Stochastic choice under risk

Pavlo Blavatskyy

June 24, 2006

Talk Outline• Introduction• Binary choice between a risky and a degenerate

lottery– Fourfold pattern of risk attitudes– Discrepancy between certainty equivalent and probability

equivalent elicitation methods– Preference reversal phenomenon

• Binary choice between two risky lotteries– Generalized common consequence effect (Allais paradox)– Common ratio effect– Violations of the betweenness

• Fit to experimental data• Conclusion

Introduction

• Repeated choice under risk is often inconsistent– In 31.6% of all cases (Camerer, 1989)– In 26.5% of all cases (Starmer and Sugden, 1989)– In ~25% of all cases (Hey and Orme, 1989)

• Stochastic nature of choice under risk is persistently documented in experimental data

• … but remains largely ignored in the majority of decision theories

Conscious randomization?

• Machina (1985) and Chew et al. (1991): stochastic choice as a result of deliberate randomization – individuals with quasi-concave preferences

(like randomization)– The most preferred lottery is outside the

choice set

• Hey and Carbone (1995): does not fit the data

Models of stochastic choice

• Core deterministic decision theory is embedded into a stochastic choice model – e.g. when estimating the parameters of

decision theory from experimental data

• Three models proposed in the literature

1. Harless and Camerer (1994)

• individuals generally choose among lotteries according to some deterministic decision theory

• …but there is a constant probability that this deterministic choice pattern reverses (as a result of pure tremble)

• Carbone (1997) and Loomes et al. (2002): fails to explain the experimental data

2. Hey and Orme (1994) Random error / Fechner model

• Deterministic decision theory → → net advantage of one lottery over another → → distorted by random errors

• independently and identically distributed errors, zero mean and constant variance – Hey (1995) and Buschena and Zilberman (2000): heteroscedasticity

– Camerer and Ho (1994) and Wu and Gonzalez (1996): choice

probability as a logit function of net advantage

• Loomes and Sugden (1998): predicts too many violations of first order stochastic dominance

3. Loomes and Sugden (1995)

• Individual preferences over lotteries are stochastic

• Represented by random utility

• Sopher and Narramore (2000): variation in individual decisions is not systematic, which strongly supports random error rather than random utility model

So…

• Different models of stochastic choice – generate stochastic choice pattern from a

deterministic core decision theory – successful in explaining some choice

anomalies – not suitable for accommodating all known

phenomena

New theory

• Explain major stylized empirical facts as a consequence of random mistakes

• …that individuals commit when evaluating a risky lottery

• Make explicit predictions about stochastic choice patterns

• …directly accessible for econometric testing on empirical data

Binary choice between a risky and a degenerate lottery

• An individual has deterministic preferences over lotteries L(x1,p1;…,xn,pn ), x1<…<xn

• represented by von Neumann-Morgenstern utility function u:R→R

• Observed binary choices of an individual are, however, stochastic

• …due to random errors that an individual commits when evaluating a risky lottery

…• An individual chooses lottery L over

outcome x for certain if U(L) ≥ u(x)• Perceived expected utility of a lottery U(L)

is equal to…– “true” expected utility of a lottery μL=Σi pi u(xi )

according to individual preferences– plus a random error ξL

• An individual always chooses lottery L over outcome x for certain if U(L) > u(x)

• An individual behaves as if maximizing the perceived expected utility

No transparent errors !

• Assumption 1 (internality axiom) An individual always chooses lottery L over outcome x for certain if outcome x is smaller than x1

• …and an individual always chooses outcome x for certain over lottery L, if outcome x is higher than xn

• → no errors in choice under certainty

Small errors are non-systematic !

• CEL is an outcome s.t. u(CEL)=μL

• Assumption 2 For any ε>0 and a risky lottery L s.t. CEL ε [x1,xn] the following events are equally likely to occur:– Lottery L is chosen over outcome CEL - ε for

certain but not over outcome CEL for certain

– Lottery L is chosen over outcome CEL for certain but not over outcome CEL + ε for certain

Results

• Assuming that individual maximizes perceived expected utility…

• …together with assumptions 1-2 about the distribution of random errors…

• we can explain:– Fourfold pattern of risk attitudes– Discrepancy between certainty equivalent and

probability equivalent elicitation methods– Preference reversal phenomenon

Fourfold pattern of risk attitudes

• Empirical observation that individuals often exhibit risk aversion when dealing with probable gains or improbable losses

• … and the same individuals often exhibit risk seeking when dealing with improbable gains or probable losses

• e.g. a simultaneous purchase of insurance and public lottery tickets

How do we explain?

