Expected Utility Theory
The most famous example of such a theory was
published by von Neumann and Morgenstern (1944)
(VNM).
They proposed it as a “normative” theory of
behaviour. That is, classical utility theory as a
“normative” theory of behaviour.
Classical utility theory was not intended to describe
how people actually behave, but how people would
behave if they followed certain requirements of
rational decision-making.
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1:33 PM
Expected Utility Theory
One of the main purposes of such a theory was to
provide an explicit set of assumptions or axioms that
underlie rational decision-making.
Once VNM specified these axioms, decision
researchers were able to compare the mathematical
predictions of expected utility theory with the
behaviour of real decision makers.
When researchers documented violations of an
axiom, they often revised the theory and made new
predictions. In this way, research on decision making
cycled back and forth between theory and
observation.
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Expected Utility Theory
What are the axioms of rational decision-making?
Most formulations of expected utility theory are
based at least in part on some subset of the
following six principles (briefly introduced last
week):
1.
Ordering of alternatives. First of all, rational
decision makers should be able to compare any two
alternatives. They should either prefer one
alternative to the other, or they should be
indifferent to them.
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Expected Utility Theory
2.
Dominance. Rational actors should never adopt
strategies that are “dominated” by other strategies
(here adopting a strategy is equivalent to making a
decision).
A strategy is weakly dominant if, when you compare
it to another strategy, it yields a better outcome in
at least one respect and is as good or better than
the other strategy in all other respects (where
“better” means that it leads to an outcome with
greater utility).
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Expected Utility Theory
A strategy is strongly dominant if, when you compare
it to another strategy, it yields an outcome that is
superior in every respect.
For example, Car A strongly dominates Car B if it is
superior in mileage, cost and looks and is weakly
dominant if it gets better mileage than Car B but is
equivalent in cost and looks.
According to expected utility theory, perfectly
rational decision makers should never choose a
dominated strategy, even if the strategy is only
weakly dominated.
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Expected Utility Theory
3.
Cancellation. If two risky alternatives include
identical and equally probable outcomes among their
possible consequences, then the utility of these
outcomes should be ignored in choosing between the
two options.
In other words, a choice between two alternatives
should depend only on those outcomes that differ,
not on outcomes that are the same for both
alternatives. Common factors should cancel out.
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Expected Utility Theory
4. Transitivity. If a rational decision maker prefers
Outcome A to Outcome B, and Outcome B to
Outcome C, then that person should prefer
Outcome A to Outcome C.
a>b and b>c ⇒ a>c
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Expected Utility Theory
5.
Continuity. For any set of outcomes, a decision
maker should always prefer a gamble between the
best and worst outcomes to a sure intermediate
outcome if the odds of the best outcome are good
enough.
This means, for example, that a rational decision
maker should prefer a gamble between £100 and
complete financial ruin to a sure gain of £10,
provided the odds of financial ruin are one to a
googol.
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Expected Utility Theory
6.
Behaviour. The Invariance Principle stipulates
that a decision maker should not be affected by the
way alternatives are presented.
For example, a rational decision maker should have no
preference between a compound gamble (e.g. a two
stage lottery with a fifty percent chance of success
on each stage and a £100 payoff if both stages yield
successful outcomes) and the simple gamble to which
it can be reduced (i.e. a one stage lottery with a
twenty five percent chance of winning £100).
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Expected Utility Theory
VNM proved mathematically that when decision
makers violate principles such as these, expected
utility is not maximised.
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Expected Utility Theory
To take an example, suppose that in violation of the
Transitivity
Principle,
you
have
intransitive
preferences for Outcomes A, B and C. You prefer
Outcome A to Outcome B, Outcome B to Outcome C
and Outcome C to Outcome A.
A►B B►C C►A
A►B B►C C►A
This means that I should be able to give you
Outcome C and offer – say, for a penny – to take
back Outcome C and give you Outcome B. Because
you prefer Outcome B to Outcome C, you would
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undoubtedly accept my offer and pay the penny.
Expected Utility Theory
A►B B►C C►A
Now you have Outcome B. In the same way, I should
be able to offer – for another penny – to take back
Outcome B and give you Outcome A. Because you
prefer Outcome A to Outcome B, you would
undoubtedly accept my offer and pay the penny.
A►B B►C C►A
Now you have Outcome A. In the same way, I should
be able to offer – for another penny – to take back
Outcome A and give you Outcome C. Because you
prefer Outcome C to Outcome A, you would
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undoubtedly accept my offer and pay the penny.
Expected Utility Theory
The result is that you are back where you started
from, with Outcome C, minus three pennies (or £3,
or £3000, or …).
In other words, I can continue to use this
intransitivity in preferences as a “money pump” until
your supply of money runs out.
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Expected Utility Theory
Kahneman and Tversky (1979) argue that utility
theory is not an adequate descriptive model and
propose an alternative account of choice under risk.
In expected utility theory risk aversion is equivalent
to the concavity of the utility function.
The prevalence of the purchase of insurance against
large and small losses has been regarded by many as
strong evidence for the concavity of the utility
function for money.
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Expected Utility Theory
Why otherwise would people spend so much money to
purchase insurance policies at a price that exceeds
the expected actuarial cost?
However,
an
examination
of
the
relative
attractiveness of various forms of insurance does
not support the notion that the utility function for
money is concave everywhere.
For example, people often prefer insurance
programmes that offer limited coverage with low or
zero deductible (the amount of expenses that must
be paid out of your pocket ) over comparable policies
that offer higher maximal coverage with higher
deductibles – contrary to risk aversion (Fuchs 1976).
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Expected Utility Theory
As Rabin and Thaler (2001) point out, expected
utility theory is used ubiquitously in theoretical and
empirical economic research.
Despite its central place, however, we will show that
this explanation for risk aversion is not plausible in
most cases where economists invoke it.
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Expected Utility Theory
To help see why we make such a claim, suppose we
know that Johnny is a risk-averse expected utility
maximiser, and that he will always turn down the
50-50 gamble of losing $10 or gaining $11.
What else can we say about Johnny?
Specifically, can we say anything about bets Johnny
will be willing to accept in which there is a 50
percent chance of losing $100 and a 50 percent
chance of winning some amount $Y?
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Expected Utility Theory
Consider the following multiple-choice quiz:
From the description above, what is the biggest Y such
that we know Johnny will turn down a 50-50 lose
$100/win $Y bet?
a) $110
b) $221
c) $2,000
Make a note of
d) $20,242
your choice.
e) $1.1 million
f) $2.5 billion
g) Johnny will reject the bet no matter what Y is.
h) We can’t say without more information
about Johnny’s utility function
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Expected Utility Theory
You are reminded that we are asking what is the
highest value of Y making this statement true for all
possible preferences consistent with Johnny being a
risk-averse expected utility maximiser who turns
down the 50/50 lose $10/gain $11 for all initial
wealth levels.
You can make no ancillary assumptions, for instance,
about the functional form of Johnny’s utility
function beyond the fact that it is an increasing and
concave function of wealth.
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Expected Utility Theory
Consider the following multiple-choice quiz:
From the description above, what is the biggest Y such
that we know Johnny will turn down a 50-50 lose
$100/win $Y bet?
a) $110
b) $221
c) $2,000
Make a note of
d) $20,242
your revised choice.
e) $1.1 million
f) $2.5 billion
g) Johnny will reject the bet no matter what Y is.
h) We can’t say without more information
about Johnny’s utility function
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Expected Utility Theory
Did you guess a, b, or c? If so, you are wrong.
a) $110
b) $221
Try again
c) $2,000
d) $20,242
e) $1.1 million
f) $2.5 billion
g) Johnny will reject the bet no matter what Y is.
h) We can’t say without more information
about Johnny’s utility function
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Expected Utility Theory
Did you guess d?
Maybe you figured we wouldn’t be asking if the
answer weren’t shocking, so you made a ridiculous
guess like e, or maybe even f. If so, again you are
wrong.
Perhaps you guessed h, thinking that the question is
impossible to answer with so little to go on. Wrong
again.
