Lecture Notes on Choice Under Uncertainty

James Andreoni

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Prospect Theory and Probability Weighting

1 Prospect Theory 1:Kahneman and Tversky, �ProspectTheory: An Analysis of Decision UnderRisk.�Econometrica, March 1979, vol 47, p263�291.

� This very in�uential critique of Expected Utility Theory. It has drawn much attention, but hasrecently been falling apart under greater scrutiny.

� Basic idea is that EU has these two properties:� Separability: U(p; x;m) =

Pni=1 piu(m + xi);where m is starting wealth, p is the probability

distribution and x are the changes in wealth from the gamble, for each state of the world i:

� Concavity: u0(z) > 0 and u00(z) < 0; which implies risk aversion.

� These have several simple properties that are easily tested. Let any gamble be a chance of anx 6= 0 with probability p and chance of x = 0 with probability 1 � p: Write such a gamble (x; p):And let u(m + 0) � u(0) = 0:

� KT provide many examples of hypothetical gambles they have presented to students in Israeland Stockholm

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Example 1:Gamble A Gamble Bx p x p

2500 .33 2400 12400 .660 .01n=18 n=72

versus

Gamble A0 Gamble B0x p x p

2500 .33 2400 .340 .67 .66

n=83 n=17

� This violates simple mathematical rearrangement:

0:33u(2500) + :66u(2400) < u(2400)

Now move :66u(2400) to the other side of the equation

0:33u(2500) < :34u(2400)

Which is violated by the data.

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Example 2:Gamble C Gamble Dx p x p

3-week 0.50 1-week 1vacation vacation

n=22 n=78

versus

Gamble C0 Gamble D0x p x p

3-week .05 1-week .10vacation vacation

n=67 n=33This violates common ratio problem.

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Example 3:Gamble E Gamble Fx p x p

4000 0.80 3000 1

n=20 n=80

versus

Gamble E0 Gamble F 0x p x p

4000 .20 3000 .25

n=65 n=35This violates betweeness. The Gambles E 0 and F 0 are each composed as a compound gambleof p = :25 of Gamble E (or F ) and a .75 chance of zero.

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� They also did all of these three with losses. All of the violations held, but this time, however, allof the choices were reversed. So, behavior in losses seems equally perverse, but mirrors thebehavior in gains.

� Example 4, gains versus losses.

Example 4:Gamble G Gamble Hx p x p

6000 0.25 4000 .252000 .25

n=18 n=82

versus

Gamble G0 Gamble H0x p x p

�6000 .25 �4000 .25�2000 .25

n=70 n=30Gambles G andH imply that u(6000) < u(4000)+u(2000) which is consistent with concavity of utilityin gains. But G0 and H 0 imply that u(�6000) > u(�4000) + u(�2000); which is consistent with utilitybeing convex in losses. That is, Utility is S � shaped with a possible kink at zero gains or losses.

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This last point is the most enduring point made by KT. It makes two main points:

� Utility on prospects seems better described by the changes in consumption rather than thelevels of consumption.

� People are risk averse on gains, but risk loving on losses. This indicates a better utility functionmay look something like this:

gainslosses

utility

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2 Prospect Theory 2, Probability Weighting:Tversky andWakker, �Risk Attitudes and Decision Weights.�Econometrica, November 1995, vol 63, p1255�1280.

� This is something of a follow up on their earlier Prospect theory paper.� Further work along these lines is summarized in the �Four-fold pattern of behavior�:� Risk-seeking over small-probability gains

� Risk-aversion over high-probability gains

� Risk-seeking over high-probability losses

� Risk-aversion over small-probability losses

� These cannot be captured in the simple model in the �gure above, in which the utility in a statefollows the S-pattern, but would need to be more complicated.

� Instead, they suggest that we can revise the standard EU model by assuming that individualsdo not behave as though the real probabilities are the true probabilities, but rather that they�weight� true probabilities. Perhaps this re�ects some bias in perception, or some �heuristic� formaking decisions.

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� Speci�cally, the standard EU model would write preferences

U = p1u(1) + p2u(2); where p1 + p2 = 1:

These authors are suggesting instead a model

U = �(p1)u(1) + �(p2)u(2); where �(p1) + �(p2) 7 1:

� In particular, the hypothesis is that the data is best described by people who over-weight smallprobabilities and underweight large probabilities. Sort of like this:

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� A re�ected-like picture for losses will also be needed.

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� Proposed theory:� Utility �ows from gains and losses, u(0) = 0:

� Two weighting functions, w+ for gains and w� for losses

� w(0) = 0; and w(1) = 1:

� Given a gamble (x1;p1; :::;xn; pn) such that x1 � ::: � xk � 0 � xk+1 � ::: � xn; then utility is

U =

kXj=1

��j v(xj) +

nXi=k+1

�+i v(xi)

where ��j = w�(p1+:::+pj)�w�(p1+:::+pj�1): Similarly, �+j = w+(pj+:::+pn)�w+(pj+1+:::+pn):Note that these weights do not necessarily sum to one.

