New Views on Risk AttitudesPeter P. Wakker

EconomicsUniversity of Amsterdam

€ 100

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or €50 for sure

What would you rather have?

Such gambles occur in games with friends.

More seriously:

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More seriously: - 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.

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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:

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Expected value

Simplest way to evaluate risky prospects:

€ 100

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½100 + ½0 = 50

General:

x1

xn

p1

pn

.

.

.... p1x1 + ... + pnxn

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Risk aversion!

Falsification of expected value.

To explain it, “expected utility.”

However, empirical observations:

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Departure from objectivity.U is subjective index of risk attitude.

Bernoulli (1738).

Expected utility is the classical economic risk theory.

x1

xn

p1

pn

.

.

.... p1 x1 + ... + pn xnU( ) U( )

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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!

x1

xn

p1

pn

.

.

.... p1x1 + ... + pnxn

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Psychologists objected:U=

sensitivity towards money≠

risk attitude.

Line (1) of this talk: the general modeling of risk attitude.

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Intuition:risk attitude (also) in terms ofprocessing of probabilities.

x1

xn

p1

pn

.

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.... p1 U(x1)+ ... + pn U(xn)w( ) w( )

w

p

00 1

1

w(0) = 0, w(1) = 1,w is increasing.

p

w(p)

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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 thanthe psychophysics of money.”

utility

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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.

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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

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Public lotteries!?!?

Friedman & Savage (1948):

Problem:

U

€

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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)

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End of seventies: renewed interest in probability weighting, a.o. because of violations of EU.

A.o. by Handa (1978, J. of Pol. Econy), Kahneman & Tversky (1979, Econometrica, "prospect theory").

Prominent economic journals ... !

Back to line (1), the general modeling of risk attitude.

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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!

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Amazing, that model could survive in the psychological literature for 30 years ...

Probability-weighting violates stochastic dominance!

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Only, one should weight the "right“ probabilities.

Not probability at: a specific outcome,butprobability at: at least an outcome.

Yet, "risk-attitude through probability weighting"is good intuition.

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Evaluation of lottery

with x1 … xn 0: w(p1)U(x1) +

(w(p2+p1) - w(p1))*U(x2) + ...

(w(pj+...+p1) - w(pj-1+...+p1))*U(xj) + ...

(w(pn+...+p1) - w(pn-1+...+p1))*U(xn)

Idea of Quiggin (1981), Rank-Dependent Utility.

x1

xn

p1

pn

.

.

....

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In the beginning, economists' views:

Risk-aversion is universal.U concave and prob. weighting wsimilar.

Impulses from empirical investigations by psychologists (Tversky and others).

Back to line 2, risk aversion.

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Small chances at large gainsLarge chances at small losses

Amazing, that “universal” risk aversion could survive in the economics literature for 30 years …

Systematic risk-seeking for:

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

Synthesis:

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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:

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1. Classical econs: Expected utility; Risk at- titude = U(€) (Bernoulli 1738, Marshall 1890).2. s: risk attitude also = w(p) (Edwards, 1954). Took wrong p’s.3. Econs: 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: