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## Stochastic choice under risk

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Presentation Transcript

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

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