Risk Aversion and Expected Utility Theory: A Field Experiment with Large and Small Stakes

Expected Utility: A Field Experiment with Large Stakes

Risk Aversion and Expected Utility Theory: A Field Experiment with

Large and Small Stakes

Matilde Bombardini (UBC) Francesco Trebbi (University of Chicago

GSB)


MotivationHow well does expected utility theory

explain the behavior of an average person? Several criticisms of expected utility theory. Here we address:

• Rabin’s critique – unreasonable rate of decline of marginal utility of wealth

• Consistency within the same individual• Bayesian updating

Why this game• High-stakes: 500,000 euros, 250,000 euros,…• Non-infinitesimal probabilities• Simple structure• No ability involved in answering questions• Ability only involved in Bayesian updating


Timing of the game20 Boxes opening and offers alternate:

1. 6 boxes opened2. First offer (always a change)3. 3 boxes opened4. Second offer (always monetary)5. 3 boxes opened6. Third offer7. 3 boxes opened8. Fourth offer9. 3 boxes opened10. Fifth offer11. Uncover content of contestant’s box


The contestants Selection

- Based on photogenic looks and personality- No questions asked about gambling and/or other games

Characteristics- Gender and age range- Marital status and children- Region/City- Urban/Rural- Occupation (to approximate income)

Self-selection- Risk lovingness does not seem to play a role- The producers: “People want to be on TV” - Data allows to reject self-selection


Information structure What the “infamous” (man on the phone) knows:

• The content of all the boxes• The past behavior of contestants

What the contestant knows:• The distribution of prizes (constant for all episodes)• The past history of the episode• The strategy of the “infamous” (empirical distribution

of offers) for E(lottery)>20,000 Euros- Stage 4: Corr (Relative offer, Relative prize in hand) = 0.32 - Stage 5: Corr (Relative offer, Relative prize in hand) = 0.36- Both correlations significant at 1 percent


What the data can tell Risk preferences

- Acceptance of monetary offers reveals a lower bound on degree of risk aversion

- Rejection of monetary offers reveals an upper bound on degree of risk aversion

- Box changes place upper bounds and lower bounds if offers are informative

Consistency of the choice set- Sequence of acceptances and rejections may reveal

inconsistencies Bayesian updating

- If consider informativeness of the signal “offer” then changes can reveal information on Bayesian updating


The contestant problem Consider only monetary offers Stage Monetary offer at stage s: Lottery at stage s: with is the set of prizes not discovered at stage s Let if the offer is accepted at stage s The problem of the contestant if offers are not informative is:

The problem of the contestant if offers are informative

s 1,2,3,4,5

Ps pi| k i Ks

Ks

Ks

ds 1

ms

maxd s0,1 dsuy m s 1 dsEsVs 1Ps 1 ,Ks 1 ,m s 1| s,Ks,Ps VsPs,Ks, m s

VsPsms,Ks,ms

maxd s0,1 dsuy m s 1 dsEsVs 1Ps 1m s 1 ,K s 1 ,m s 1| s,Ks,Psm s


Prize probabilities Non-informative offers

• Priors are not affected by offers

• Where is the number of boxes in set

Informative offers• Offers affect probabilities according to empirical likelihood:

• The posterior, conditional on the offer is calculated by Bayes rule:

nKs Ks

Ps p i 1nKs

| k i Ks

f sm s|hs,k i k i

Psm s

pims fs m s |h s,k i k i

p im s1,h s

j

fs m s|h s,k j k j p jm s1,h s

| k i KsPsm s


Solution of the optimization problem Finite horizon dynamic optimization problem Recursive solution:

• At stage 5 given two prizes left • Given offer m5

• Compute expected utility and compare it to utility from offer• Decision to accept the offer:

• Compute for all possible offers at s=5 and all possible paths• Obtain continuation value at s=4 and compare to utility from offer

m4

• Solve recursively

K5 k i,k j

d5 Iuy m5 uy k ip im5 uy k jpjm5

maxd 50,1 d5uy m5 1 d5uy k ipim5 uy k jpjm5V5P5 ,K5 ,m5


Empirical ModelNon informative offers (NIO) Take CRRA utility function:

Calculate continuation value at each stage s as function of and:1. Possible future box openings2. Future empirical distribution of offers3. Optimal behavior in future offers4. Empirical probability of change vs. monetary offer5. Empirical distribution of offers as a function of expected value of the

lottery Compare continuation value with utility from monetary offer and find

threshold • Acceptance: • Rejection:

uc c1

1

L

H


Empirical ModelInformative offers (IO) Same basic structure as for NIO Obtain empirical signal (offer) likelihood from the data:

• For each stage s = 3, 4, 5• For each box held by participant at stage s (relative to remaining boxes)• Segment offers in terms of percentage of expected value (5 segments:

0-20, 20-40, 40-60, 60-80, 80-100) • Obtain frequency of offers at stage s as function of box held by contestant

(three likelihood matrices L3, L4, L5) Bayesian updating of priors conditional on offer observed at stage s Calculate continuation value at each stage s as function of and:

• Updated probabilities• Future Bayesian updating and optimal behavior


Empirical ModelMaximum Likelihood: Assume unobserved heterogeneity in risk preferences: for

individual j:

Partition the set of individuals in 3 subsets:- NA : individuals for which only observed

- NR : individuals for which only observed

- NAR : observations for which both and observed

j x j j

j N 0, j2

L

H

L H


Empirical Model Multiple rejections:

Log Likelihood function

Hj min H,s

j | s 3,4,5

logL jNAR

ln H

j

j

x j j

L

j

j

x j j

jNR

ln H

j

j

x j j

jNA

ln 1 L

j

j

x j j


Expected value of the lottery 52560.00Average Final Payoff (AFP) 29436.09Average income of contestant 17875.82 for which AFP is CE 0.53CE for =1 18003.50

TABLE 1CBack-of-the-envelope Calculation - 252 episodes,

CRRA

Note: Monetary amounts expressed in Euros. Consider the lottery at the beginning of the game with equi-probable payoffs of [50 50 50 50 50 50 50 100 250 500 5000 10000 15000 20000 25000 50000 75000 100000 250000 500000]. All prizes nominally below 50 Euros in Table 1A are paid 50 Euros.

Coef. Std. Err. Coef. Std. Err. Coef. Std. Err.MEAN

0.8834 0.0473 0.8843 -- 0.8883 --Individual Covariates:Age -0.0165 0.2431 -0.0681 0.2116(Age)2 -0.0007 0.0057 0.0010 0.0050(Age)3 0.00001 0.00004 -3.29e-6 -3.87e-5Female -0.0250 0.1300 0.0062 0.0959Constant -20.4564 27.8139 -15.6332 18.9354Regional F.E. (p-value) -- [0.7566] -- [0.2904]Income(/ 10,000) 0.4187 0.4037 0.2756 0.3487Time Covariates:Time trend 12.7240 16.7935 10.2451 12.0591Season of the show F.E. (p-value) -- [0.1975] -- [0.0688]

STANDARD DEVIATION 0.5341 0.0329 0.4633 0.0434 0.4161 --

Individual CovariatesAge 0.4883 0.6889(Age)2 -0.0110 0.0156(Age)3 0.0001 0.0001Female -0.0044 0.2453Constant 13.1877 17.8708Income(/ 10,000) 0.6414 0.9319Time CovariatesTime trend -13.5666 6.5326

Log Likelihood -177.7120 -158.9505 -152.1448No. Thresholds* -- -- --No. Obs. 237 237 237No. Left-censored Obs. 129 130 129No. Right-censored Obs. 21 21 21No. Interval Obs. 87 90 87

TABLE 2AMLE: Unobserved Risk Preferences in the Static Model

CRRA Dep. Variable: Dep. Variable: Dep. Variable: (1) (2) (3)


1.0879 0.1014 1.0871 -- 1.0875 --Individual Covariates:Age -0.0332 0.4920 -0.0477 0.4318(Age)2 -0.0009 0.0115 0.0004 0.0102(Age)3 0.00002 0.0001 1.17e-6 0.0001Female -0.0545 0.2518 -0.0742 0.1813Constant -44.9222 50.2362 -9.1447 32.4073Regional F.E. (p-value) -- [0.6548] -- [0.1815]Income(/ 10,000) 0.3281 0.7407 0.0944 0.4973Time Covariates:Time trend 28.7294 30.1198 7.1336 20.2197Season of the show F.E. (p-value) -- [0.0738] -- [0.0053]


Individual CovariatesAge 0.6716 0.6452(Age)2 -0.0172 0.0149(Age)3 0.0001 0.0001Female 0.4635 0.3132Constant 6.0922 14.8282Income(/ 10,000) 0.7477 0.8617Time CovariatesTime trend -9.8800 6.8247

