Rational Choice under Risk and
Uncertainty
Choice under uncertainty
 Imagine that you are about to leave your house and have to decide
whether to take an umbrella or leave it at home. You are concerned that it
might rain. If you do not take the umbrella and it does not rain, you will spend
the day dry and happy; if you do not take the umbrella and it does rain, you
will be wet and miserable. If you take the umbrella, you will be dry no matter,
but carrying the cumbersome umbrella will infringe on your happiness.
 Represent your decision problem in the table.
 Solve for maximin criterion, maximax criterion and minimax-risk criterion.
 Critics of maximin reasoning: Harsanyi’s challenge
 Suppose you live in New York City and are offered two jobs at the same
time. One is a tedious and badly paid job in NYC itself, while the other is a
very interesting and well-paid job in Chicago. The catch is that, if you wanted
the Chicago job, you would have to take a plane from New York to Chicago.
Therefore there would be a very small but positive probability that you might
be killed in a plane accident.
 Maximin reasoning would favor the tedious NYC job. Does it sound quite
right? What do you think?
Choice under risk
 Assume that it is both meaningful and possible to assign probabilities to
outcomes
 The expected value of a gamble is what you can expect to win on the
average, in the long run, when you play the gamble
 Example: You are considering whether to park legally or illegally and
decide to be rational about it. Use negative numbers to represent costs in
your calculations. A) Suppose that a parking ticket costs 30 USD and that the
probability of getting a ticket if you park illegally is 1/5. What is EV of parking
illegally? B) Assuming that you use EV calculations as a guide in life, would it
be worth paying 5 USD in order to park legally?
 St Petersburg paradox: A gamble is resolved by tossing an unbiased coin
as many times as necessary to obtain heads. If it takes only one toss, the
payoff is 2, if it takes two tosses it is 4, if it takes three tosses it is 8, and so
forth. What is EV of the gamble?
 St Petersburg paradox: A gamble is resolved by tossing an unbiased coin
as many times as necessary to obtain heads. If it takes only one toss, the
payoff is 2, if it takes two tosses it is 4, if it takes three tosses it is 8, and so
forth. What is EV of the gamble?
 EV = 1 + 1 + 1 + 1 + … = infinity
 This means that if you try to maximize EV you should be willing to pay any
finite price for this gamble. That does not seem right, which is why the result
is called paradox.
 Expected utility concept
 Assume that marginal utility for money is diminishing
 Suppose u(x) = log(x). Now consequences can be expressed in utilities
instead of dollars. We can now compute the expected utility of the gamble.
 EU is the amount of utility you can expect to gain on the average, in the
long run, when you play the gamble.
 In St Petersburg paradox we have: ½*log(2) + ¼*log(4) + 1/8*log(8) + … =
0.602
 EU maximization is a better guide to behavior than EV maximization
 Expected utility theory and attitudes toward risk
 We can explain people’s attitudes toward risk in terms of the character of
their utility functions
 Suppose you own 2 USD and are offered a gamble giving you a 50 percent
chance of winning a dollar and a 50 percent chance of losing a dollar. Your
utility function is u(x)=x^(1/2), so the marginal utility of money is diminishing.
Should you take a gamble?
 What about utility function u(x)=x?
 What about utility function u(x)=x^2?
 Risk averse – utility function bends downwards
 Risk prone – utility function bends upwards
 Risk neutral – utility function is a straight line
 Certainty equivalent of a gamble G is the number CE that satisfies the
equation: u(CE)=EU(G). The amount of money such that you are indifferent
between receiving the amount for sure and playing the gamble.
