Psychology and Economics Ec Lecture 4 Prospect Theory

Transcription

1 Psychology and Economics Ec 2030 Lecture 4 Prospect Theory Andrei Shleifer September 28, 2011

2 Outline: 1. Prospect Theory 2. A Dutch Auction 3. Loss Aversion and status quo bias 4. Risk aversion in the gain domain 5. Risk seeking in the loss domain 6. Probability weighting function 7. Applications 8. Readings 2

3 1. PROSPECT THEORY 1 Prospect Theory social scientists look for parsimonious models that predict human behavior Prospect Theory (Kahneman and Tversky 1979) is an example of such a descriptive model positive application (summarizes what people actually do) not normative or prescriptive probably the most widely cited (influential) social science paper ever published 3

4 1. PROSPECT THEORY starting idea: people care about changes not levels people use a weighted value function to think about risky decisions π(p) is the subjective weight on an event with objective probability p v(x) is the value of a change of magnitude x so a gamble (60% +$10, 40% -$10) would have prospect theoretic value π(.6)v(10) + π(.4)v( 10) contrast with expected utility theory (.6)u(w +10)+(.4)u(w 10) 4

5 3. LOSS AVERSION AND STATUS QUO BIAS 3 Loss Aversion and status quo bias consider small changes around the current reference point loss aversion: a loss of size is 2 as important (in terms of utility consequences) as a gain of size this implies that people are indifferent between A : status quo B : 2/3 chance of +$10; 1/3 chance of -$10 this also implies that people are indifferent between A : status quo B : 1/2 chance of +$20; 1/2 chance of -$10 6

6 3. LOSS AVERSION AND STATUS QUO BIAS the 1030 subject pool rejected a riskier gamble with the same expected value as the last gamble on the previous slide A : status quo B : 50% chance of +$110; 50% chance of -$ total respondents: 18% accept and 72% reject Ben Schoefer has been selected to receive this lottery 7

7 3. LOSS AVERSION AND STATUS QUO BIAS 3.1 Classical economic analysis: Classical economic theory assumes that people have log utility u(c) =ln(c) and that people integrate gains and losses with the rest of their consumption (or wealth). These assumptions are inconsistent with loss aversion. We ll show that classical economic theory predicts that people are roughly risk neutral when faced with small gambles. In other words, decision makers should prefer B to A as long as is just slightly greater than zero. A : a guarantee of no change B : p =1/2+, +$10; p =1/2, -$10 8

8 3. LOSS AVERSION AND STATUS QUO BIAS To show this, note that indifference between A and B implies u(x) =(1/2+ ) u(x +10)+(1/2 ) u(x 10), where x is the current level of wealth or consumption. UseaTaylorexpansiontoexpandthetermsontheRHS around x : u(x) = (1/2+ ) u(x)+u 0 (x)(10) u00 (x)(10) + (1/2 ) u(x)+u 0 (x)( 10) u00 (x)( 10) Gathering terms on the RHS, yields, u(x) = [(1/2+ )+(1/2 )] u(x) [(1/2+ ) (1/2 )] u 0 (x)(10) + [(1/2+ )+(1/2 )] 1 2 u00 (x)(100) = u(x)+2 u 0 (x)(10) u00 (x)(100) Crossing out terms yields, 0=2 u 0 (x)+ 1 2 u00 (x)(10). 9

9 3. LOSS AVERSION AND STATUS QUO BIAS So, = 10 4 u00 (x) u 0 (x) = 10 4x x u00 (x) u 0. (x) Note that, x u 00 (x) u 0 = γ(x) (x) represents the coefficient of relative risk aversion. With log utility, γ(x) =1for all x, since u(x) =ln(x) u 0 (x) = 1 x u 00 (x) = 1 x 2 x u 00 (x) u 0 (x) =1 10

10 3. LOSS AVERSION AND STATUS QUO BIAS Recall that the expression for simplifies to (stakes in gamble) (coefficient of relative risk aversion). 4 (total consumption) If γ =1, and x =$25, 000, then = According to classical theory, you should indifferent between the following two options: A : guarantee of no change B : chance of +$10; chance -$10 Moreover, lim (1/2+ ) =1/2 x But in the lab, subjects require that p =2/3 to get them to accept the lottery. 11