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Expected Utility Theory
Recall
a) $110
b) $221
c) $2,000
d) $20,242
e) $1.1 million
f) $2.5 billion
g) Johnny will reject the bet no matter what Y is.
h) We can’t say without more information
about Johnny’s utility function
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Expected Utility Theory
The correct answer is g (Johnny will reject the bet
no matter what Y is.). Johnny will turn down any bet
with a 50 percent risk of losing at least $100, no
matter how high the upside risk.
Johnny would, of course, have to be insane to turn
down bets like d, e, and f. So, what is going on here?
In conventional expected utility theory, risk aversion
comes solely from the concavity of a person’s utility
defined over wealth levels (Rabin 2000).
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Expected Utility Theory
Johnny’s risk aversion over the small bet means,
therefore, that his marginal utility for wealth must
diminish incredibly rapidly.
This means, in turn, that even the chance for
staggering gains in wealth provide him with so little
marginal utility that he would be unwilling to risk
anything significant to get these gains.
The problem here is much more general.
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Expected Utility Theory
Using expected utility to explain anything more than
economically negligible risk aversion over moderate
stakes such as $10, $100, and even $1,000 requires a
utility-of-wealth function that predicts absurdly
severe risk aversion over very large stakes.
Since this theory and its generalisation, the
cumulative prospect theory (Tversky and Kahneman
1992) are essentially mathematical theories they will
not be further explored here.
See case 22 on the module web site.
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Expected Utility Theory
See case 22
on the module
web site.
7.27
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Expected Utility Theory
Conventional expected utility theory is simply not a
plausible explanation for many instances of risk
aversion that economists study.
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Comparison Of Utility And
Prospect Theory
This discussion closely follows that of Kahneman
(2011). As we have seen, prospect theory differs
from utility theory in the relationship it suggests
between probability and decision weight. In utility
theory, decision weights and probabilities are the
same.
Kahneman (2011) and Tversky found out that a much
more accurate description of human decision making
can be obtained if we assume that, instead of
maximizing the expected gain, people maximize a
weighted gain, with weights determined by the
corresponding probabilities.
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Comparison Of Utility And
Prospect Theory
The decision weight of a sure
thing is 100, and the weight
that corresponds to a 90%
chance is exactly 90, which
is 9 times more than the
decision weight for a 10%
chance. Simply linear.
The empirical transformation
from probabilities to
weights takes the following
form
Probability
Weight
0 1 2 5 10 20 50 80 90 95 98 99 100
0 5.5 8.1 13 19 26 42 60 71 79 87 91 100
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Comparison Of Utility And
Prospect Theory
probability
weight
probability
weight
0
0
1
5.5
80
60.1
2
8.1
90
71.2
5
13.2
95
79.3
10
18.6
98
87.1
20
26.1
99
91.2
50
42.1
100
100
In prospect theory, variations of probability have
less effect on decision weights. An experiment that
was mentioned earlier found that the decision weight
for a 90% chance was 71.2 and the decision weight
for a 10% chance was 18.6.
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Comparison Of Utility And
Prospect Theory
The ratio of the probabilities was 9.0 (90/10), but
the ratio of the decision weights was only 3.83
(71.2/18.6), indicating insufficient sensitivity to
probability in that range. In both theories, the
decision weights depend only on probability, not on
the outcome. Both theories predict that the decision
weight for a 90% chance is the same for winning
$100, receiving a dozen roses, or getting an electric
shock. This theoretical prediction turns out to be
wrong.
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Allais Paradox
The following paradox was developed by Allais
(1953).
Which do you prefer?
Experiment 1
Gamble 1A
Winnings
$1 million
Chance
100%
Gamble 1B
Winnings
Chance
$1 million
89%
Nothing
1%
$5 million
10%
Most prefer 1A
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Allais Paradox
In notation choosing 1A over 1B because
U(1 mill.) > 0.89 U(1 mill.) +0.01 U(0) + 0.1 U(5 mill.)
So 0.11 U(1 mill.) > 0.01 U(0) + 0.1 U(5 mill.)