� The rest of this paper goes on to derive conditions under which these preferences can besensible, and looks at the theoretical aspects of the model.

� The main point to draw from this paper is the claim that people's misperceptions of the realodds of a gamble is what could be responsible for some of the failures of expected utilitytheory found by KT in 1979.

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More Critiques of Expected Utility and a Critique of theCritique

3 Rabin's Critique and Calibration Theorem: Matthew Rabin,�Risk Aversion and Expected Utility Theory:A CalibrationTheorem,�Econometrica, September 2000, vol 68,1281-1292

See also, Rabin and Thaler, �Anomalies: Risk Aversion,� J. Econ Perspectives,Winter 2001.

� The basic claim here is that risk aversion over small gambles implies �absurd� amounts of riskaversion over large gambles.

� The example is this: If you would reject a 50-50 gamble of winning 11 or losing 10, then youwould reject almost any gamble of losing $100 and winning Y , no matter how high Y:

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� Here is some logic: Rejecting the small gamble means that

u(m + 11)� u(m) < u(m)� u(m� 10)

This means that winning the 11th dollar is worth at most 10/11 of the 10th dollar lost. That is,for this $21 swing in wealth, marginal utility is declining at about 10%. If a person would rejectthe gain-11-or-lose-10 gamble at every level of income, then as keep increasing wealth on thewinning side by $21 and we must decrease marginal utility by 10/11ths. This adds up quicklyto imply that even for somewhat small losses you would end up rejecting even very large gains.

� Graphically, it implies a minimial level of concavity at each m to m + 11 as illustrate next.

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m ­ 10m + 11m m + 10

Minimal Risk Aversion

Risk Neutral

Utility

� This concavity accumulates incredibly quickly, making people very risk averse over big gambles.� This seem to make RA over small gambles inconsistent with RA over large gambles.

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� The table from both papers makes these claims:

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� The argument hinges on people rejecting the small gamble for all income levels. One criticismof this is analysis is that perhaps we need a different model for small gambles than for largegambles. That is, suppose there are �decision costs� that you need to incur to decide whether agamble is worth taking. For the win 11 lose 10 gamble the difference is so trivial that it is easierto simply say �no� when offered this gamble. But, the problem with this is that the argumentis about marginal utilities being constant. This is the same as assuming constant absolute riskaversion. Suppose people have decreasing absolute risk aversion. Then the marginal utilitiescan shrink to zero.

� Another way to put this is: Rabin's observation either implies that the theory of EU is badlywrong, or experiments on small gambles are so prone to other in�uences as to be uninformative.

� For instance, their could be decision costs, hypothetical bias in behavior, inexperience, ex-perimenter effects. We don't know whether the situations in the lab are representative of thekinds of decisions made in the real world.

� What we need to settle this is an experiment with real gambles that are high enough to makepeople pay attention to their choices, and high enough that these effects will swamp any conta-minating effects of the experimental procedure itself.

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4 Experiments with Large Stakes: Charles Holt and SusanLaury, �Risk Aversion and Incentive Effects� AmericanEconomic Review, 2002, 92, 1644-1655.

� These authors look at a couple of questions:� Can we reconcile the Rabin Critique

� Do hypothetical gambles really matter?� Kahneman and Tversky claim that they don't and base many of their criticisms of EU on them.

� No real experiments on large stakes�it's simply too expensive.

� It is important to know whether real stakes are different for both methodological reasons andas a basic scienti�c questions.

� Can a reasonable EU function �t the data for both small and large gambles?� Can we reconcile data with the Rabin Critique?� If one extrapolates from choices over small gambles using Rabin's logic and �nd �absurd�results, but can still �t a more general utility function to the data, then that would be important.

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� The basic experiment:

� Subjects pick either option A or option B for each of the 10 choices. Then the experimenterchooses one at random to carry out.

� Notice, option A is the �safe� option and B is the �risky� option

� At some point a RA person should cross from choosing A to choosing B.

� Thus, the crossing point is an index or Risk Aversion.

� A risk lover should have the opposite pattern, but no one in the sample does.

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� Constant relative risk averse utility functions are:

u(x) = x1�r (CRRA)

has been estimated on other data to �nd r � 0:5: Under these preferences, then such aperson would choose 6 safe and 4 risky choices above.

� The conditions of the study� Also used hypothetical questions of 20x, 50x and 90x the payoffs in table 1, each time with a�return to the baseline.�

� This was to check whether shifting wealth had an effect on choices. (it didn't)� They also did the same gambles in which people were paid.

� This is very expensive. In the 90x treatment the winning payoff is $391 in the highestoutcome.