11 3. LOSS AVERSION AND STATUS QUO BIAS 3.2 Endowment effect: Loss aversion discourages trade. If gains are valued discretely less than losses, than it is hard to conduct exchange that makes both parties feel better off One demonstration of this is Kahneman, Knetch, and Thaler s mugs experiment Half of the class is randomly given mugs Those with mugs are asked to value them Those without mugs are asked to bid for them The market clears 12

12 3. LOSS AVERSION AND STATUS QUO BIAS If nobody had loss aversion, this market for mugs would result in trade equal to half of the quantity of the mugs In practice the median Willingness to Pay (WTP) is $2.50 and the median Willingness to Accept (WTA) is $5.25. Few mugs change hands (about 10%) 13

13 4. RISK AVERSION IN THE GAIN DOMAIN 4 Risk aversion in the gain domain Subjects are risk averse when facing gambles with outcomes in the gain domain. For example: Imagine that the US is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows: If Program A is adopted, 200 people will be saved. If Program B is adopted, there is a one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved. K&T gave this problem to subjects and found that 67% chose Program A. 17

14 5. RISK SEEKING IN THE LOSS DOMAIN 5 Risk seeking in the loss domain (Same introduction.) If Program A is adopted, 400 people will die. If Program B is adopted, there is a one-third probability that nobody will die and a two-thirds probability that 600 will die. Note that A on previous slide = A on this slide: 200 saved = 400 die Note that B on previous slide = B on this slide: 1/3 chance 600 saved = 2/3 chance 600 die But, now preference switches: 33% prefer A. 18

15 5. RISK SEEKING IN THE LOSS DOMAIN Note what happened here: Reframing the description of the problem changed preference. First question (in gain domain): v(200) > 2 3 v(0) v(600) Second question (in loss domain): v( 400) < 2 3 v( 600) v(0) 19

16 5. RISK SEEKING IN THE LOSS DOMAIN Another example (with fewer confounds): In addition to whatever you own, you have been given $1,000. You are now asked to choose between: A : A 50% chance of $1,000 (16%) B : A sure gain of $500 (84%) v(500) > 1 2 v(0) v(1000) In addition to whatever you own, you have been given $2,000. You are now asked to choose between C : A 50% chance of -$1,000 (69%) D : A sure loss of -$500 (31%) v( 500) < 1 2 v( 1000) v(0) 20

17 5. RISK SEEKING IN THE LOSS DOMAIN Properties of the value function, v(x): v take changes as its argument (not levels), so framing matters v has a kink at zero slope to the right of zero is one slope to the left of zero is two v is concave (risk averse) in the gain domain v is convex (risk seeking) in the loss domain 21

18 6. PROBABILITY WEIGHTING FUNCTION 6 Probability weighting function Motivation: Choose one of the following two lotteries: A : An 80% chance of winning $4000. (28%) B : A 100% chance of winning $3000. (72%) Choose one of the following two lotteries: C : An 20% chance of winning $4000. (59%) D : A 25% chance of winning $3000. (41%) 22

19 6. PROBABILITY WEIGHTING FUNCTION Whyisthisacontradiction? Expected utility theory (coupled with the results of the first lottery) implies that: Hence, (.8)u(x ) + (.2)u(x) <u(x ) (.2)u(x ) + (.05)u(x) < (.25)u(x ) Now add, (.75)u(x) to both sides, yielding (.2)u(x+4000)+(.8)u(x) < (.25)u(x+3000)+(.75)u(x) But, this contradicts the results of the second lottery. 23

20 6. PROBABILITY WEIGHTING FUNCTION To see this more easily, shift the utility function additively, so that u(x) =0. Then lottery one implies, (.8)u(x ) <u(x ) and lottery two implies (.2)u(x ) > (.25)u(x ) 24

21 6. PROBABILITY WEIGHTING FUNCTION Experiments like this led K&T to conclude that people display a certainty effect. π(1) = 1 π(p) < p for p close to 1 π(.8)v(4000) + π(.2)v(0) <π(1)v(3000) Their experiments also revealed that decision probabilities are high when p is close to zero. π(0) = 0 π(p) > p for p close to 0 25

22 6. PROBABILITY WEIGHTING FUNCTION To see this last effect, consider the following two experiments: Choose one of the following two lotteries: A : A.001 probability of winning $5000. (72%) B : A 100% chance of winning $5. (28%) π(.001)v(5000) + π(1.001)v(0) >π(1)v(5) Choose one of the following two lotteries: A : A.001 probability of -$5000. (17%) B : A 100% chance of -$5. (83%) π(.001)v( 5000) + π(1.001)v(0) <π(1)v( 5) 26

23 6. PROBABILITY WEIGHTING FUNCTION Summarizing the properties of the probability weighting function: π(p) is an increasing function π(0) = 0 π(p) >pfor p close to 0 π(p) crosses p around p.4 π(p) <pfor p close to 1 π(1) = 1 27

24 7. APPLICATIONS 7 Applications equity premium (stocks are unattractive unless they have a very high return, since stocks expose investors to potential losses) target earnings (cab drivers adjust hours to meet their target daily earnings) asymmetric price elasticities (purchases more sensitive to price increases than price cuts) status quo bias (people don t change their investment allocations, appeal of defaults) long-shot effect (long-shots overbet at horse races) end of the day-effect (shift to long-shots at the end of the day) 28

25 7. APPLICATIONS over-insurance effect (overpurchase overpriced insurance for low p events) lottery betting (lotteries are very popular) 29

26 Salience Theory of Choice under Risk Pedro Bordalo, Royal Holloway, University of London Nicola Gennaioli, UPF, CREI Andrei Shleifer, Harvard University September, 2011

27 Introduction In the past 60 years, a range of important violations of Expected Utility Theory have been identified. Risk preferences seem to be systematically unstable: In real life, people participate in unfair gambles, pick risky professions and invest without diversification. Yet they buy insurance. In the lab, subjects switch from risk seeking to risk averse choices upon normatively irrelevant changes in the choice problem (Allais paradox) In comparing two lotteries with similar expected value, subjects often pick the safer one but are willing to pay more for the riskier one (preference reversals) We propose a new psychologically founded model of choice under risk, whereby risk attitudes are driven by the salience of different lottery payoffs. Salience refers to the phenomenon that when one s attention is differentially directed to one portion of the environment, the information contained in that portion will receive disproportionate weighting in subsequent judgments Taylor and Thompson, In our model the decision maker focuses on the most salient payoffs: he is risk seeking when a lottery s upside is salient and risk averse when its downside is salient. 1

28 Introduction Salience is a property of stimuli in context. Insight from Weber s law: changes and differences are more accessible to a decision maker than absolute values Kahneman, We apply this idea to choice under risk. When presented with two lotteries, the agent: focuses on the most salient states of the world, in which lottery payoffs are most different. transforms probabilities into decision weights, with more salient states receiving greater weight. computes value of lotteries using endogenous decision weights. Our model: gives an intuitive account of shifts in risk attitudes, without relying on curvature of value function. has many distinctive implications which we confirm experimentally: * identifies conditions leading to Allais paradoxes, and conditions where independence is observed. * provides a tractable model of context dependence that accounts for preference reversals. is easily applicable to the standard asset pricing framework, and may help explain the well-known growth/value anomaly in finance. 2

29 Introduction Prospect Theory is the main alternative addressing many of the violations of Expected Utility Theory. It is based on the four assumptions: Probabilities are distorted into decision weights. Narrow Framing: Carriers of utility are gains and losses around a reference point. Value function is S-shaped: concave for gains, convex for losses. Loss aversion: losses loom larger than gains in the value function. Distortion of decision weights in Prospect Theory is the least psychologically founded assumption, but it is crucial to explain anomalies. In our model, distortions of decision weights are based on the endogenous salience of payoffs. Moreover: We keep narrow framing, as salience applies to payoffs as they are presented. Some properties of salience (such as diminishing sensitivity) are embodied in Prospect Theory s value function. We keep loss aversion in the value function. We note, however, that the principle that losses loom larger than gains can also be captured by the salience function. 3