 Example: finding CE (graphical method)
 Standard theory of decision under risk – Expected Utility Theory (EUT)
 EUT – generally accepted as a normative model of rational choice and
widely applied as a descriptive model of economic behavior
 Critique of EUT – preferences systematically violate the axioms of EUT
Axioms of EUT
 A prospect (x1,p1;…;xn,pn) – contract that yields outcome xi with
probability pi, where p1+p2+…+pn=1
 Tenets of EUT:
1. U(x1,p1;…;xn,pn)=p1u(x1)+…+pnu(xn) – overall utility of a prospect is the
expected utility of its outcomes
2. U(w+x1,p1;…;w+xn,pn)>u(w) – a prospect is acceptable if the utility
resulting from integrating the prospect with one’s assets exceeds the utility
of those assets alone
3. u is concave function – a person is risk averse if she prefers the certain
prospect (x) to any risky prospect with expected value x
Experiments – demonstrate violations of EUT axioms
(responses of students and university faculty)
Example – hypothetical choices
 Which of the following would you prefer?
A.
50% chance to win 1000
50% chance to win nothing
B.
450 for sure
Certainty Effect
Experiments
 Problem 1: Choose between
A.
2500 with probabilty .33
2400 with probability .66
0 with probability .01
B.
2400 with certainty
Experiments
 Problem 1: Choose between
A. [18]
2500 with probabilty .33
2400 with probability .66
0 with probability .01
B. [82]
2400 with certainty
N=72
Experiments
 Problem 2: Choose between
C.
2500 with probabilty .33
0 with probability .67
D.
2400 with prob. .34
0 with prob. .66
Experiments
 Problem 2: Choose between
C. [83]
2500 with probabilty .33
0 with probability .67
D. [17]
2400 with prob. .34
0 with prob. .66
N = 72
Experiments
 82 percent of the subjects chose B in Problem 1 and 83 percent chose C in
Problem 2
 This result violates EUT
u(2400)>.33u(2500)+.66u(2400) or .34u(2400)>.33u(2500)
Experiments – non-monetary outcomes
 Problem 5: Choose between
A.
50% chance to win a three-week tour of England, France and Italy
B.
A one-week tour of England with certainty
Problem 6: Choose between
C.
5% chance to win a three-week tour of England, France and Italy
D.
10% chance to win a one week tour of England
Experiments – non-monetary outcomes
 Problem 5: Choose between
A. [22]
50% chance to win a three-week tour of England, France and Italy
B. [78]
A one-week tour of England with certainty
Problem 6: Choose between
C. [67]
5% chance to win a three-week tour of England, France and Italy
D. [33]
10% chance to win a one week tour of England
 Certainty effect – people overweight outcomes
that are considered certain, relative to outcomes
which are merely probable
Possibility Effect
Experiments
 Problem 7: Choose between
A.
(6000, .45)
B.
(3000, .90)
Problem 8: Choose between
C.
(6000, .001)
D.
(3000, .002)
Experiments
 Problem 7: Choose between
A. [14]
(6000, .45)
B. [86]
(3000, .90)
Problem 8: Choose between
C. [73]
(6000, .001)
D. [27]
(3000, .002)
 Possibility effect – where winning is possible
but not probable, most people choose the
prospect that offers the larger gain
Reflection Effect
 Reflection effect – the previous slides
discussed
preferences
between
positive
prospects. What happens when the gains are
replaced by losses?
 In the table the preference between negative
prospects is the mirror image of the preference
between positive prospects, hence we talk about
reflection effect. The reflection of prospects
around 0 reverses the preference order.
 Reflection effect implies:
1. risk aversion in the positive domain,
2. risk seeking in the negative domain.
Overweighting of certainty (which violates EUT)
favors risk aversion in the domain of gains and
risk seeking in the domain of losses.
Hypothetical value function
Isolation Effect
 Isolation effect: in order to simplify the choice
between alternatives, people often disregard
components that the alternatives share and focus
on the components that distinguish them; a pair of
prospects can be decomposed into common and
distinctive components in more than one way
which can lead to different preferences – this
phenomenon is called isolation effect.
 Problem 10: Consider two-stage game. In the
1st stage there is a prob. of .75 to end a game
with winning nothing and the prob. of .25 to move
into the second stage. In the 2nd stage you have
a choice between
(4000,.80) and (3000)
Your choice must be made before game starts!
