary Choice as a Function of Payment Distributions Across Natural Reference Points in Shanghai The participants were asked to indicate their minimally required first-job salary for living in Shanghai (MR) as well as their desired salary for the first job (G) The roundup average was 2,400 renminbi (RMB; approximately $350) per month for MR and 6,400 RMB (over $900) per month for G.5 A second independent survey of 117 senior students from the same student population identified the average expected (most likely) first salary in Shanghai as being 3,500 RMB (a little over $500); this value was used as the SQ We then designed choice problems as fixed and variable salary programs with payments across these reference points One pair of options had a mean expected salary below the estimated MR with a fixed salary of 1,700 RMB and a variable salary option ranging from 1,000 to 2,400 RMB with equal probabilities of 0.5 Another pair of options had a mean expected salary between the MR and SQ with a fixed salary of 3,100 RMB and a variable salary option ranging from1,850 (below MR) to 4,350 RMB (above SQ); these choice options are stylized by the pairs {AЈ, A} and {BЈ, B}, respectively, in Figure 1B A total of 56 (33 women) student volunteers with an average age of 21.4 years were recruited from the same student population in Shanghai Participants were randomly assigned to one of the two salary conditions (mean salary above or below MR) Each participant had to make only one binary choice between two salary options (fixed or variable) described as job offers available to them after graduation They were asked to assume that other aspects of the job offers (e.g., job location, organizational culture) were comparable Only after the participants made their choices did we asked them to provide their subjective SQ, G, and MR to check if their stated reference points were consistent with our a priori estimates used in guiding the design of the choice options Hypothesis and Predictions As illustrated in Figure 2, the TRP theory predicts that people will be risk seeking when choosing between a low-variance (or a fixed choice) option that is below MR and a risky gamble with a higher outcome variance ranging from below MR to above MR (e.g., A Ͼ AԺ in Figure 2) In addition, the TRP theory predicts that people will be risk averse when choosing between a relatively safe gamble that will leave them between MR and SQ and a relatively risky gamble that may take them below MR or above SQ (i.e., BԺ Ͼ B in Figure 2) Note that these specific predictions are in contrast to a general risk-seeking preference that prospect theory would predict in conditions where expected outcomes of choice options are all below the SQ In particular, we predict that a variable salary program would be preferred when the mean expected value of salary options is below the MR for survival, whereas a fixed option would be preferred when the mean is between the MR and the expected SQ of the first job offer Method Following the idea that the setting of a reference point is likely to be variable around a fixed or mean value, we started to estimate the mean values of the three reference points and then designed the choice options around the means This allowed us to make predictions about overall trends of risk preference of the experimental participants without inducing the reference points beforehand We first surveyed a group of senior students (20 men and 20 women) Results and Discussion Table summarizes the choice data of Experiment When the mean was below the MR, 72% of participants chose the variable option that had a chance to reach the estimated MR However, when the mean was between the estimated MR and SQ, the majority (67%) of the participants chose the fixed option that did not contain a chance of falling below the MR; the difference between these conditions was significant, 2(1, N ϭ 56) ϭ 8.59, p Ͻ 01 At the end of the experiment, we asked the participants for their MR (2,510 RMB), SQ (3,990 RMB), and G (6,400 RMB) for monthly salary income in Shanghai after graduation These values are comparable to the estimated MR (2,400 RMB), SQ (3,500 RMB), and G (6,400 RMB) used for designing the salary options In contrast, prospect theory predicts a different choice pattern We calculated, on the basis of the cumulative prospect theory (CPT) parameters estimated by Tversky and Kahneman (1992), the CPT values for each option If zero is assumed as the reference point, the CPT values predict risk aversion in favor of the fixed options in both conditions If the actual value of the average SQ (3,990 RMB) provided by the participants is used as the reference In this choice situation, each participant was given different job offers, and thus options such as “no job” or “living with parents” were excluded from consideration TRI-REFERENCE POINT THEORY