new views on risk attitudes peter p. wakker economics university of amsterdam € 100 € 0€ 0 ½...
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New Views on Risk AttitudesPeter P. Wakker
EconomicsUniversity of Amsterdam
€ 100
€ 0
½
½
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
€ 0
½
½
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:
€ 100
€ 0
½
½
€ 50
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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
.
.
.... 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: