New Views on Risk Attitudes
Peter P. Wakker
Economics
University of Amsterdam
½
€100
or
€50 for sure
€0
½
What would you rather have?
Such gambles occur in games with friends.
More seriously:
2
In public lotteries, casinos, and horse races.
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.
3
Two questions/lines-of-talk:
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?
4
Simplest way to evaluate risky prospects:
Expected value
½
½
€100
€0
General:
p1
x
. .1
. ..
.
pn xn
 ½100 + ½0
= 50
 p1x1 + ... + pnxn
5
However, empirical observations:
½
½
€100

€50
€0
Risk aversion!
Falsification of expected value.
To explain it, “expected utility.”
Expected utility is the
classical economic risk theory.
p1
x
. .1
. ..
 p1U(x1) + ...
.
pn xn
6
+ pn U(
xn )
Departure from objectivity.
U is subjective index of risk attitude.
Bernoulli (1738).
Risk aversion in general:
p1 x
. .1
. ..

p1x1 + ...
.
xn
pn
7
+ pnxn
Theorem (Marshall 1890). Risk aversion holds
if and only if
utility U is concave.
U
€
U is used as the
subjective index of risk attitude!
8
Line (1) of this talk:
the general modeling of risk attitude.
Psychologists objected:
U
=
sensitivity towards money
≠
risk attitude.
Intuition: risk attitude (also) in terms of
processing of probabilities.
p1 x
. .1
. .. w( p) U(x )+ ... +w( p ) U(x )
1
1
n
n
.
xn
pn
w 1
w(0) = 0,
w(1) = 1,
w is increasing.
w(p)
0
0
p
1
p
9
10
Lola Lopes (1987):
“Risk attitude is more than
the psychophysics of money.”
utility
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.”
11
Line (2) of this talk: risk aversion.
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;
12
Marshall, A. (1920)
Principles of Economics
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.
13
Problem:
Public lotteries!?!?
Friedman & Savage (1948):
U
€
14
Arrow (1971, p.90) (about lotteries)
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: ?????
15
Back to line (1), the general modeling
of risk attitude.
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 ... !
16
To Handa (1978), the JPE received some 10
comments!
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.
17
Probability-weighting violates
stochastic dominance!
Amazing, that model could survive in the
psychological literature for 30 years ...
18
Yet,
"risk-attitude through probability weighting"
is good intuition.
Only, one should weight the "right“
probabilities.
Not
probability at: a specific outcome,
but
probability at: at least an outcome.
Evaluation of lottery
with x1  …  xn  0:
p1
.
.
.
pn
x1
..
.
xn
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.
19
20
Back to line 2, risk aversion.
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).
21
Systematic risk-seeking for:
Small chances at large gains
Large chances at small losses
Amazing, that “universal” risk
aversion could survive in the
economics literature for 30 years
…
22
Synthesis:
Tversky, A. & D. Kahneman (1992),
“Advances in Prospect Theory: Cumulative
Representation of Uncertainty,"
Journal of Risk and Uncertainty 5,
297-323.
Cumulative prospect theory:
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.
23
24
Summary:
1. Classical econs: Expected utility; Risk attitude = 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