Table Choice Between Options Under Two Levels of Expected Mean Salary Fixed Variable Mean Salary n % Salary n % N Below MR Above MR 1,700 RMB 3,100 RMB 18 28 67 1,000 or 2,400 RMB 1,850 or 4,350 RMB 21 72 33 29 27 Note MR ϭ minimum requirements; RMB ϭ renminbi Cell entries show the number of participants selecting fixed option or its gamble equivalent under the below MR and above MR conditions point, we obtained the net gain or loss for each choice outcome (e.g., the net loss for the fixed option of 1,700 RMB would be 3,990 Ϫ 1,700 ϭ 2,290 RMB) If the actual MR (2,510 RMB) is used as the reference point, the CPT values predict risk aversion under both conditions These CPT values predict a preference for the variable option under both mean below MR and mean above MR conditions.6 Thus, in both situations, the CPT predicted risk preference patterns are inconsistent with the observed pattern that the participants would prefer the between option to the spreadover option to avoid the risk of (x Յ MR), even if it meant forfeiting a chance to reach over the G Second, we predicted that the participants would prefer the fixed option that has passed the SQ to the between option that varied around the SQ but would not reach the G Experiment 2: Choices Involving MR and G Trade-Offs The fixed-pay option (4,350 RMB) was between the G and MR and above the SQ The extreme variable option (the spread-over option) was either 1,800 or 6,900 RMB with equal probabilities The intermediate variable option (the between option) was either 2,800 or 5,900 RMB with equal probabilities A total of 81 (57 women) student volunteers were recruited from a large university in Shanghai; the average age of the participants was 22.7 years The scenario was the same as Experiment Binary choices were created by producing each of the three pairwise combinations of salary options Each participant was presented with one pair of options The purpose of this experiment is twofold First, we investigated the situation where the outcome distributions of variable-salary options spread across not only MR but also G (the “spread-over” option) Second, we examined the effects of the SQ in a choice situation where the fixed option is above SQ but below G and a variable option ranging between MR and G (the “between” option) As in Experiment 1, the three reference points were estimated from an independent sample rather than externally imposed by the choice task (see Figure for an illustration of the salary options used in Experiment 2) To be more precise, the estimated reference points are then replaced by the actual reference points obtained from the participants after the choices were made Hypothesis and Predictions When a trade-off between G and MR is involved, the TRP theory assumes an MR priority (see Assumption 4) We predicted Figure Salary options used in Experiment with a wider cylinder representing the mean of a salary option and a narrower cylinder representing the variance of the option MR ϭ minimum requirements; SQ ϭ status quo; G ϭ goals Method Results and Discussion Supporting the MR priority assumption, the participants overwhelmingly (90%) preferred the between variable option to the spread-over variable option, 2(1, N ϭ 20) ϭ 12.80, p Ͻ 01 The fixed option was also clearly (85%) preferred to the spread-over option, 2(1, N ϭ 33) ϭ 16.03, p Ͻ 01 The participants refused to incur the possible failure of falling below MR for the sake of reaching over a desired G (see Table 2) Second, the SQ reference point was also a driving force in making salary choices as illustrated by the choice preference of the fixed option over the between option, 2(1, N ϭ 28) ϭ 7.00, p Ͻ 01 This most likely salary (SQ) also contains information of social comparison as to what constitutes the average level of salary If 3,990 RMB is used as the reference point, for the “below MR” pair of options, the net gain or loss values are Ϫ2,290 versus (.5, Ϫ2,990; 5, Ϫ1,590) and the CPT values are Ϫ2,036 and Ϫ1,976 RMB, respectively For the “above MR” pair of options, the net gain or loss values are Ϫ890 RMB versus (.5, Ϫ2,140; 5, 360) and the CPT values are Ϫ886 and Ϫ796 RMB, respectively On the basis of Tversky and Kahneman (1992), we used a power utility function with exponents for gains, ␣ ϭ 88, and losses,  ϭ 88; the loss aversion multiplier, ϭ 2.25; and the probability weighting exponents for gains, ␥ ϭ 0.61, and losses, ␦ ϭ 0.69 Predictions for CPT were derived using the calculator at http://psych.fullerton.edu/ mbirnbaum/calculators/cpt_calculator.htm WANG AND JOHNSON Table Choice Frequencies and Percentages of Salary Options in Experiment Salary options Choice Choice Choice Choice Between Spread over Fixed Between Fixed Spread over Frequency % 18 90% 10% 21 75% 25% 28 85% 15% for the peer group of a person and thus serves as a motivational anchor for salary choice (see Hill & Buss, 2010, for empirical tests) As in Experiment 1, we elicited participants’ G and MR after they finished making choices as a validity check The averages of the subjective MR, SQ, and G of the participants were 2,520 RMB, 3,680 RMB, and 6,380 RMB, respectively As in Experiment 1, we used the participants’ subjective SQ to obtain the values of net gains and losses for the choice options and then plugged the values into the CPT value functions It is interesting that the CPT values predicted the same choice pattern as the TRP did, although for different reasons and mechanisms For example, on the basis of the TRP theory, a strong aversion to the spread-over option was predicted as a result of an MR–G trade-off with an MR priority In contrast, this same choice pattern can also be well accounted for by prospect theory in terms of loss aversion To further test the rival accounts of loss aversion versus MR–G trade-off, we examined the effects of a variable option that reaches over G and stays above MR in the next experiment, thus the negative variance stays in the loss region rather than the failure region Experiment 3: Variable Outcomes Straddle Both MR and SQ or Straddle Both SQ and G Hypothesis and Predictions were designed on the basis of previously determined average values of MR, SQ, and G salaries (see Figure 4) After the participants completed the choice tasks, they were again asked to provide their subjective MR, SQ (most likely), and G salaries The average G, SQ, and MR values were 6,200, 3,600, and 2,600 RMB, respectively Results and Discussion As shown in Table 3, the participants exhibited significantly different choice preferences under the two conditions Consistent with the TRP predictions, a majority of the participants preferred the fixed option when the gamble option straddled both SQ and MR, 2(1, N ϭ 68) ϭ 7.12, p Ͻ 01 However, the risk preference of the participants reversed in favor of the variable option when it straddled over both SQ and G, 2(1, N ϭ 68) ϭ 7.12, p Ͻ 01, indicating a kink at the G level (see Table 3) As in Experiments and 2, we used the participants’ subjective SQ to obtain the values of net gains and losses for the choice options and then plugged the values into the CPT value function The CPT value of the variable option spreading over MR–SQ is higher than that of its fixed equivalent, but the CPT value of the variable option that spreads over SQ–G is lower than its fixed equivalent In both conditions, the CPT values predict risk preferences that are the opposite of the TRP predictions and thus were inconsistent with the observed choice preferences If zero or the mean expected value is assumed to be the reference point, the CPT values predict risk aversion in favor of the fixed option over the variable option in both conditions (see Table 4) The same predictions can be derived from the basic assumptions of prospect theory without specific parametric calculations The TRP theory makes straight testable predictions regarding valuation and choice In Experiment 1, the TRP predictions were partially different from the CPT predictions; in Experiment 2, the predictions of the two theories coincide, although they are based on different mechanisms; and in Experiment 3, the predictions of the TRP and CPT were divergent In an attempt to reconsider prospect theory in light of the data, we explored three possible reference points that can be theoretically assumed to separate mere gains from mere losses in our task: zero, the mean expected value Our mathematical formulation shows that there is a kink in the value function when expected outcomes cross a reference point, and it furthermore assumes a psychological priority order of MR Ͼ G Ͼ SQ When comparing a variable option spanning both MR and SQ to its certainty equivalent, we predict risk aversion in favor of the certain (fixed) option because of an MR–G trade-off, also shown in Experiment However, in the case of a variable option spanning SQ and G, we predict that the convexity of the TRP value curve in the success region would increase the choice of the variable option over its certainty equivalent once MR is no longer a locus of concern (see Figure for an illustration) In contrast, prospect theory would not predict an additional kink in the value function when outcomes are already above the reference point SQ Method A total of 68 participants (33 men and 35 women) participated in Experiment Each participant was given two choice problems with a counterbalanced order of presentation The choice questions Figure Salary options used in Experiment with a wider cylinder representing the mean of a salary option and a narrower cylinder representing the variance of the option MR ϭ minimum requirements; SQ ϭ status quo; G ϭ goals TRI-REFERENCE POINT THEORY Table Choice Between Fixed