1. Axiomatic Theory of Probabilistic Decision Making under Risk Pavlo R. Blavatskyy University of Zurich April 21st, 2007

2. Outline Introduction Framework Axioms Representation Theorem Implications Conclusions

3. Introduction Experimental studies of repeated decision making under risk => individual choices are often contradictory –Camerer (1989) reports that 31.6% of subjects reversed their choices –Starmer and Sugden (1989) find that 26.5% of all choices are reversed –Hey and Orme (1994) report an inconsistency rate of 25% –Wu (1994) finds that 5% to 45% of choice decisions are reversed –Ballinger and Wilcox (1997) report a median switching rate of 20.8%

4. Introduction continued Majority of decision theories are deterministic –Exception Machina (1985) and Chew et al. (1991) They predict that repeated choice is always consistent (except for decision problems where an individual is exactly indifferent) Common approach is to embed a deterministic decision theory into a model of stochastic choice –tremble model of Harless and Camerer (1994) –Fechner model of random errors (e.g. Hey and Orme, 1994) –Random utility model (e.g. Loomes and Sugden, 1995)

5. This Paper Individuals do not have a unique preference relation on the set of risky lotteries Individuals possess a probability measure that captures the likelihood of one lottery being chosen over another lottery A related axiomatization of choice probabilities –Debreu (1958) –Fishburn (1978)

6. Framework A finite set of all possible outcomes (consequences) –Outcomes are not necessarily monetary payoffs A risky lottery is a probability distribution on X A compound lottery The set of all risky lotteries is denoted by Λ

7. Framework, continued An individual possesses a probability measure on –Choice probability denotes a likelihood that an individual chooses L 1 over L 2 in a repeated binary choice A deterministic preference relation can be easily converted into a choice probability –If an individual strictly prefers L 1 over L 2, then –If an individual strictly prefers L 2 over L 1, then –If an individual is exactly indifferent, then

8. Axioms Axiom 1 (Completeness) For any two lotteries there exist a choice probability and a choice probability – for any –Only two events are possible: either choose L 1 or choose L 2

9. Axioms, continued Axiom 2 (Strong Stochastic Transitivity) For any three lotteries if and then Axiom 3 (Continuity) For any three lotteries the sets and are closed

10. Axioms, continued Axiom 4 (Common Consequence Independence) For any four lotteries and any probability : If two risky lotteries yield identical chances of the same outcome (or, more generally, if two compound lotteries yield identical chances of the same risky lottery) this common consequence does not affect the choice probability

11. Axioms, continued Axiom 5 (Interchangeability) For any three lotteries if then If an individual chooses between two lotteries at random then he or she does not mind which of the two lotteries is involved in another decision problem

12. Representation Theorem Theorem 1 (Stochastic Utility Theorem) Probability measure on satisfies Axioms 1-5 if and only if there exist an assignment of real numbers to every outcome,, and there exist a non-decreasing function such that for any two risky lotteries :

13. Implications Function has to satisfy a restriction for every, which immediately implies that If a vector and function represent a probability measure on then a vector and a function represent the same probability measure for any two real numbers a and b,

14. Special cases Fechner model of random errors –function is a cumulative distribution function of the normal distribution with mean zero and constant standard deviation Luce choice model –function is a cumulative distribution function of the logistic distribution where is constant Tremble model of Harless and Camerer (1994) –function is the step function

15. Empirical paradoxes Unlike expected utility theory, stochastic utility theory is consistent with systematic violations of betweenness and a common ratio effect …but cannot explain a common consequence effect

16. Conclusions Individuals often make contradictory choices –Either individuals have multiple preference relations on Λ (random utility model) –or individuals have a probability measure on Choice probabilities admit a stochastic utility representation if and only if they are complete, strongly transitive, continuous, independent of common consequences and interchangeable Special cases: Fechner model of random errors, Luce choice model and a tremble model of Harless and Camerer (1994)