Certainty Equivalent and Stochastic Preferences June 2006 FUR 2006, Rome

1. Certainty Equivalent and Stochastic PreferencesJune 2006FUR 2006, Rome Pavlo Blavatskyy Wolfgang Köhler IEW, University of Zürich

2. Introduction Most decision theories are deterministic although observed choices are stochastic. Most (economic) theories of decision making do not explain failure of procedure invariance. Instead: strong assumptions on reasoning abilities of individuals. Our model has two ingredients: Individuals have stochastic preferences. Individuals solve only binary decision problems. If faced with a complex decision problem, they split it into a sequence of binary decision problems.

3. (Introduction) We are interested in the interpretation of observed choices.We consider two types of decision problems: - binary choice - the determination of certainty equivalents (matching).We assume that subjects have correct incentives to reveal their preferences.

4. The Model Let X be a convex and bounded subset of R. A lottery L is defined on a finite subset Let and Let ≿ be a preference relation defined on the set of lotteries. Assumptions: ≿ is: complete transitive monotone satisfies Convexity Let P be the set of preferences relations. Let be a probability measure on P.

5. Binary choice:Individual draws ≿ according to and chooses accordingly.

6. Binary choice:Individual draws ≿ according to and chooses accordingly. Determination of Certainty Equivalent Different elicitation methods in experiments. Here: Consider situation where subjects are asked to state certainty equivalent. Let be the elicited certainty equivalent. From Convexity follows that

7. (Determination of certainty equivalent) Idea: Individuals split determination of certainty equivalent into sequence of binary decisions. I.e., individuals compare some amount to lottery. Depending on whether amount or lottery is preferred, individuals adjust amount and make new comparison, and so on.

8. (Determination of certainty equivalent) Idea: Individuals split determination of certainty equivalent into sequence of binary decisions. I.e., individuals compare some amount to lottery. Depending on whether amount or lottery is preferred, individuals adjust amount and make new comparison, and so on. Formally: 1. Step Draw 2. Step Draw ≿ according to and compare L and . 3. Step If ~ L then If L (if L) then is replaced by and step 2 is repeated. If the preferred alternative switches, is equal to average of the last two amounts to which lottery has been compared.

9. (The Model) To relate results to empirical findings, we need two technical assumptions on the distribution of preferences. A5 (Symmetry): ≿ ≿

10. (The Model) To relate results to empirical findings, we need two technical assumptions on the distribution of preferences. A5 (Symmetry): ≿ ≿ If let n be the unique integer s.t. and and similar for A6 (Preferences sufficiently stochastic): If , then ≿ and similar for

11. (The Model) Theorem 1 If A1-A6 are satisfied then for any L with If , then Theorem 1 refers to a situation where subjects are asked to state certainty equi-valent (e.g., willingness to pay under BDM-mechanism).

12. Other elicitation procedures Elicitation via second-price auction Price increases/decreases with step-size . Lemma 1 Suppose A1-A6 are satisfied. If the auction is ascending and starts at then If the auction is descending and starts at , then

13. (Other elicitation procedures) Elicitation via a sequence of observed choices Suppose that amounts are equally spaced with distance , that is one of the amounts, that ~ and that a computer program prevents inconsistent choices. Lemma 2 If A1-A6 are satisfied and if subjects start with one of the choices at random and then solve adjacent choice problems, then if and if

14. Explanation of empirical observationsFourfold pattern of risk-attitudes Tversky and Kahneman (1992), Cohen et al. (1985): Subjects make choices between lottery and list of amounts for certain. Find fourfold pattern: - most decisions are riskaverse if likely gain or unlikely loss - most decisions are riskseeking if unlikely gain or likely loss

15. Explanation of empirical observationsFourfold pattern of risk-attitudes Tversky and Kahneman (1992), Cohen et al. (1985): Subjects make choices between lottery and list of amounts for certain. Find fourfold pattern: - most decisions are riskaverse if likely gain or unlikely loss - most decisions are riskseeking if unlikely gain or likely loss Likely gain or unlikely loss Unlikely gain or likely loss Hence Lemma 2 implies that elicitation via list of observed choices generates the fourfold pattern of risk-attitudes.

16. (fourfold pattern of risk-attitudes) Harbaugh et al. (2003): six lotteries (low, medium, and high probability of gain/loss) pricing task: elicitation of certainty equivalents via BDM-procedure choice task: subjects choose between lottery and its expected value

17. (fourfold pattern of risk-attitudes) Harbaugh et al. (2003): six lotteries (low, medium, and high probability of gain/loss) pricing task: elicitation of certainty equivalents via BDM-procedure choice task: subjects choose between lottery and its expected value Find fourfold pattern only in pricing task but not in choice task. Choice behavior is statistically indistinguishable from risk-neutrality. Only 4 of 64 subjects choose according to predictions of fourfold pattern. Shows difference between (single) choice and elicitation of certainty equivalent Our model: predicts fourfold pattern in pricing task (Theorem 1) predicts stochastic choice in choice task but no systematic bias

18. Preference Reversal Standard preference reversal: and Non-standard preference reversal: and Tversky et al. (1990) use ordinal payoff schemes. For each lottery pair, fix amount X (equal or slightly smaller than expected values). Two Tasks: binary choice between $-bet vs. P-bet, $-bet vs. X, and P-bet vs. X. state certainty equivalent for $-bet and P-bet. 45% of response patterns are standard preference reversals. 4% of response patterns are non-standard preference reversals.

19. Distribution of response patterns for standard reversals in Tversky et al. (for decisions withand ) (Preference Reversal) Pattern percent Diagnosis 10.0 Intransitivity 65.5 Overpricing of $-bet 6.1 Underpricing of P-bet 18.4 Over- and Underpricing procedure invariance

20. Distribution of response patterns for standard reversals in Tversky et al. (for decisions withand ) (Preference Reversal) Pattern percent Diagnosis 10.0 Intransitivity 65.5 Overpricing of $-bet 6.1 Underpricing of P-bet 18.4 Over- and Underpricing procedure invariance 1) Since we assume that preferences are stochastic, our model predicts that some fraction of choices violates transitivity. 2) Our model explains why procedure invariance is violated (e.g., Theorem 1 predicts that subjects are likely to overprice the $-bet and to underprice the P-bet). 3) Over-/underpricing explains why standard reversal observed more frequently.

21. Conclusion • Model has two ingredients: • Stochastic preferences • Complex decision problems are split into sequence of binary decision problems • Binary choice can immediately infer preferences • Complex decision problem cannot directly infer preferences, details of • decision problem matter • Model offers explanation for failure of procedure invariance (e.g., preference reversal).