 Problem 10: Look that one has a choice between .25x.80=.20 to win 4000
and a .25x1=.25 to win 3000.
(4000,.20) and (3000, .25)
The majority chose the latter
Explanation: Evidently, people ignored the 1st stage of the game and
considered Problem 10 as a choice between (3000) and (4000, .80). Many
people cannot transform properly sequential form of the game into standard
form.
PROSPECT THEORY
Presented experiments appear to invalidate EUT
 An alternative account of individual decision making under risk –
prospect theory
Prospect theory includes two phases in the choice process:
1. an early phase of editing
2. a subsequent phase of evaluation
 Editing phase – the function of this phase is to organize and
reformulate the options so as to simplify subsequent evaluation and
choice
Operations of the editing phase:
1. Coding – people perceive outcomes as gains and losses which
are defined to some neutral reference point (e.g. current asset
position)
2. Combination – e.g. (200, .25; 200, .25) reduced to (200, .50)
3. Segregation – e.g. (300, .80; 200, .20) is decomposed to a sure
gain of 200 and the risky prospect (100, .80)
4. Cancellation – e.g. respondents ignored the first stage of the
sequential game presented earlier
 Editing phase – the function of this phase is to organize and
reformulate the options so as to simplify subsequent evaluation and
choice
Operations of the editing phase:
5. simplification – rounding probabilities or outcomes, e.g. (101, .49) is
likely to be recorded as an even chance to win 100
6. detection of dominated alternatives – scanning of offered prospects
to reject dominated options without further evaluation
 Phase of evaluation – the overall value of an edited prospect, V, is
expressed in terms of two scales, π and v
 π associates with each probability p a decision weight π(p), which
reflects the impact of p on the overall value of prospect (π is not a
probability measure)
 v assigns to each outcome x a number v(x), which reflect the
subjective value of that outcome; v measures the value of deviations
from the reference point, i.e. gains and losses
Example: the prospect (x,p; y,q) is valued at
V(x,p; y,q) = π(p)v(x) + π(q)v(y)
The Value Function
 Value should be treated as a function in two arguments: the asset
position that serves as reference point, and the magnitude of the
change from the reference point.
 Kahneman and Tversky expect that for most people the difference
in value between a gain of 100 and a gain of 200 appears to be
greater than the difference between a gain of 1100 and a gain of
1200. Similarly, the difference between a loss of 100 and a loss of 200
appears greater than the difference between a loss of 1100 and a loss
of 1200.
 Simply put, K&T hypothesize that the value function for changes of
wealth is concave above the reference point and convex below it.
Experiments
 Problem 13: Choose between
(6000, .25)
(4000, .25; 2000, .25)
Problem 13’: Choose between
(-6000, .25)
(-4000, .25; -2000, .25)
Experiments
 Problem 13: Choose between
(6000, .25) [18]
(4000, .25; 2000, .25) [82]
Problem 13’: Choose between
(-6000, .25) [70]
(-4000, .25; -2000, .25) [30]
Experiments
 Solving problems 13 and 13’:
π(.25)v(6000)<π(.25)[v(4000)+v(2000)]
and
π(.25)v(-6000)>π(.25)[v(-4000)+v(-2000)]
These preferences are in acord with the hypothesis that the value function is
concave for gains and convex for losses
Hypothetical value function
 Value function is:
1. defined on deviations from the reference point,
2. generally concave for gains and commonly convex for losses,
3. steeper for losses than for gains – the aggravation that one
experiences in losing a sum of money appears to be greater than
the pleasure associated with gaining the same amount
Extensions
 The theory is concerned mainly with monetary outcomes. However,
we may also apply the theory to choices involving other attributes,
e.g. quality of life.
 Prospect theory is a cognitive model – question arises: what is the
influence of emotional state, personality traits on individual decision
making under risk?