and Variable Options With Outcome Variation Spanning MR–SQ or SQ–G Fixed Variable Outcome variation Salary n % Salary n % N Spanning MR–SQ Spanning SQ–G 3,000 RMB 5,000 RMB 45 23 66 34 1,900–4,100 RMB 3,000–7,000 RMB 23 45 34 66 68 68 Note RMB ϭ renminbi; MR ϭ minimum requirements; SQ ϭ status quo; G ϭ goals of the choice options, and the subjective SQ as measured by the most likely salary value estimated by the actual decision makers (see Table 4) However, none of them give rise to prospect (CPT) values that consistently predict the actual choice data from the three experiments The results from the above three experiments are also difficult to account for by notions of shifting SQ, replacing SQ with G or MR, or diminishing sensitivity, given that all of these remedies of single reference point theory would require changes in the shape of its value function It should be pointed out that given the success that prospect theory had in predicting various decision phenomena, it is not surprising that the TRP theory and prospect theory partially overlap in their predictions The TRP theory may better predict some choice situations where the expected outcomes are beyond the scope of a single reference point based prospect theory In sum, TRP analyses provide clear and parsimonious predictions without resorting to parametric calculations Converging empirical evidence from previous studies (e.g., Koop & Johnson, 2012) and our original experiments are largely supportive of the TRP assumptions supplemental material) This value function is steeper and convex below MR or G but flatter and concave when an outcome exceeds MR or G In Experiment 4, we test key predictions derived from the double-S-shaped value function across its four outcome regions That is, choice preference between a sure option and its gamble equivalent would go from risk seeking to risk averse, then back to risk seeking, then back to risk averse, as a constant is added three times to a sure option so that the expected value of the sure option and its gamble equivalent moves from below MR to between MR and SQ, to between SQ and G, to above G (see Figure 5; see also Figure 1B) In other words, the subjective value of a gamble would be higher than its certainty equivalent when the certain option is below MR or between SQ and G, because the gamble offers a prospect of exceeding MR (surpassing the most steep and convex portion of the value curve) or G (passing over into the second convex portion of the value curve), respectively In contrast, when the certain option is between SQ and MR or above G, the gamble offers a prospect of passing over into the flatter convex portion of the value curve between SQ and G or the concave portion above G, respectively; in this case, the certainty option would be preferred (see also the figure in the supplemental material) This risk-averse preference would be greatly strengthened if a worse outcome of a gamble falls down the steeper, convex portions of the value curve Experiment 4: Risk Preference Determined by the Double-S-Shaped Value Function Across Four Outcome Regions Hypothesis and Predictions Method The TRP value function suggests a double-S-shaped value function (see also our mathematical formulation provided in the online A total of 60 senior-level managers (46 men and 14 women) recruited from the executive master of business administration Table Theoretical Predictions and Actual Choice Pattern in Two Binary Choice Tasks Choice (across MR and SQ) Choice (across SQ and G) Theory or value Fixed Variable Fixed Variable Salary CPT–zero CPT–EV CPT–SQ TRP Actual 3,000 RMB √ √ 1,900–4,100 RMB 5,000 RMB √ √ √ 3,000–7,00 RMB √ √ √ √ √ Note MR ϭ minimum requirements; SQ ϭ status quo; G ϭ goals; RMB ϭ renminbi; CPT–zero ϭ choice preferences predicted by prospect theory assuming zero as the reference point; CPT–EV ϭ choice preferences predicted by prospect theory assuming the mean expected value of the choice options as the reference point; CPT–SQ ϭ choice preferences predicted by prospect theory assuming subjective SQ (the most likely salary) as the reference point; TRP ϭ tri-reference point theory predicted choice preferences; Actual ϭ actual choice preferences between the fixed and variable options in two choice tasks WANG AND JOHNSON 10 Figure The design framework of Experiment MR ϭ minimum requirements; SQ ϭ status quo; G ϭ goals; EV ϭ expected value program of an international business school in China participated in the study The average age of the participants was 40.6 years, ranging from 28 to 55 years Each participant was given eight independent choice tasks, presented in a random order.7 Each choice was made between a sure option and a gamble of equal expected value (EV) As shown in Figure 5, the participant in each choice task started with 20 performance points (their SQ), and each