30 A Simple Example Problem 1: L 1 = { $1, 0.95 $381, 0.05 vs L 2 = { $20, 1 Problem 2: L 1 = { $301, 0.95 $681, 0.05 vs L 2 = { $320, 1 Problem 2 adds $300 to all payoffs of Problem 1. In both problems: The two options have the same expected value. Small probability of large payoff and large probability of a $19 loss relative to the sure payoff. 4

31 A Simple Example Experimental Results Problem 1: L 1 = { $1, 0.95 $381, 0.05 vs L 2 = { $20, 1 Problem 2: L 1 = { $301, 0.95 $681, 0.05 vs L 2 = { $320, 1 Conducted online survey with 120 subjects. Problem 1: 83% of subjects chose safe option L 2 Problem 2: 65% of same subjects chose risky option L 1 A majority of subjects chose L 2 in Problem 1 and L 1 in Problem 2 5

32 Salience Function (I) We model the choice between two lotteries L 1 and L 2. The salience of state s (x, y) where L 1 pays x and L 2 pays y is σ(x, y) = x y x + y Tractable functional form, with four key properties. For any payoffs x, y > 0 and ɛ > 0: Ordering: if [x, y ] is a subset of [x, y], then σ(x, y) > σ(x, y ). Salience is higher when payoffs are more different relative to the average payoff level. Diminishing sensitivity: σ(x + ɛ, y + ɛ) < σ(x, y). Salience decreases as payoff levels rise (as implied by Weber s law). Reflection: if σ(x, y) > σ(x, y ) then σ( x, y) > σ( x, y ). Salience increases in change relative to reference of 0 regardless of direction of change. 6

33 Salience Function (II) 7

34 Salience Function (III) Example: A local thinker chooses between L 1 = { x + g, p x l, 1 p and L 2 = { x, 1 Salience of gain state is σ(x + g, x) = g 2x+g Salience of loss state is σ(x l, x) = l 2x l Loss is more salient than gain when l 2x l > g 2x+g, or l > g x l x since x l x as x increases, x l x < 1, losses are ceteris paribus more salient than gains (diminishing sensitivity). increases and gains get a better chance to be salient. 8

35 Probability weighting x If Loss is salient, g < l x l, then Local Thinker underweights the odds of x + g by δ (0, 1]: p LT 1 p LT = δ p 1 p x If Gain is salient, g > l x l, then Local Thinker overweights the odds of x + g by 1/δ 1: The resulting decision weights add up to 1 p LT 1 p LT = 1 δ p 1 p Local Thinker evaluates lotteries using these endogenous decision weights. u LT (L 1 ) = p LT u(x + g) + (1 p LT ) u(x l) u LT (L 2 ) = u(x) For δ < 1, the agent overweights the salient lottery outcome For δ = 1, we have p LT = p, and agent is a (loss averse) Expected Utility maximizer. 9

36 Shifts in Risk attitudes: Suppose L 1 = (x + g, p; x l, 1 p) and L 2 = (x, 1) have the same expected value, i.e. pg = (1 p)l also, agent has linear utility and salience function σ(x, y) = x y /( x + y ) Then L 1 L 2 iff gain is salient, i.e. g > l x x l so that x(1 2p) > l(1 p). Local Thinker is always risk averse for p 0.5 (from g l and diminishing sensitivity). For given p < 0.5, increasing x increases risk seeking: loss becomes less salient. Explain large shifts in risk attitudes independently of shifts in wealth. 10

37 Experimental Evidence on Shifts in Risk attitudes (I) Conducted online surveys, giving subjects binary choices between: L 1 = { x + g, p x l, 1 p and L 2 = { x, 1 Set parameters to: l = $20, g = l 1 p p (L 1, L 2 have the same expected value x) x {$20, $100, $400, $2100, $10500}, p {.01,.05,.2,.33,.4,.5,.67} For each choice problem, collected at least 70 responses. 11

38 Experimental Evidence on Shifts in Risk attitudes (II) Proportion of subjects choosing risky lottery L 1 $ $ $ $ $ Risk seeking drops for p > 0.5. We explain risk aversion for small stakes even at low p Risk seeking for moderate probabilities p = 0.2, 0.33 and 0.4, but only for large x. Standard Prospect Theory calibration v(x) = x 0.88 implies risk seeking when π(p) > p, i.e. p

39 Shifts in Risk Attitudes and Probability Weighting For given expected value x, as p increases the gain g decreases, giving rise to p LT as a function of p: Reproduce main features of KT s inverse S-shaped weighting function π(p) At low p agents choose longshot lotteries with salient upside, at high p the reverse occurs. Weighting depends on payoffs! Crucial for Allais paradox and certainty effect. 13

40 Summary so far By endogenizing decision weights as a function of states payoffs, our model unifies explanation of risk averse and risk seeking attitudes Small probabilities are overweighted but only if associated with salient states In the paper, we show that a local thinker prefers long-shot lotteries to their expected value, but only if the latter is small. the reflection effect We now turn to the Allais paradoxes. This illustrates another key property of the model: Context dependence, illustrated through the role of correlations among lotteries. 14

41 Allais paradox (I) Common consequence paradox, compares choices L 1 (z)= (2500, 0.33; z, 0.66; 0, 0.01) vs L 2 (z)= (2400, 0.34; z, 0.66) for different values of z. Typical example of failure of Independence: 2500, 0.33 L 1 (2400) = 2400, , 0.01 L 2 (2400) = (2400, 1) but L 1 (0) = { 2500, , 0.67 L 2 (0) = { 2400, , 0.66 The intuition from Local Thinking: For z = 2400, L 1 has a much worse downside than L 2, and subjects are risk averse. For z = 0, the downsides are the same, so subjects focus on the higher upside of L 1 Get Allais paradox for small δ, because most salient state matters most. 15

42 Allais Paradox (II) Formally, for z = 2400, salience ranking payoff of L 1 (2400) payoff of L 2 (2400) For z = 0, salience ranking payoff of L 1 (0) payoff of L 2 (0) Changing z affects the state space and its salience ordering. For z = 2400, L 2 dominates L 1 in the most salient state. For z = 0, the expected value of L 1 is higher than that of L 2 in the most salient states. Get Allais paradox for δ small enough. With linear utility, upper bound is δ <

43 Allais Paradox (III): Correlations Consider a correlated version of the Allais choice problem: Probability L 1 (z) z L 2 (z) z Difference from previous formulation: z occurs in single common consequence state. In Prospect Theory, correlations do not affect preferences because evaluation depends only on each lottery s payoffs and probabilities. Drastic effect of correlation for Local Thinker s preferences: common consequence state has same weight for L 1 (z) and L 2 (z) and cancels out in evaluation. For any δ, a local thinker s preferences do not depend on z. If δ is sufficiently low, our model predicts L 2 (z) L 1 (z) for any z. 17

44 Allais Paradox (IV): Experimental Results on Correlations Choices in correlated Allais problem with z = 0 and z = 2400: L 1 (2400) L 2 (2400) L 1 (0) 7% 9% L 2 (0) 11% 73% Consistent with our model: Choices do not change with z. Subjects prefer the safer lottery L 2. Inconsistent with Prospect Theory, which predicts that Allais paradox also occurs here: namely, that L 1 (0) L 2 (0), L 1 (2400) L 2 (2400), irrespective of correlation structure. this follows from probability weighting function being context independent. Local Thinkers satisfy independence axiom when common consequence is made evident by correlation. 18

45 Context dependence Salience is defined on lottery payoffs as presented to the agent. This implies the following forms of context dependence: Evaluation of a lottery in the context of choice depends on the alternative lottery being considered. Evaluation of a lottery depends on whether it is compared to another lottery or priced in isolation. This can account for preference reversals between choice and pricing in isolation. Preference reversals between choice and pricing should subside if pricing is done in the context of choice. We now examine these features in turn. 19

