Is Expected Utility a Good Descriptive (Positive) Theory?

1. Is Expected Utility a Good Descriptive (Positive) Theory? Based on “Prospect Theory” by Daniel Kahneman and Amos Tversky, Econometrica 1979; David Kreps “Notes on The Theory of Choice”

2. Von Neumann Morgenstern Axioms • Let L be the set of (simple) lotteries over the set of consequences (outcomes) C={x_1,..,x_N}. • L=(p_1,..,p_N) in Δ^{N-1} is a typical lottery. • Let a degenerate lottery with associated probability 1 on outcome x_i be denoted by L_{x_{i}}.

3. Von Neumann Morgenstern Axioms (Z,X,Y in L) • Reduction of compound lotteries: Z=pX+(1-p)Y implies qX+(1-q)Z~(q+(1-q)p)X+(1-q)(1-p)Y 2. Continuity: Z≽X≽Y implies there is a unique p:X~pZ+(1-p)Y 3. Independence: X~Y implies pX+(1-p)Z~pY+(1-p)Z

4. Von Neumann Morgenstern Axioms (Z,X,Y in L) 4. Existence of the Best and the worst lottery There are lotteries B,W: for any lottery X B≽X≽W 5. Monotonicity p>q implies pB+(1-p)W≻qB+(1-q)W

5. Choose between A: 2500 with probability .33 2400 with probability .66 0 with probability .01 B 2400 with certainty 1. A variation of Allais’s ParadoxWhich Axiom is Violated Here?(N=72) 82%

6. Choose between C: 2500 with probability .33 0 with probability .67 D 2400 with probability .34 0 with probability .66 1. A variation of Allais’s ParadoxWhich Axiom is Violated Here?(N=72) 82%

7. 2. Which Axiom is Violated Here? • Consider a two-stage game. Stage 1. • With probability .75 prize is 0 • With probability .25 stage 2 takes place. Stage 2. Choose between • 4000 with prob .8 • 3000 for sure 78%

8. 2. Which Axiom is Violated Here? • The same subjects were to choose between two lotteries • C: (4000, .2) • D: (3000, .25) 65%

9. In addition to whatever you own you have been given 1000. Choose between A: (1000, ½) B: (500,1) In addition to whatever you own you have been given 2000. Choose between C: (-1000, ½) D: (-500,1) 3. Representation of Outcomes 69% 84%

10. 4. Another Axiom Violation? • Choose from A: $10 for sure B: • $1000 with probability p • Death with probability 1-p Is there such 1>p>0 that will make you choose B?

11. HW assignment • You start up a company named “Normative, Inc.” • You can get a loan from a bank (a start-up fund) and a randomization device (a roulette). • Construct a sequence of lotteries for the subjects described in Examples 1-3 (who violate the Axioms) to make sure you “win” money with certainty.