Journal of experimental psychology case study

Captured by a double-S shaped value function with 3 inflection points, risk preferences switched between risk seeking and risk aversion when the distribution of a gamble straddled a diff

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

A Tri-Reference Point Theory of Decision Making Under Risk

X T Wang

University of South Dakota

Joseph G Johnson

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 3 reference points demarcate risky outcomes and

risk perception into 4 functional regions: success (expected value of xⱖ G), gain (SQ ⬍ ⫻ ⬍ G), loss (MRⱕ x ⬍ SQ), and failure (x ⬍ MR) The psychological impact of achieving or failing to achieve these

reference points is rank ordered as MR⬎ G ⬎ 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

3 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

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

func-tion 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

risk-sensitive foraging theory on survival-related minimum

require-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 math-ematical formalization of the TRP value function (see the supple-mental material) Third, we report empirical tests of specific pre-dictions derived from key assumptions of the TRP theory in comparison to predictions of other decision theories, particularly prospect theory (Kahneman & Tversky, 1979; Tversky & Kahne-man, 1992) In closing, we briefly analyze the potential signifi-cance of the TRP theory in risk analysis and risk management and limitations that call for future research

Reference Points and Payoff Distributions

in Risky Choice

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

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

3 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

1

consideration of the variance and distribution of expected

out-comes 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:

EU⫽i冘⫽1n

p i 䡠 u共x i兲

where p i is the probability of outcome i, x i 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

evalua-tion 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),

re-sulting in value functions with a kink at the reference level (e.g.,

Tversky & Kahneman, 1992) and EU models that measure

target-oriented 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

out-comes; 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 evi-dence; 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 refer-ence 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 parsimoni-ous 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 typi-cally 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 &

1There 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 do (e.g., outcome variance)

Tversky, 1979, 2000) Second, many studies in management

sci-ence 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

Tver-sky’s (1972) elimination-by-aspects model, which assumes that

people make choices by gradually eliminating less attractive

alter-natives 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

exoge-nously 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

experi-ence

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

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 mone-tary 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 prior-itized 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

TRP Theory: An Introduction

Assumptions of TRP-Dependent Evaluation

Assumption 1. 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.2Furthermore, 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

2This 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

Assumption 2. The magnitude of the three reference points

follows the order3of MR⬍ SQ ⬍ G and effectively divides the

value scale of choice outcomes into four functional regions:

suc-cess (expected value, x, is above G, thus xⱖ G), gain

(improve-ment from the SQ, thus SQ⬍ x ⬍ G), loss (deterioration from the

SQ, thus MRⱕ x ⬍ SQ), and failure (falling below the MR, thus

x⬍ MR).4Note 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 3. 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

subjec-tively greater when it passes across a reference point, en route to

a different region, than when it remains in the same region

Assumption 4. 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

long-standing security-first principle in financial investment and

busi-ness 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

aver-sion (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– 4 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⬎ G ⬎ SQ The supplemental material shows

one possible way such a double-S-shaped function could be

formed

Assumption 5. 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

self-efficacy) 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 3 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

C to D) produces a larger subjective value difference than an increase from B to C

In Figure 1B, the importance ordering established in Assump-tion 4 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 differ-ence between achieving success and lack thereof The differdiffer-ence 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

3There are other possible orders of reference points besides the most common order of MR ⬍ SQ ⬍ G For example, one possible order is

MR⬍ G ⬍ SQ after successfully completing a task and before goal updating, whereas another possible order is SQ⬍ MR ⬍ 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⬎ G, and risk seeking when SQ⬍ MR)

4Note that these four regions do not explicitly include the value x⫽ SQ This could technically be defined as a fifth region, such as mere preser-vation or stagnation, although we refrain from doing so; this does not affect subsequent interpretations or analyses

Figure 1. Effects of reference points on outcome evaluation Each neigh-boring pair of points are equidistant, producing an equal change in psy-chological 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

gamble (A in Figure 2) and one low-variance gamble (A⬘) 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 2

When encountering a choice between the risky option A and the

safer option A⬘, we predict a strong risk (variance)-seeking

pref-erence for A because it offers the only chance of staying above the

MR In contrast, when making a choice between options B and B⬘,

we predict a strong risk-averse preference for the safe option B⬘ to

avoid a disastrous potential outcome of B Although the small

upper tail of B offers a potential gain that B⬘ does not, the

possibility of failure outweighs this potential gain (Assumption 4)

Similarly, a weak risk-seeking preference for C over C⬘ 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 D⬘ is expected because the safer

option D⬘ is well above the goal, whereas for the risky option D,

the cumulative probability of getting better than D⬘ 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 C⬘ and D⬘ 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 2 and 3

In addition to the qualitative predictions made using the

TRP-dependent 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 situation-ally 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 em-phasizes 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 thresh-old For a potential-minded individual, the two-factor model pre-dicts risk seeking for losses with a higher aspiration level How-ever, 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 B⬘ in Figure 2

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 assump-tions 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 empha-size 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 prefer-ences 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 aspi-ration 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 distribu-tions 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

Figure 2. Pairs of choice options with identical mean but different

variances High-variance options (A, B, C, D) are paired with low-variance

options (A⬘, B⬘, C⬘, D⬘, respectively) of equal expected value MR ⫽

minimum requirements; SQ⫽ status quo; G ⫽ goals

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

pre-dictions 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

with-out 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

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⬎ Aⱊ 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.,

Bⱊ ⬎ 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

ex-pected 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

pre-dictions 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)

in Shanghai The participants were asked to indicate their mini-mally 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