Log Likelihood -172.9480 -153.8920 -144.0787No. Thresholds* 518 518 518No. Obs. 241 241 241No. Left-censored Obs. 130 130 130No. Right-censored Obs. 21 21 21No. Interval Obs. 90 90 90

TABLE 2MLE: Unobserved Risk Preferences in the Uninformative Offers Model

Dep. Variable: Dep. Variable: Dep. Variable: CRRA (1) (2) (3)


1.3688 0.1254 1.3514 -- 1.3140 --Individual Covariates:Age -0.1809 0.5628 -0.8064 0.4436(Age)2 0.0043 0.0130 0.0192 0.0106(Age)3 -0.00003 0.0001 -0.0001 0.0001Female -0.1239 0.2934 -0.3107 0.1862Constant -51.5455 65.7595 -48.8788 27.3729Regional F.E. (p-value) -- [0.6293] -- [0.0303]Income(/ 10,000) 0.7785 0.8427 0.3331 0.4227Time Covariates:Time trend 32.9423 39.7442 36.6249 17.5469Season of the show F.E. (p-value) -- [0.0748] -- [0.0028]


Individual CovariatesAge 2.1016 0.7522(Age)2 -0.0498 0.0177(Age)3 0.0004 0.0001Female 0.3636 0.2871Constant -24.1237 13.3156Income(/ 10,000) 1.7102 0.6557Time CovariatesTime trend -4.2717 5.8627

Log Likelihood -190.1919 -171.0048 -163.8269No. Thresholds* 517 517 517No. Obs. 233 233 233No. Left-censored Obs. 126 126 126No. Right-censored Obs. 21 21 21No. Interval Obs. 86 86 86

TABLE 3MLE: Unobserved Risk Preferences in the Informative Offers Model

CRRA Dep. Variable: Dep. Variable: Dep. Variable: (1) (2) (3)


1000

0030

0000

Offe

r

0 100000 200000 300000 400000 500000E(Lottery)

Rejected Offer Accepted OfferLowess fit offer CE gamma=1CE gamma=2 CE gamma=0

Figure 1aNonparametric Fit for Reject/ Accept Decision: Stage 5


050

000

1000

0015

0000

2000

00O

ffer

0 50000 100000 150000 200000E(Lottery)

Rejected Offer Accepted OfferLowess fit offer Lowess fit CE gamma=1Lowess fit CE gamma=2 Lowess fit CE gamma=0

Figure 1bNonparametric Fit for Reject/ Accept Decision: Stage 4

Rejection Threshold < Acceptance Threshold (Upper bound < Lower bound)

Rejection Threshold at 0

Large Inconsistencies (Upper bound -Lower bound < -1)

Total Number of Choices in Sample

CRRA

No. Inconsistent choices 14 2 6 518Fraction inconsistencies 0.0270 0.0039 0.0116 --Average payoffs at acceptance 47,666 -- 13,718 31,775

CRRA

No. Inconsistent choices 30 10 22 517Fraction inconsistencies 0.0580 0.0193 0.0426 --Average payoffs at acceptance 49,147 -- 22,987 31,775

Note: Payoff amounts are reported in Euros. The choice set includes decisions on monetary offers only. The total number of stages is 756 (=252 episodes*3 stages). Missing and changes amount to 224 (see Table A1 in Appendix). Of the remaining 532 monetary offers, 13 were made at s=5 with 2 prizes of identical value and are dropped, leaving 519 choices. The threshold-finding routine did not converge in 1 (resp. 2) occasion for the uninformative (resp. informative) offers model producing the totals in the last column. Only observations that present inconsistent behavior are dropped.

TABLE 4Consistency of the Choice Set

Informative Offers

Uninformative Offers

CRRA

Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err.

MEAN 1.0879 0.1014 1.1631 0.1404 1.1527 0.1528 0.4299 0.5378 0.4604 0.5705

Stage 4,5: -- -- 1.0986 0.0808 1.0653 0.0834 -- -- -- --

STD. DEV. 1.0765 0.0523 0.8192 0.1367 0.8704 0.1374 2.2830 0.4289 1.9228 1.0782

Stage 4,5: -- -- 0.4741 0.1097 0.4614 0.1272 -- -- -- --

Log Likelihood -172.95 -52.71 -53.38 -29.13 -13.14No. Obs. 241 99 94 76 35Stage 4,5: log L -- -39.86 -34.65 -- --Stage 4,5: No. Obs -- 67 56 -- --

Small stakes (Min)

Note: The distributional assumption is that the unobserved is N(). OPG asymptotic standard errors in column on the right. We define 'Large stakes' : 1) a lottery including at least one prize in excess of 250,000 Euros (max option); 2) a lottery with mean payoffs of at least 75,000 Euros (mean option). We define 'Small stakes' : 1) a lottery including no prize in excess of 25,000 Euros (min option); 2) a lottery with mean payoffs of at most 20,000 Euros (mean option). Stage 4,5 indicates that the sample is restricted to those stages.