Discrepancy between certainty equivalent and probability equivalent

elicitation methods• Consider lottery L(x1,0.5;x2,0.5 )

• Outcome c is a minimum outcome that an individual is willing to accept in exchange for lottery L

• Probability p is the highest probability s.t. an individual is willing to accept outcome c for certain in exchange for lottery L’(x1,1-p;x2,p )

…

• Any deterministic decision theory predicts that p = 0.5

• Hershey and Schoemaker (1985): individuals, who initially reveal high c also declare that p > 0.5 one week later

• Robust finding both for gains and losses

Explanation (rather logic behind it)

• An individual makes random mistakes when evaluating a risky lottery L

• → the perceived CE of L is equally likely to be below or above certain outcome ML

– For risk-neutral guy, ML is simply (x1+x2)/2

• Accidentally, an individual has too high realization of the perceived CE, c >> ML

• Now he or she searches for PE of this high outcome c

Explanation, continued

• An individual is most likely to associate the sure outcome c with a lottery L’

• …whose perceived certainty equivalent is equally probable to be below or above c

• For such lottery p>0.5– If it were exactly 0.5 lottery L’ coincides with

original lottery L– Median of distribution of CE of L is ML

– But c >> ML

The preference reversal phenomenon

• 2 lotteries of similar expected value • R yields a relatively high outcome with

low probability (a dollar-bet)• S yields a modest outcome with

probability ~1 (a probability-bet) • Individuals often choose S over R in a

direct binary choice• … and simultaneously reveal a higher

min selling price for R

Binary choice between two risky lotteries

• Individual chooses lottery L over lottery L’ if μL+ξL ≥ μL’+ξL’ or μL+ξL,L’ ≥ μL’

• The same choice rule as in the Fechner model• But different assumptions about the

distribution of an error term ξL,L’

• Large positive errors ξL,L’ ≥u(xn)-u(y1)+μL’ – μL

large negative errors ξL,L’ ≤ u(x1)-u(ym)+μL’ – μL

are not possible due to A1

Small errors are non-systematic (A2)

• Error term ξL,L’ is symmetrically distributed on the utility scale

• Assumption 2a For any ε>0 and any

lotteries L(x1,p1;…xn,pn) & L’(y1,q1;…ym,qm)

such that ε≤u(xn)-u(y1)+μL’ – μL and

-ε≥u(x1)-u(ym)+μL’ – μL :

prob(-ε ≤ ξL,L’ ≤ 0)=prob( 0 ≤ ξL,L’ ≤ ε)

No error for “almost sure things”

• A1 implies that an individual makes no errors when choosing among degenerate lotteries

• When choosing between “almost sure things”, the dispersion of random errors is progressively narrower

• … the closer are risky lotteries to the degenerate lotteries

Formally…

Results

• “Fechner” choice rule together with assumptions 1, 2a and 3 explains:– Common consequence effect (Allais paradox)– Common ratio effect– Violations of betweenness (Blavatskyy, EL, 2006)

Fit to experimental data

• Estimate: – stochastic decision theory (errors drawn

from truncated normal distribution) and – RDEU (CPT) + standard Fechner error

• on experimental data:– Loomes and Sugden (1998), 92 subjects

make 46 binary choices twice– Hey and Orme (1994), 80 subjects make

100 binary choices twice

0

2

4

6

8

10

12

14

16

Vuong's likelihood ratio statistic

Number of subjects

Predictions of stochastic decision theory and RDEU are not significantly different

Prediction of RDEU is significantly better at significance level...

p≤0.1% p≤1% p≤5% p≤10%

Prediction of stochastic decision theory is significantly better at significance level...

p≤10% p≤5% p≤1% p≤0.1%

Loomes and Sugden (1998)

0

2

4

6

8

10

12

14

16

18

20

Vuong's adjusted likelihood ratio statistic

Number of subjects

Akaike information criterion Schwarz information criterion

Predictions of stochastic decision theory and RDEU are not significantly different

Prediction of RDEU is significantly better at significance level...

p≤0.1% p≤1% p≤5% p≤10%

Prediction of stochastic decision theory is significantly better at significance level...

p≤10% p≤5% p≤1% p≤0.1%

Hey and Orme (1994)

Conclusion

• Individuals often make inconsistent decisions in repeated choice under risk

• => preferences are stochastic (random utility) => observed randomness is due to random errors

• Existing error models: – prob. of error is constant (Harless and Camerer, 1994) – distribution of errors is constant (Hey and Orme, 1994)

• Too simple:– No errors are observed in choice between “sure things”– 20% - 30% of inconsistencies in non-trivial choice