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 {A⬘, A} and {B⬘, 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 partic-ipant 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

Results and Discussion

Table 1 summarizes the choice data of Experiment 1 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⫽ 56) ⫽ 8.59,

p⬍ 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

5In 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

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⫺ 1,700 ⫽ 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.6Thus, in both situations, the CPT predicted risk

preference patterns are inconsistent with the observed pattern

Experiment 2:

Choices Involving MR and G Trade-Offs

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”

op-tion) As in Experiment 1, the three reference points were

esti-mated from an independent sample rather than externally imposed

by the choice task (see Figure 3 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

that the participants would prefer the between option to the

spread-over option to avoid the risk of (x ⱕ 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

Method

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 1 Binary choices were created by producing each of the three pair-wise combinations of salary options Each participant was pre-sented with one pair of options

Results and Discussion

Supporting the MR priority assumption, the participants over-whelmingly (90%) preferred the between variable option to the spread-over variable option,␹2(1, N ⫽ 20) ⫽ 12.80, p ⬍ 01 The

fixed option was also clearly (85%) preferred to the spread-over option,␹2(1, N ⫽ 33) ⫽ 16.03, p ⬍ 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 ⫽ 28) ⫽ 7.00, p ⬍

.01 This most likely salary (SQ) also contains information of social comparison as to what constitutes the average level of salary

6If 3,990 RMB is used as the reference point, for the “below MR” pair

of options, the net gain or loss values are⫺2,290 versus (.5, ⫺2,990; 5,

⫺1,590) and the CPT values are ⫺2,036 and ⫺1,976 RMB, respectively For the “above MR” pair of options, the net gain or loss values are⫺890 RMB versus (.5,⫺2,140; 5, 360) and the CPT values are ⫺886 and ⫺796 RMB, respectively On the basis of Tversky and Kahneman (1992), we used a power utility function with exponents for gains,␣ ⫽ 88, and losses,

␤ ⫽ 88; the loss aversion multiplier, ␭ ⫽ 2.25; and the probability weighting exponents for gains,␥ ⫽ 0.61, and losses, ␦ ⫽ 0.69 Predictions for CPT were derived using the calculator at http://psych.fullerton.edu/ mbirnbaum/calculators/cpt_calculator.htm

Figure 3. Salary options used in Experiment 2 with a wider cylinder

representing the mean of a salary option and a narrower cylinder

repre-senting the variance of the option MR⫽ minimum requirements; SQ ⫽

status quo; G⫽ goals

Table 1

Choice Between Options Under Two Levels of Expected Mean Salary

Mean

N

Note. MR⫽ minimum requirements; RMB ⫽ renminbi Cell entries show the number of participants selecting fixed option or its gamble equivalent under the below MR and above MR conditions

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

inter-esting 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

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⬎

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 2 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 4 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 3 Each participant was given two choice problems

with a counterbalanced order of presentation The choice questions

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 ⫽ 68) ⫽ 7.12, p ⬍ 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 ⫽ 68) ⫽ 7.12, p ⬍ 01,

indicating a kink at the G level (see Table 3)

As in Experiments 1 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 prefer-ences 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 predic-tions can be derived from the basic assumppredic-tions 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

Figure 4. Salary options used in Experiment 3 with a wider cylinder representing the mean of a salary option and a narrower cylinder repre-senting the variance of the option MR⫽ minimum requirements; SQ ⫽ status quo; G⫽ goals

Table 2

Choice Frequencies and Percentages of Salary Options in

Experiment 2

Choice

Salary options Choice 1 Choice 2 Choice 3

Between

Spread over Fixed Between Fixed

Spread over

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

over-lap 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

predic-tions without resorting to parametric calculapredic-tions Converging

empirical evidence from previous studies (e.g., Koop & Johnson,

2012) and our original experiments are largely supportive of the

TRP assumptions

Experiment 4: Risk Preference Determined by the

Double-S-Shaped Value Function

Across Four Outcome Regions

Hypothesis and Predictions

The TRP value function suggests a double-S-shaped value

func-tion (see also our mathematical formulafunc-tion provided in the online

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 re-gions 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, respec-tively; 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

Method

A total of 60 senior-level managers (46 men and 14 women) recruited from the executive master of business administration

Table 3

Choice Between Fixed and Variable Options With Outcome Variation Spanning MR–SQ or SQ–G

Outcome variation

N

Spanning MR–SQ 3,000 RMB 45 66 1,900–4,100 RMB 23 34 68 Spanning SQ–G 5,000 RMB 23 34 3,000–7,000 RMB 45 66 68

Note. RMB⫽ renminbi; MR ⫽ minimum requirements; SQ ⫽ status quo; G ⫽ goals

Table 4

Theoretical Predictions and Actual Choice Pattern in Two Binary Choice Tasks

Theory or value

Choice 1 (across MR and SQ) Choice 2 (across SQ and G)

Salary 3,000 RMB 1,900–4,100 RMB 5,000 RMB 3,000–7,00 RMB

Note. MR⫽ minimum requirements; SQ ⫽ status quo; G ⫽ goals; RMB ⫽ renminbi; CPT–zero ⫽ choice preferences predicted by prospect theory assuming zero as the reference point; CPT–EV⫽ choice preferences predicted by prospect theory assuming the mean expected value of the choice options as the reference point;

CPT–SQ⫽ choice preferences predicted by prospect theory assuming subjective SQ (the most likely salary) as the reference point; TRP⫽ tri-reference point theory predicted choice preferences; Actual ⫽ actual choice preferences between the fixed and variable options in two choice tasks