All stakes Large stakes (Mean)

Large stakes (Max)

Small stakes (Mean)

TABLE 5MLE: Uninformative Offers Model at Large and Small Stakes

Dep. Variable: Dep. Variable: Dep. Variable: Dep. Variable: Dep. Variable:

CRRA

Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err.

MEAN 1.3688 0.1254 1.3306 0.1207 1.2865 0.1303 0.7986 0.5399 0.6757 0.6660

Stage 4,5: -- -- 1.2732 0.1002 1.2189 0.1058 -- -- -- --

STD. DEV. 1.2800 0.0842 0.7375 0.1378 0.7585 0.1293 2.0675 0.5399 1.7232 1.0432

Stage 4,5: -- -- 0.5874 0.1395 0.5812 0.1474 -- -- -- --

Log Likelihood -190.19 -53.36 -54.95 -24.81 -8.41No. Obs. 233 95 90 73 32Stage 4,5: log L -- -43.65 -38.57 -- --Stage 4,5: No. Obs -- 66 55 -- --

TABLE 6MLE: Informative Offers Model at Large and Small Stakes

Dep. Variable: Dep. Variable: Dep. Variable: Dep. Variable: Dep. Variable:

All stakes Large stakes (Mean)

Small stakes (Min)

Note: The distributional assumption is that the unobserved is N(). OPG asymptotic standard errors in column on the right. We define 'Large stakes' : 1) a lottery including at least one prize in excess of 250,000 Euros (max option); 2) a lottery with mean payoffs of at least 75,000 Euros (mean option). We define 'Small stakes' : 1) a lottery including no prize in excess of 25,000 Euros (min option); 2) a lottery with mean payoffs of at most 20,000 Euros (mean option). Stage 4,5 indicates that the sample is restricted to those stages.

Large stakes (Max)

Small stakes (Mean)


010

000

2000

030

000

Offe

r

0 10000 20000 30000E(Lottery)

Rejected Offer Accepted OfferLowess fit offer CE gamma=1CE gamma=2 CE gamma=0

Figure 2Nonparametric Fit for Reject/ Accept Decision: Stage 5 Low Stakes

Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err.MEAN

0.6543 0.2022 0.9168 0.0595 0.6461 -- 0.9001 --Individual Covariates:Age 0.5543 6.0880 -0.0010 2.3017(Age)2 -0.3650 13.8660 -0.2195 6.3862(Age)3 -0.1945 10.7007 0.3854 4.9588Female -0.5149 2.1262 -0.0303 0.2942Constant -1.1427 16.9433 0.3076 5.0913Income(/ 10,000) -0.0759 0.7665 0.2467 0.7202Time Covariates:Time trend 0.4454 8.1389 0.1470 3.6673

STD. DEV. 0.2421 0.1407 0.0878 0.1042 0.3120 2.9204 0.1007 0.5762 -0.0087 0.0342 -- -- 0.0150 0.1763 -- --

Log Likelihood -259.7074 -336.7649No. Obs. 250 250

TABLE 7Static Model and CPT with Power Function.

CRRA Dep. Variable: Dep. Variable: Dep. Variable:

Note: The distributional assumption is that the unobserved () are joint N(). OPG asymptotic standard errors in column on the right for columns (1) and (2). For columns (2) the mean of the distribution of and are calculated at the mean of the covariates.

Dep. Variable: (1) (2)


Conclusions People are approximately risk neutral when stakes are

small Reasonable rate of decline of marginal utility of wealth

Rabin’s critique:• “Data sets dominated by modest-risk investment opportunities are

likely to yield much higher estimates of risk aversion than data sets dominated by larger-scale investment opportunities”-- (Rabin, Econometrica 2000) We obtain the opposite, consistently with expected utility theory

• We find that it is reasonable to use the same underlying expected utility function for small and large stakes

People make fewer mistakes when stakes are high

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Risk Aversion and Expected Utility Theory: A Field Experiment with Large and Small Stakes