choice would result in one of the four outcomes: out (fail), stay (loss), advance (gain), and win (success) The instruction read as follows: For every choice you will be given 20 endowment points and your choice will result in one of the following four outcomes (1) If the net outcome point of your choice is equal or less than 0, you fail (out of the game) (2) If the net outcome of your choice is greater than points and less than 20 points, you are in a loss condition but will be allowed to stay in the game (3) If the net outcome of your choice is equal or greater than 20 points and less than 40 points, you are gaining points and will advance in the game (4) If the net outcome of your choice is equal or greater than 40 points, you win the game Your performance will be evaluated in terms of number of Fail, Stay, Advance, and Win by coin flipping to determine the outcome of any gamble choices The eight choice questions were derived from the four hypothetical conditions (two choice questions in each condition), where the EV of choice options falls in the outcome regions below (MR), between MR and 20 (SQ), between SQ and 40 (G), and above G, respectively The four EVs differ by a constant number (20 points), so that EV2 ϭ EV1 ϩ 20, EV3 ϭ EV2 ϩ 20, and EV4 ϭ EV3 ϩ 20 (see also Table for details, and compare to Figures 1B and 5) We used multiple gamble options in each EV condition to reduce the likelihood that the results were driven by specific features of a few gambles Results and Discussion Table summarizes the results of Experiment As a constant (20 points) was added to the EV so that it moved from below MR (EV1) to between MR and SQ (EV2) to between SQ and G (EV3) to above G (EV4), four experimental conditions were created Each condition consisted of 120 choices (2 questions ϫ 60 participants) The risk preference of the participants, as indicated by the average choice percentage of the gamble option in each condition, changed from risk seeking to risk averse, to risk seeking, to risk averse in the four consecutive experimental conditions, respectively The choice preference patterns in the four experimental conditions are all in the above TRP predicted directions Chi-square analyses of goodness of fit showed that the risk preference was significantly different from a 50/50 distribution in choice tasks of 1, 2, 4, 6, and at the p Ͻ 05 level These results have important theoretical implications First, the three reversals observed when the location of the EV was changed across the four functional regions of outcomes matched well with the double-shaped TRP value function where the value function is steeper and convex before reaching MR and G but flatter and concave after passing each of the reference points Participants showed a consistent risk-seeking preference in the EV1 Ͻ MR condition as well as an overall risk-averse preference in the MR Ͻ EV2 Ͻ SQ condition to avoid the gambles that may fall below the MR Furthermore, they displayed failure aversion, indicated by a larger change in choice proportions when crossing the MR (75.8% to 36.7%) than when crossing the G (63.3% to 40%)—that is, the same change in value showed a larger impact when it avoided failure than when it earned success Second, these results challenge some contemporary risk–value models, which assume that risk is location free (see Sarin & Weber, 1993, for a review) The location-free models presume that if a constant amount of wealth is added to all outcomes of a gamble, then the relative utilities are unchanged Violations of the location-free axiom can be seen, for instance, in the opposite risk preferences in Choice Tasks versus 3, versus 4, versus 7, or versus 8, where a constant of 20 points is added to all outcomes of the former gamble to form the latter gamble Third, unlike in Experiments 1–3, the three reference points in Experiment were explicitly specified rather than implicitly elicited The results thus add to the credibility of TRP predictions Fourth, the observed pattern of risk preference is unlikely to be the result of choices based on only one or two reference points The single reference point based CPT valuations of the 12 pairs of binary choices, for instance, yield a different pattern of risk preference Most noticeably, the risk-averse preference predicted by CPT values for the choices of SQ Ͻ EV4 Ͻ G is the opposite of the observed choice preference of risk seeking One alternative account of the results in Experiment is that the observed choice pattern is a result of aggregated choices based on a single reference point Accordingly, each participant uses one and only one reference point at any one time, but different participants use different reference points—some use MR, some SQ, some G To test this alternative, we conducted a within-subject analysis to examine the extent to which multiple reference