� A summary of treatments is below

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Findings:

� Hypothetical versus Real. Figure1 shows hypothetical versus the real low payment. Thereis no difference. But Figure 2 shows all real payments. The lines shift out, which is consistentwith increasing risk aversion with larger gambles.

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� Can a Reasonable EU function �t the data? Figure 3 shows three straight lines from the topto the bottom. The �rst is a risk neutral function. The second is a CRRA utility function that�ts the data for small real gambles. The last is that same CRRA utility function for the largegambles. This surely shows an absurd amount of RA that is not captured by the data.

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� It is clear from this that a utility function that has Decreasing absolute risk aversion will beneeded. Note that a constant absolute RA utility function has decreasing relative RA. Such afunction is

u(x) = �e�ax (CARA)

It turns out that this also is too strict to �t the data. Instead the authors do two things. Firstthe assume that there are some �decision errors�. As such, we assume that the greater thedifference in utility from A and B, the more likely the person is to choose the alternative with thehigher utility. In particular, consider a error parameter � such that

Pr(choose Option A) =u1=�A

u1=�A + u

1=�B

:

Next assume that utility is a more �exible form:

u(x) =1� exp(�ax1�r)

a(Flex)

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u(x) =1� exp(�ax1�r)

a

� This has both CRRA and CARA as special cases. Note the Arrow-Pratt Coef�cient of relativerisk aversion is

�u00(x)

u0(x)x = r + a(1� r)x1�r;

so when a = 0 we have CRRA and when r = 0 we have CARA.

� The authors estimate (Flex) under the assumption of decision errors. They get coef�cients� = 0:134(0:0046); r = 0:269(0:017); a = 0:029(0:0025). The �tted values and the data are shownin �gure 6. The ability to �t the data is quite remarkable. The poorest �t is on 50x, where thedata is also fairly variable. And even with only 18 observations on the 90x payoffs, the function�ts quite nicely.

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Conclusion:

� Real payoffs matter quite a bit.� Rabin's critique is easily overcome by being more �exible in the utility function used, and RAover small gambles need not lead to �absurd� conclusions about large gambles.

Criticisms and Replies

� Harrison, Johnson, McInnes, and Ruström (2003) Risk Aversion and Incentive Effects: Com-ment,�American Economic Review, June 2005, 95(3), 897-901.

� Argue that the Holt and Luary paper suffers from order effects. Because of the cost, HL useda within-subjects design. In the end, they needed to spend more money to get it right:

� Holt and Laury, �Risk Aversion and Incentives: New Data Without Order Effects� AmericanEconomic Review, June 2005, 95(3), 902-912.

� They conduct new studies that give all the budgets at once for 1� or 20� and real or hypothet-ical, in random order.

� They show their effect holds: hypothetical q's under-state the amount of risk aversion.

� In the end, Holt-Laury mechanisms have become the industry standard.

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Other Interesting Papers

Glenn W. Harrison, Morten I. Lau, E. Elisabet Rutström, �Estimating RiskAttitudes in Denmark: A Field Experiment� Scand. J. of Economics 109(2),341�368, 2007

� Subjects are real humans (rather than college students) drawn from a nationally representativesample of Danish people.

� Met in hotels across Denmark.

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� First give a Holt-Laury �Multiple Price List" or MPL

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� Extend the MPL by iterating it, the iMPL� The iMPL format extends MPL by �rst asking the subject to simply choose the row at which hewants to �rst switch from option A to option B (or where they are indifferent)

� don't allow switching back and forth.

� Then present them a second MPL that re�nes the options from the last choice.� That is, if someone decides at some stage to switch from option A to option B between proba-bility values of 0.1 and 0.2, the next stage of an iMPL would then prompt the subject to makemore choices within this interval, to re�ne the values elicited.

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Results:

mean value of the CRRA parameter is 0.67

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Fun regression:

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Kerstin Preuschoff, Peter Bossaerts, and Steven R. Quartz, �NeuralDifferentiation of Expected Reward and Risk in Human SubcorticalStructures� Neuron 51, 381�390, August 3, 2006Summary:

� Economic studies emphasize the importance of risk in addition to expected reward. Studies inneuroscience focus on expected reward and learning rather than risk.

� Combined functional imaging with a simple gambling task to vary expected reward and risksimultaneously and in an uncorrelated manner.

� Drawing on �nancial decision theory, we modeled expected reward as mathematical expectationof reward, and risk as reward variance.

� Activations in dopaminoceptive structures correlated with both mathematical parameters. Theseactivations differentiated spatially and temporally.

� Temporally, the activation related to expected reward was immediate, while the activationrelated to risk was delayed.

� These results suggest that the primary task of the dopaminergic system is to convey signalsof upcoming stochastic rewards, such as expected reward and risk, beyond its role in learning,motivation, and salience.

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� Required brain scan image:

The end