Experiment 1
Gamble 1A
Winnings
$1 million
Chance
100%
Gamble 1B
Winnings
Chance
$1 million
89%
Nothing
1%
$5 million
10%
Most prefer 1A
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Allais Paradox
The following paradox was developed by Allais
(1953).
Which do you prefer?
Experiment 2
Gamble 2A
Gamble 2B
Winnings
Chance
Winnings
Chance
Nothing
89%
Nothing
90%
1 million
11%
$5 million
10%
Most prefer 2B
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Allais Paradox
In notation choosing 2B over 2A because
0.9 U(0) + 0.1 U(5 mill.) > 0.89 U(0)+ 0.11 U(1 mill.)
So 0.01 U(0) + 0.1 U(5 mill.) > 0.11 U(1 mill.)
Experiment 2
Gamble 2A
Gamble 2B
Winnings
Chance
Winnings
Chance
Nothing
89%
Nothing
90%
1 million
11%
$5 million
10%
Most prefer 2B
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Allais Paradox
From 1
From 2
0.11 U(1 mill.) > 0.01 U(0) + 0.1 U(5 mill.)
0.01 U(0) + 0.1 U(5 mill.) > 0.11 U(1 mill.)
Whoops!
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Allais Paradox
There have been many tests of the descriptive
validity of the axioms of expected utility theory
using money outcomes. Such tests are relatively
uncommon with respect to health outcomes. This is
unfortunate, because the standard gamble –
considered by many health economists to be the gold
standard for cardinal health state value assessment –
is implied from the axioms of expected utility.
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Allais Paradox
The classic Allais paradox, which predicts a
systematic violation of the independence axiom, was
tested in the context of health outcomes, as
described below. Seventeen of 38 participants
demonstrated strict violations of independence, with
14 of these violating in the direction predicted by
Allais. The violations were thus significant and
systematic. Moreover, the participants’ qualitative
explanations for their behaviour show seemingly
rational and not inconsistent reasoning for the
violations. This evidence offers a further challenge
to the descriptive validity of expected utility (Oliver
2003).
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Allais Paradox
What appeared to happen is that when the
participants were asked to rate an option that gives a
positive outcome for certain against an option that
has a small probability of immediate death, they
often focused upon the small probability of
immediate death, which may have induced risk averse
behaviour.
For example
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Allais Paradox
With an identical difference in the percentage
chance of immediate death between the two options,
but with a large chance of death in both options, the
participants often appeared to attach less weight to
the probability of death and were more likely to base
their preference on the option that gave the best
possible outcome, which resulted in seemingly risk
seeking behaviour.
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Allais Paradox
These explanations are consistent with both loss
aversion and the over weighting of small
probabilities. By increasing the probability of death
in both treatment options across contexts, and thus
by omitting certainty (and negating the certainty
effect), the prominent attribute in the contexts
switched, for many participants, from probability to
outcome, which appears to explain at least some of
the violations of independence (Oliver 2003a).
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Expected Utility Theory
The study of risky decision making has long used
monetary gambles to study choice, but many everyday
decisions do not involve the prospect of winning or losing
money. Monetary gambles, may be processed and
evaluated differently than gambles with nonmonetary
outcomes. Whereas monetary gambles involve numeric
amounts that can be straightforwardly combined with
probabilities to yield at least an approximate
“expectation” of value, nonmonetary outcomes are
typically not numeric and do not lend themselves to easy
combination with the associated probabilities. Compared
with monetary gambles, the evaluation of nonmonetary
prospects typically proves less sensitive to changes in the
probability range, which, among other things, can yield
preference reversals (McGraw et al. 2010).
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Expected Utility Theory
For a list of useful references, see Smith and von
Winterfeldt (2004).
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Expected Utility Theory
Gollier and Muermann (2010) propose a new decision
criterion under risk in which individuals extract both
utility from anticipatory feelings before the event
and disutility from disappointment after the event.
They propose a new decision criterion under
uncertainty by allowing individuals to extract utility
from dreaming about the future and disutility from
being disappointed after the event. With such an
approach, optimal beliefs balance the benefits of
higher expectations against the costs of worse
decision making and are necessarily biased toward
optimism. They showed that a larger weight on
savouring (prior to the event) increases risk aversion
and hence reduces the allocation in the risky asset. 7.45
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Expected Utility Theory
This is an alternative to “the classical expected
utility model, decision makers are assumed to be
stonemen”. In their work they take into account of
both anticipatory feelings and disappointment.
Disappointment theory was first introduced by Bell
(1985). Bell observes that the effect of a salary
bonus of $5,000 on the worker’s welfare depends
upon whether the worker anticipated no bonus or a
bonus of $10,000. Bell (1985) builds a theory of
disappointment on this observation, taking the
anticipated payoff as external. The new model
combines Bell’s (1985) disappointment theory with
Akerlof and Dickens’ (1982) notion of anticipatory
feelings.
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Expected Utility Theory
There exists experimental support for the trade-off
between optimism and disappointment that underlies
the formation of beliefs in the model. Mellers et al.
(1997) and Shepperd and McNulty (1998, 2002)
provide evidence that people’s feeling about an
outcome is determined in part by counterfactual
thinking — such as disappointment — which can come
from expectations about the future. In an
experiment with recreational basketball players,
McGraw et al. (2004) show that overconfidence while
making shots is negatively correlated with the
average pleasure of the outcome.
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The Six Laws of Experienced Utility
Satisfaction in experiencing the future depends on
decisions made today. Baucells and Sarin (2013)
consider six well-known psychological laws governing
satisfaction – “The Six Laws of Experienced Utility”.
A very brief summary is given here, see the original
paper for an extensive bibliography and much more
mathematical detail.
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The Six Laws of Experienced Utility
1
Relative Comparison - The first law states
that experienced utility is determined by reality
minus expectations. Frijda (1988) calls relative
comparison the law of change. Kahneman and Tversky
(1979)
and
Tversky
and
Kahneman
(1991)
demonstrate that the carrier of utility is the change
from a reference level and not the final outcome.
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The Six Laws of Experienced Utility
2
Motion of Expectations - The second law of
experienced utility asserts that expectations change.
Three
factors
affect
expectations.
First,
expectations can be a function of the consumption of
our peer group (social comparison). Second,
expectations can be a function of our past
consumption (adaptation). Third, expectations can be
changed by seeing reality in a new way (reframing).
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The Six Laws of Experienced Utility
3
Aversion to Loss - The third law of
experienced utility says that losses are felt more
keenly than equivalent gains.
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The Six Laws of Experienced Utility
4
Diminishing Sensitivity - Experienced utility is
not proportional to the difference between
consumption and the habituation level; rather, the
increase in experienced utility slows as consumption
moves further from the habituation level. This
implies that each additional unit of consumption gives
a smaller increment of satisfaction. Simply put, this
law states that experienced utility cannot be easily
taken to blissful or tragic extremes. The increase in
experienced utility slows as reality moves further
from the reference point. It is well known in
psychophysics that doubling the stimulus does not
always double the intensity of the response.
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The Six Laws of Experienced Utility
5
Satiation - Satiation is one of the lingering
effects that recent consumption has on current
consumption. Recent consumption reduces the
experienced utility of subsequent consumption, and
recent abstinence increases the experienced utility
of subsequent consumption.
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The Six Laws of Experienced Utility
6
Projection - People forecast that future
preferences and feelings will be more similar to their
current preferences and feelings than they actually
will be. This implies that when we predict the
satisfaction of future consumption, our prediction is
not very different from the satisfaction we would
experience if we were to realize this consumption
now. For example, shoppers who are hungry overbuy
food because when they are in a hungry state, food
— even if it is for future consumption — seems more
attractive.
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The Six Laws of Experienced Utility
Similarly, they may choose a variety of snacks today
for consumption in future periods. Projecting today’s
static preferences, the same snack over and over
seems very satiating, and the way to counteract this
is by choosing a high level of variety. But satiation
wanes with the passage of time, and when the choice
is made sequentially, consumers may stick with the
same snack.
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Next Week
Preferences In Ambiguity Aversion
7.56
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