points were considered In particular, we examined for each participant the number of risk preference reversals between outcome regions (categories) created by multiple reference points and within each outcome region We did this analysis with the two independent chains of choice questions (i.e., Choices 1, 3, 5, and and Choices 2, 4, 6, and 8), created by adding 20 points three times The cross-category reversals in risk preference would occur between Choices and 3, and 5, or and in Chain and between Choices and 4, and 6, or and in Chain The withincategory reversals would occur between Choices and 2, and 3, and 4, and 5, and 6, and 7, or and We also checked the direction of each categorical reversal to verify if it was consistent with a TRP prediction stated previously Each participant completed 12 choice questions Because of an error in expected value, four questions were dropped and subsequently excluded from data analysis TRI-REFERENCE POINT THEORY 11 Table Choice Questions, Choice Frequencies, and Percentages in Experiment MR Ͻ EV2 Ͻ SQ EV1 Ͻ MR Choice Property Outcome Surea Gamble Frequency (%) of sure Frequency (%) of gamble 2 (p) M gamble Risk preference Choice Choice Ϫ25 Ϫ25 0, Ϫ50 Ϫ10, Ϫ40 17 (28.3%) 12 (20.0%) 43 (71.7%) 48 (80.0%) 11.27 (Ͻ.01) 21.60 (Ͻ.01) 75.8% Risk seeking SQ Ͻ EV3 Ͻ G Choice Ϫ5 Ϫ5 20, Ϫ30 10, Ϫ20 36 (60.0%) 40 (66.7%) 24 (40.0%) 20 (33.3%) 2.40 (.12) 6.67 (.01) 36.7% Risk averse Choice Choice 15 15 40, Ϫ10 30, 24 (40.0%) 20 (33.3%) 36 (60.0%) 40 (66.7%) 2.40 (.12) 6.67 (.01) 63.3% Risk seeking EV4 Ͼ G Choice Choice 35 35 60, 10 50, 20 34 (56.7%) 38 (63.3%) 26 (43.3%) 22 (36.7%) 1.07 (.30) 4.27 (.04) 40.0% Risk averse Note EV ϭ expected value; MR ϭ minimum requirements; SQ ϭ status quo; G ϭ goals; Frequency (%) of sure ϭ choice frequency and percentage of participants selecting the sure option; Frequency (%) of gamble ϭ choice frequency and percentage of participants selecting the gamble option; M gamble ϭ the avenge choice percentage of gamble options in each outcome region a Binary choice between a sure option and a 50 –50 chance gamble of equal expected value The frequency and percentage data of preference reversals are shown in Table First, of the total of 183 categorical reversals, 137 (74.9%) were in the cells of multiple (two or three) reversals, 2(1, N ϭ 183) ϭ 44.26, p Ͻ 0001 This is unlikely a result of a uniformly distributed number of reversals Second and more important, of the total of 183 categorical reversals, 153 (83.6%) were consistent with the TRP predictions, 2(1, N ϭ 183) ϭ 81.34, p Ͻ 0001 This result suggests that the observed group choice pattern in Experiment is unlikely to be the result of a combination of single reference point strategies It is possible that a decision maker considers only one reference point at a given time However, it is unlikely that a decision maker considers one reference point and only that reference point entirely in making risky choices Experiment 5: Loss and Gain With or Without MR and G Constraints and Outcome Distributions Within or Across a Reference Point Hypothesis and Predictions The purpose of Experiment is twofold First, we compare risk preference in mere gains and mere losses with absence of MR and G information against the identical gains and losses with explicit MR and G, under the condition where the outcome distribution of a gamble option is well within the boundaries of SQ and G for a gain and well within the boundaries of SQ and MR for a loss This comparison thus also tests the extent to which the TRP theory and prospect theory overlap in their prediction space We predict that risk preference for mere gains or mere losses would be consistent with that derived from the S-shaped value function of prospect theory However, the very existence of MR or G, even without intersecting with choice outcomes, may shift risk preference toward the opposite directions We tested a strong prediction that the existence of MR even outside the range of outcome distributions of choice options would push risk preference toward risk aversion (away from the MR) while the existence of a G outside of the range of outcome distributions would pull the risk preference toward risk seeking (toward the goal) Second, in Experiment 5, we examined the effects of three types of outcome distributions on risk preference while EV is located in each of the four outcome regions (fail, loss, gain, or success) The outcome distribution of a gamble can either be well within a region (within condition), barely touch a reference point (border condition), or clearly cross over a reference point (across condition) We predicted that the across conditions in outcome regions of failure, Table Within-Subject Analysis of Frequencies and Percentages of Preference Reversals In Experiment Reversal Category N Cross-category, Chain Frequency % Cross-category, Chain Frequency % Within-category Frequency % 15.0 26 43.3 15 25.0 10 16.7 60 100 8.3 20 33.3 24 40.0 11 18.3 60 100 11 18.3 23 38.3 21 35.0 6.7 1.7 Total 60 100 WANG AND JOHNSON 12 loss, gain, and success would yield risk seeking, risk averse, risk seeking, and risk averse preferences, respectively, as found in Experiment Such effects of reference points would be weaker in the within and border conditions Method A total of 30 students (13 men and 17 women) from the University of South Dakota participated in the study for extra course credit The average age of the participants was 25.4 years, ranging from 21 to 32 years Eighteen binary choice questions were presented on paper (one question per page) to the participants in two separate classroom settings Each participant first received six binary choice questions (three mere gain and three mere loss questions without introducing MR or G requirements), sorted in a random order On completing the six questions, each participant was then given the next 12 binary choice questions sorted in a random order Similar to Experiment 4, each choice was made between a sure option and a gamble of equal EV Four EV values were located in the following outcome regions (fail, loss, gain, and win) The instructions read as follows: Your choice will result in one of the following four outcomes (1) If the outcome of your choice is equal to or less than Ϫ$200, you fail (2) If the outcome of your choice is greater than Ϫ$200 and less than $0, you are in a loss condition (3) If the outcome of your choice is equal to or greater than $0 and less than $200, you are in a gain condition (4) If the outcome of your choice is equal or greater than $200, you win the game Your performance will be evaluated in terms of number of Fail, Gain, Loss, and Win by coin flipping to determine the outcome of any gamble choices For each EV value, there were three questions The three questions differed in their outcome distribution of the gamble option: within an outcome region, at a border of an outcome region, or across a reference point Results and Discussion Figure summarizes the results of Experiment First, as predicted, without the constraints of MR or G, participants were risk averse in mere gains, 2(1, N ϭ 30) ϭ 7.50, p Ͻ 01, across Choice Tasks 4Ј, 5Ј, and 6Ј, and they were risk seeking in mere losses, 2(1, N ϭ 30) ϭ 5.64, p Ͻ 02, across Choice Tasks 7Ј, 8Ј, and 9Ј When the identical losses were presented with the presence of MR, participants, across the three outcome distribution conditions, became more risk averse (the average percentage of gamble choices fell from 75.6% to 43.3%) This difference remained even in the within condition where the negative end of gamble distribution was clearly above the MR (86.7% to 53.3% gamble choice), as well as in the border condition (66.7% to 40.0% gamble choice) When mere gains were placed under the constraints of MR and G, only in the across condition did participants switch from risk averse (16.7% gamble choice), 2(1, N ϭ 30) ϭ 13.33, p Ͻ 001, to risk seeking (66.7% gamble choice), 2(1, N ϭ 30) ϭ 4.80, p Ͻ 03 The choice patterns in the within and border conditions were not significantly different from the corresponding mere gains The finding of risk aversion in mere gains and risk seeking in mere losses are consistent with prospect theory’s S-shaped value function Furthermore, partially supporting our predictions, the existence of MR significantly shifted the choice preference toward risk aversion, even when the MR was beyond the reach of the outcome distribution However, the existence of G did not shift choice preference when it was beyond the reach of the outcome distribution This result also suggests that MR has a bigger impact than G on risky choice, supporting our Assumption The risk preference patterns in the four across conditions spanning the four outcome regions are consistent with the TRP predic- Figure Choice questions and choice preference under different conditions in Experiment The reference points are status quo (SQ) ϭ 0, minimum requirement (MR) ϭ Ϫ$200, and goal (G) ϭ $200 Questions in Choice Set are identical to the corresponding questions in Choice Set 2, except no MR or G is explicitly introduced Outcome distribution is either within an outcome region, at the border of a region, or over a reference point Choice of Gamble shows the number and percentage of the participants choosing the gamble over the sure option (n ϭ 30) Av.