46 Preference Reversals Problem 1: choose between the lotteries { 2x, p/2 L 1 = 0, 1 p/2 and L 2 = { x, p 0, 1 p As in the common ratio effect, if p is large the agent values L 2 more than L 1, and chooses L 2 Problem 2: price lotteries L 1, L 2 in isolation. the local thinker evaluates lottery L i w.r.t. the status-quo lottery L 0 = (0, 1) of not having L i Lottery gains are always salient, so the evaluation is given by: u sq (L 1 ) = 2x p/2 p/2+(1 p/2)δ, and u sq(l 2 ) = x p p+(1 p)δ For any δ < 1, a local thinker prices the risky lottery L 1 higher than the safer lottery L 2. Thus, in Problem 1 L 2 is valued more than L 1 but in Problem 2 the reverse holds. This can explain Preference Reversals (related to the Compatibility principle, Tversky et al 1990). 20

47 Preference Reversals Choice and pricing follow the same fundamental principle of context-dependent evaluation. Preferences based on choice differ from those based on pricing in isolation because they represent evaluations made in different contexts. When choice and pricing are made in the same context, namely that of choice, preference reversals should subside. Empirical evidence supports both predictions: Systematic preference reversals occur between choice and pricing in isolation, and are due to the vast overpricing in isolation of the riskier lottery L 1 (Tversky et al 1990) no systematic preference reversals occur between choice and pricing in the context of choice (Cox and Epstein, 1989) 21

48 Extension to Many Lotteries choice set {L 1,..., L N }, with vector of payoffs x s = (x 1 s,..., x N s ) in state s. salience of payoff x i s is measured relative to the average of alternative payoffs in x s : σ(x i s, x s ) σ x i s, 1 N 1 j i x j s Implications: the same state of the world may have different salience for different lotteries. preference ranking among any two lotteries depends not only on the contrast between their payoffs but also on the remaining alternatives. this can account for the decoy effect and other violations of the weak axiom of revealed preference. 22

49 Summing Up Presented a parsimonious model of decision-making under risk, in which salience of payoffs drives distortion of decision weights. Departures from Expected Utility arise when salience and probability of a state are very different. The model yields many stylized facts of choice under risk, as a result of endogenous decision weights. Failures of invariance follow from changes in the (average) salience of payoffs. The model develops new predictions which we test experimentally. Risk loving behavior when gains are salient (even for moderate probabilities) Correlations affect choice and violations of invariance (determining under which conditions the Allais paradox occurs) The model predicts context dependent evaluation and can account for preference reversals described by Slovic and Lichtenstein (1971). The model is extended to choice over many lotteries, and can account for well known context effects. 23

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51 Summary of biases Attitudes towards Risk Risk seeking in MPS for small probabilities of large gains Risk aversion in MPS for moderate probabilities of gains Shifts in Risk attitudes from translation of payoffs Reflection Loss aversion Prospect Theory Regret Theory Salience Theory Probability Weighting π(p) (contributes to risk seeking at any payoff level) Concavity of value function for gains Consistent if value function has D.A.R.A Value function is concave for gains and convex for losses Loss averse value function Convexity of Regret function Q (with linear utility, implies risk seeking for any p < 1/2) Convexity of Regret function (for p > 1/2) Inconsistent for linear utility (no reference point) Regret function Q is antisymmetric on the difference in utilities in each state N.A. (no point) reference Gain is salient, its probability is overweighted Loss is salient, its probability is overweighted Convexity of salience Salience ranking is preserved under reflection of payoffs Loss averse value function

52 Anomalies Prospect Theory Regret Theory Salience Theory Sub-additivity of π Consistent with For large p, safer a convex Regret lottery s payoff Common Ratio function Q more salient than riskier lottery s. Converse for small p Sub-certainty of π Consistent with Common payoffs Common a convex Regret have different Consequence function Q salience and do not cancel out Framing Narrow Framing N.A. Narrow Framing for evaluation for evaluation Common branch is Correlated state Correlated state Isolation effect edited out cancels out in cancels out in evaluation evaluation N.A. Regret evaluation Salience evaluation Correlations depends on State Space depends on State Space N.A. Pricing elicits true Pricing entails Preference Reversals evaluation, different from choice. parison with zero, evaluation in com- Follows from intransitive and overvaluation choice of lottery