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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

Stochastic choice under risk

Pavlo Blavatskyy

June 24, 2006

talk outline
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
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
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
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
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
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
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
slide9
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
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
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
slide12
  • 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
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
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
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
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
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)
slide19
  • 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
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
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
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
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
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
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
results1
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
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
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