% G is the average choice percentage of gamble options in each outcome region EV ϭ expected value TRI-REFERENCE POINT THEORY tions The risk preference was consistently risk seeking under the EV1 Ͻ MR condition, 2(1, N ϭ 30) ϭ 13.33, p Ͻ 001, and consistently risk averse in the EV4 Ͼ G condition, 2(1, N ϭ 30) ϭ 6.53, p Ͻ 01 In sum, the reversals in risk preference happened at the three reflection points as suggested by the doubleS-shaped value function with one restriction: that in the gain region, the reversal only occurred when the outcome distribution clearly passed the G reference point These results were consistent with the main finding of Experiment and showed that the predicted preference reversals also were observed under new conditions where we did not give any endowment; changed the unit of outcome from points to dollar amounts; and labeled the outcome categories more factually in terms of fail, loss, gain, and win instead of out, stay, advance, and win On the basis of TRP assumptions and the results of Experiments 1–5, we present in Table a summary of the predicted risk preferences under different choice conditions The table shows when and under what combination of parameters the theory predicts risk seeking or risk aversion The left column indicates different locations of the expected value of a choice option, the middle column specifies the outcome variance distributions of gambles intersected by reference points, and the right column gives predicted risk preference Notice that in a between condition where a loss is between (within) the boundaries of MR and SQ or a gain is between the boundaries of SQ and G, the model predicts weak risk aversion, depending on how close the outcome variances are to MR or G One is expected to become less risk averse or even risk seeking when one sees a clear distance between outcome variances and the relevant reference point, MR or G In such a situation, the average risk preference may become risk neutral as shown in the within condition of MR Ͻ EV2 Ͻ SQ in Experiment 5, where 53.3% of choices were risk seeking Closing Remarks The TRP evaluation scheme casts new light on some fundamental assumptions of decision-making theories, such as the classification of payoff distributions into four functional regions beyond the gain–loss dichotomy, the use of reference point priority of MR Ͼ G Ͼ SQ to account for the curvature of the value functions beyond diminishing sensitivity, and the notion of failure aversion as an extension of the concept of loss aversion These new theoretical developments invite further and more systematic examinations Some researchers suggest that the choice behaviors that are usually explained in terms of loss aversion can be attributed to diminishing sensitivity to the distance from zero (e.g., Erev, Ert, & Yechiam, 2008) This alternative explanation can be tested against the notion of failure aversion in future studies The TRP theory extends prospect theory to cover four functional regions that give rise to risk preference attributed to a potentially double-S-shaped value function In this function, psychological values of choice outcomes are convex before reaching MR and G but become concave after passing each of the reference points with a slope that is greatest near MR, less steep near G, and shallowest near the SQ It is conceivable that a three reference point model may be reduced to two reference points or even one reference point in situations where only one or two reference points are relevant or when a reference point is clearly untouchable by distributions of 13 expected choice outcomes Researchers conducting future studies should also aim to systematically identify relative impact of failure, loss, gain, and success in different risk domains, using choice data similar to the way decision weights (e.g., Wu & Gonzalez, 1999) and the coefficient of loss aversion (e.g., Tversky & Kahneman, 1992) were measured In the present article, our main theoretical comparison has been prospect theory One of the future directions in terms of testing validity of the TRP theory is to establish the need for the three reference points by systematically comparing TRP theory with models with either fewer or more reference points The finding in Experiment that barely reaching the G did not significantly increase risk-seeking choice suggests that G is not absolutely fixed but fluctuating within a small range (see also the incorporation of this notion in the supplemental materials) The idea of reference point variation can be traced to Harry Helson’s (1947, 1964) adaptation level theory, which suggests that a decision maker’s adaptation level (a reference value or price) is not fixed but variable and is constantly updated on the basis of past experiences 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The psychology of risk Advances in Experimental Social Psychology, 20, 255–295 doi:10.1016/ S0065-2601(08)60416-5 Mao, J C T (1970) Survey of capital budgeting: Theory and practice Journal of. .. points Cognitive Psychology, 38, 79 –109 doi:10.1006/cogp.1998.0708 Helson, H (1947) Adaptation-level as frame of reference for prediction of psychophysical data The American Journal of Psychology,