CHAPTER 1100
c00010 Decisions Under Uncertainty: Psychological, Economic, and Neuroeconomic Explanations of Risk Preference Elke U. Weber and Eric J. Johnson
OUTLINE s0010 Risk Preference: The Historical Context 127 Modeling Decision-making Under Uncertainty 133 s0130 s0020 Expected Value Theory 128 Risk-taking and Risk Attitudes in EU and PT 133 s0140 s0030 Expected Utility Theory 128 Risk-taking and Risk Attitude in Psychological s0190 s0040 Risk–return Models 128 Risk–return Models 137 s0050 Limitations of Economic Risky Choice Models 129 Process-tracing Methods and Process Data 138 s0200 s0060 Prospect Theory 130 Neuroimaging Studies and Data 139 s0210 s0070 Decisions Under Uncertainty 130 Summary and Implications 140 s0220 s0080 Uncertainty 131 Acknowledgments 142 s0230 s0120 Multiple Processing Systems and the Resolution of References 142 s0240 Uncertainty 132
s0010 RISK PREFERENCE: THE HISTORICAL unpredictability and uncertainty of decision outcomes CONTEXT has increased as the result of ever faster social, institu- tional, environmental and technological change. It is no surprise, then, that the topic of decision- p0020 p0010 Democratic and libertarian societies ask their citizens making under risk and uncertainty has fascinated to make many decisions that involve uncertainty and observers of human behavior. From philosophers risk, including important choices about pension invest- charged with providing tactical gambling advice to ments, medical and long-term care insurance, or medi- noblemen, to economists charged with predicting cal treatments. Risky decisions, from barely conscious people’s reactions to tax changes, risky choice and the ones when driving (“ Should I overtake this car?” ) to selection criterion that people seek to optimize when carefully deliberated ones about capital investments making such decisions has been the object of theoreti- (“ Do I need to adjust my portfolio weights?” ) abound. cal and empirical investigation for centuries (Machina As citizens have taken on more decision responsibility, 1987 ; Glimcher, 2003 ).
Neuroeconomics: Decision Making and the Brain 127 © 2008, Elsevier Inc.
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0.5 1/2 u(x)
u(3000)
0.4 u(2000)
u(1000)
0.3
1/4 Probability u(u) 0.2 0 1000 2000 3000 4000 Wealth, x 5 1/8 FIGURE 10.2 Concave utility function u(x) x which converts f0020 wealth, x, into its utility u(x). An increase in wealth from $0 to $1000 0.1 is shown to result in a greater increase in utility than an increase in wealth from $2000 to $3000. 1/16
1/32
0.0 248163264128 256 512 1024 2048 4092 postulating that money and wealth are diminishing in Payoff value, as shown in Figure 10.2 . The function that maps f0010 FIGURE 10.1 Payoff distribution for St Petersburg paradox actual wealth (x) on the x-axis into utility for wealth game, where a fair coin is tossed until the first “ head ” is scored. The (u(x)) is no longer linear but “ concave. ” An increase payoff depends on the trial at which the first “ head ” occurs, with in wealth of $1000 is worth a lot more at lower initial $2 if on the first trial, $4 if on the second trial, and $2n if on the nth trial. levels of wealth (from $0 to $1000) than at higher ini- tial levels (from $2000 to $3000). In power functions, θ u(x) x , for example, the exponent θ is a param- s0020 Expected Value Theory eter that describes the function’s degree of curvature θ p0030 The maximization of expected (monetary) value ( .50 in Figure 10.2 ) and serves as an index of an (EV) of gamble X , individual’s degree of risk aversion. Such an indi- vidual difference parameter has face validity, as some individuals seem to resolve choices among options EV() X ∑ p () x x (10.1) that differ in risk in very cautious ways (θ 1), while x others seem willing to take on great risks in the hope first considered in the mid-seventeenth century, was of even greater returns (θ 1). rejected as a universally applicable decision criterion Von Neumann and Morgenstern (1947) provided p0050 based on the so-called St Petersburg paradox, where an intuitively appealing axiomatic foundation for people are willing to pay only a small price (typically expected utility (EU) maximization, which made it a between $2 and $4) for the privilege of playing a game normatively attractive decision criterion not only for with a highly skewed payoff distribution that has repeated decisions in the long run but also for unique infinite expected value, as shown in Figure 10.1 . risky decisions, and made EU maximization the dom- inant assumption in the economic analysis of choice under risk and uncertainty. See Chapter 3 of this vol- s0030 Expected Utility Theory ume for more detail on EU theory and its variants. p0040 To resolve the St Petersburg paradox, Bernoulli (1954/1738) proposed that people maximize expected Risk–return models s0040 utility (EU) rather than expected value, In parallel to these developments in econom- p0060 EU( X ) ∑ p ()() x u x (10.2) ics, Markowitz (1959) proposed a somewhat differ- x ent solution to the St Petersburg paradox in finance,
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100 X1 (EV 100, variance 15) X 1 X2 (EV 60, variance 6) 80
60 X2
WTP (X) WTP 40
20 FIGURE 10.3 Willingess-to-pay (TWP) for f0030 risky investment options X (for X1 (EV 100, Variance 15) and X2 (EV 60, Variance 6)) as 0 predicted by risk-return model in Equation 10.3, for 123 different values of b. b
modeling people’s willingness to pay (WTP) for risky Limitations of Economic Risky Choice Models s0050 option X as a tradeoff between the option’s return p0080 V(X) and its risk R(X), with the assumption that A central assumption of all economic risky choice people will try to minimize level of risk for a given models described above is that the utility of decision level of return: outcomes or the risk and return of choice options are determined entirely by the objective value of possi- XXbX (10.3) ble outcomes (and the final wealth they generate) in a WTP() V () R () “ reference-independent” way, i.e., in a way that does Traditional risk–return models in finance equate V(X) not depend on what the outcome can be compared to. with the EV of option X and R(X) with its variance. Thus the receipt of a $100 is assumed to have the same Model parameter b describes the precise nature of effect on the decision of an individual, whether is it the tradeoff between the maximization of return and the top prize in the office basketball pool or the con- minimization of risk, and serves as an individual solation prize in a lottery for $1 m dollars. Decision- difference index of risk aversion. Figure 10.3 shows makers’ evaluation of outcomes and choice options, how WTP varies for two gambles as a function of the however, appears to be influenced by a variety of rela- tradeoff parameter b. This risk–return tradeoff model tive comparisons ( Kahneman, 2003 ). is widely used in finance, e.g., in the Capital Asset In fact it is now widely known that people often p0090 Pricing Model (CAPM; Sharpe, 1964; see Bodie and compare the outcome of their chosen option with the Merton, 1999, for more detail), and can be seen as a outcome they could have gotten, had they selected a quadratic approximation to a power or exponential different option ( Landman, 1993 ). Such comparisons utility function (Levy and Markowitz, 1979 ). Other have an obvious learning function, particularly when classes of utility functions also have risk–return inter- the “ counterfactual ” outcome (i.e., the outcome that pretations, where returns, V(X), are typically mod- could have been obtained, but wasn’t) would have eled as the EV of the risky option, and different utility been better. This unfavorable comparison between functions imply different functional forms for risk, what was received and what could have been received R(X) ( Jia and Dyer, 1997 ). with a different (counterfactual) action under the same p0070 Despite their prescriptive and normative strengths, state of the world is termed regret . When the realized both EU maximization and risk–return optimization outcome is better than the alternative, the feeling is have encountered problems as descriptive models for called rejoicing . Consistent with the negativity effect decisions under risk and uncertainty. Experimental found in many judgment domains (Weber, 1994), feel- evidence as well as choice patterns observed in the ings of regret are typically stronger than feelings of real world suggests that individuals often do not rejoicing. Regret theory, independently proposed by behave in a manner consistent with either of these Loomes and Sugden (1982) and Bell (1982) , assumes classes of models (McFadden, 1999 ; Camerer, 2000). that decision-makers anticipate these feelings of regret Human choice behavior deviates in systematic ways, and rejoicing, and attempt to maximize EU as well as captured originally in two classical demonstrations as minimizing anticipated post-decisional net regret. referred to as the Allais (1953) and Ellsberg (1961) Susceptibility to regret is a model parameter and paradoxes, described below. an individual difference variable that dictates the
