1. New Views on Risk Attitudes Peter P. Wakker Economics University of Amsterdam € 100 € 0€ 0 ½ ½ or € 50 for sure What would you rather have? Such gambles occur in games with friends. More seriously:

2. 2 - Whether you can study medicine in the Netherlands; - In the US in the 1960s, whether you had to serve in Vietnam (only for men …) Even more seriously: Investments, insurance, medical treatments, etc. etc. This lecture is on the history of risk-theory. In public lotteries, casinos, and horse races.

3. 3 1)General modeling of risk attitude. Is it determined by: - sensitivity towards outcomes (utility); - sensitivity towards chance (probability weighting)? 2) Particular form of risk attitude. Is risk-aversion - universally valid (modulo noise); - systematically violated? Two questions/lines-of-talk:

4. 4 Expected value Simplest way to evaluate risky prospects: € 100 € 0€ 0 ½ ½  ½  100 + ½  0 = 50 General: x1x1 xnxn p1p1 pnpn............  p 1 x 1 +... + p n x n

5. 5 Risk aversion! Falsification of expected value. To explain it, “expected utility.” However, empirical observations:  € 100 € 0€ 0 ½ ½ € 50

6. 6 Departure from objectivity. U is subjective index of risk attitude. Bernoulli (1738). Expected utility is the classical economic risk theory. x1x1 xnxn p1p1 pnpn............  p 1 x 1 +... + p n x n U( )

7. 7 Theorem (Marshall 1890). Risk aversion holds if and only if utility U is concave. Risk aversion in general: U € U is used as the subjective index of risk attitude! x1x1 xnxn p1p1 pnpn............  p 1 x 1 +... + p n x n

8. 8 Psychologists objected: U = sensitivity towards money ≠ risk attitude. Line (1) of this talk: the general modeling of risk attitude.

9. 9 Intuition:risk attitude (also) in terms of processing of probabilities. x1x1 xnxn p1p1 pnpn............  p 1 U(x 1 )+... + p n U(x n ) w( ) w p 0 01 1 w(0) = 0, w(1) = 1, w is increasing. p w(p)

10. 10 Prob. weighting already considered in 1950s (Ward Edwards). Called subjective expected utility (unfortunate term).  's argument intuitive, not theoretical. economists: Such argumentation is an error! Subj. exp. ut. theory never became “big.” Lola Lopes (1987): “Risk attitude is more than the psychophysics of money.” utility

11. 11 Economic arguments for universal risk aversion: 1)diminishing marginal utility is intuitively plausible; 2)concave utility needed for existence of equilibria; 3)no concave U  market for lotteries; Line (2) of this talk: risk aversion.

12. 12 about risk-seeking individuals:... since experience shows that they are likely to engender a restless, feverish character, unsuited for steady work as well as for the higher and more solid pleasures of life. Marshall, A. (1920) Principles of Economics

13. 13 Public lotteries!?!? Friedman & Savage (1948): Problem: U €

14. 14 I will not dwell on this point extensively, emulating rather the preacher, who, expounding a subtle theological point to his congregation, frankly stated: Brethren, here there is a great difficulty; let us face it firmly and pass on. Psychologists: ????? Arrow (1971, p.90) (about lotteries)

15. 15 End of seventies: renewed interest in probability weighting, a.o. because of violations of EU. A.o. by Handa (1978, J. of Pol. Econ y ), Kahneman & Tversky (1979, Econometrica, "prospect theory"). Prominent economic journals... ! Back to line (1), the general modeling of risk attitude.

16. 16 Of those, Fishburn (1978, JPE) was published. (Among non-published reactions, one by the unknown Australian John Quiggin.) Prospect theory is an exceptionally big succes; theoretically problematic. To Handa (1978), the JPE received some 10 comments!

17. 17 Amazing, that model could survive in the psychological literature for 30 years... Probability-weighting violates stochastic dominance!

18. 18 Only, one should weight the "right“ probabilities. Not probability at: a specific outcome, but probability at: at least an outcome. Yet, "risk-attitude through probability weighting" is good intuition.

19. 19 Evaluation of lottery with x 1  …  x n  0: w(p 1 )U(x 1 ) + ( w(p 2 +p 1 ) - w(p 1 ) ) *U(x 2 ) +... ( w(p j +...+p 1 ) - w(p j-1 +...+p 1 ) ) *U(x j ) +... ( w(p n +...+p 1 ) - w(p n-1 +...+p 1 ) ) *U(x n ) Idea of Quiggin (1981), Rank-Dependent Utility. x1x1 xnxn p1p1 pnpn............

20. 20 In the beginning, economists' views: Risk-aversion is universal. U concave and prob. weighting w similar. Impulses from empirical investigations by psychologists (Tversky and others). Back to line 2, risk aversion.

21. 21 Small chances at large gains Large chances at small losses Amazing, that “universal” risk aversion could survive in the economics literature for 30 years … Systematic risk-seeking for:

22. 22 Tversky, A. & D. Kahneman (1992), “Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty 5, 297-323. Synthesis:

23. 23 Risk-attitudes in terms of - utilities ánd - probability weighting (- ánd loss aversion). Risk-aversion prevailing, but, systematic deviations. Reference point ("framing"). Theory combines - descriptive force of prospect theory - theoretical force of econ. theories. Cumulative prospect theory:

24. 24 1. Classical econ s : Expected utility; Risk at- titude = U( € ) ( Bernoulli 1738, Marshall 1890). 2.  s : risk attitude also = w(p) (Edwards, 1954). Took wrong p’s. 3. Econ s : Take right ("cumulative“) p’s (Quiggin, 1981). Thought universal risk aversion; convex/cave. 4.  s : diminishing sensitive iso risk aversion (Tversky & Kahneman, 1992); S-shaped. Synthesis: Cumulative prospect theory Summary: