Stochastic choice under risk

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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

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## Stochastic choice under risk

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### 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
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’ – μLand -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
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
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