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specifics of the tradeoff between the two choice cri- Another noteworthy characteristic of PT’s value p0110 teria. Minimization of anticipated decision regret function is the asymmetry in the steepness of the is a goal frequently observed, even if it results in function that evaluates losses and gains, with a much lower material profitability (Markman et al ., 1993 ). steeper function for losses (“ losses loom larger ” ), Extending these ideas, Braun and Muermann (2004) also shown in Figure 10.4a. The ratio of the slope of proposed a formulation of regret theory that can be the loss function over the slope of the gain function is applied to decisions that have more than two possible referred to as loss aversion, and is another individual choice options. While post-decisional regret undoubt- difference parameter, which is reflected by param- edly plays an important learning function, the impor- eter λ . Empirical studies have consistently confirmed tance to pre-decisional, anticipated regret is less clear. loss aversion as an important aspect of human choice A recent set of choice simulations by Laciana et al . behavior (Rabin, 1998 ; Camerer, 2005 ). It is also a (2007) showed that the incorporation of anticipated likely explanation for real-world phenomena such regret into EU maximization did not result in risky as the endowment effect (Thaler, 1980), the status quo choices that were significantly differently from those bias ( Samuelson and Zeckhauser, 1988 ; Johnson and of EU maximization in a real-world risky decision Goldstein, 2003), and the equity premium puzzle domain, namely precision agriculture. In contrast, the (Benartzi and Thaler, 1995), which describe behavior actions prescribed by prospect theory value maximi- that deviates from the normative predictions of classi- zation, a theory described next, were considerably dif- cal EU theory and risk–return models. ferent from those prescribed by EU maximization. Just as PT suggests a subjective transformation p0120 of objective outcomes, it also suggests a psychologi- cal transformation of objective probabilities, p, into π s0060 Prospect Theory subjective decision weights, (p) , which indicates the impact the event has on the decision. The original p0100 Prospect theory (PT; Kahneman and Tversky, 1979 ; PT decision weight function, shown in Figure 10.4b, Tversky and Kahneman, 1992 ) introduced a different formalized empirical observations showing that small type of relative comparison into the evaluation of risky probability events receive more weight than they choice options, related to the $100 example above. As should, based on their likelihood of occurrence, while shown in Figure 10.4a , PT replaces the utility func- large probabilities receive too little weight. More tion u of EU theory with value function v , which is recently, a more complex continuous function has defined not over absolute outcomes (and resulting been substituted (Tversky and Kahneman, 1992 ). See wealth levels) but in terms of relative gains or losses, Chapter 11 of this volume for more details on PT. i.e., as changes from a reference point, often the PT also suggests that decision-makers will simplify p0130 status quo. PT’s value function maintains EU’s assump- complex choices by ignoring small differences or elim- tion that outcomes have decreasing effects as more is inating common components of a choice. These editing gained or lost (a property referred to by economists as process are not well understood, or easily captured by “ decreasing marginal sensitivity ” ). A person’s degree a formal model. While PT is significantly more com- of marginal sensitivity is measured by the parameter plex than EU, its psychological modifications explain α α in PT’s power value function v(x) x . However, many anomalous observations that have accrued over because outcomes are defined relative to a neutral ref- many years. Risk–return models have also under- erence point, the leveling off of increases in value as gone similar psychological modifications in recent gains increase (“ good things satiate” ) leads to a “ con- years (Sarin and Weber, 1993 ), and are discussed in cave” shape of the value function only in the domain ‘ Modeling decision-making under uncertainty, ’ below. of gains, as shown in Figure 10.4a . This concave shape is associated with risk-averse behavior, e.g., preferring the sure receipt of an amount much smaller than the DECISIONS UNDER UNCERTAINTY s0070 expected value of a particular lottery over the oppor- tunity to play the lottery. In contrast, the leveling off of increases in disutility as losses increase (“ bad things The models of risk preference introduced above, p0140 lead to psychic numbing ” ) leads to a “ convex ” shape will be more formally revisited in the following sec- of the value function in the domain of losses. This tion. In this section, we examine some distinctions convex shape is associated with risk-seeking behav- between different types of uncertainty and different ior, e.g., preferring a lottery of possible losses over the ways of reducing or resolving uncertainty. In the proc- sure loss of an amount of money that is much smaller ess of doing so, we discuss recent suggestions that than the expected value of the lottery. dual-processing systems are involved in risky choice.
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500
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0 x, losses x, gains 100