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FUNDAMENTAL ECONOMICS – Vol. I - Decision Making Under Uncertainty - David Easley and Mukul Majumdar

DECISION MAKING UNDER UNCERTAINTY

David Easley and Mukul Majumdar

Department of Economics, Cornell University, USA

Keywords: uncertainty, decision, utility, risk, insurance, games, learning

Contents

1. Introduction

2. Expected Utility

2.1 Objective Expected Utility

2.2. Risk Aversion

2.3 Subjective Expected Utility

3. Sequential Decision Making

3.1 Discounted Dynamic Programming

3.2 Characterization of Optimal Policies

3.3 Learning

4. Games as Multi-Person Decision Theory

4.1 Nash Equilibrium

4.2 Bayes Nash Equilibrium

5. Uses and Extensions

Glossary

Bibliography

Biographical Sketch

Summary

Often decision makers are uncertain about the consequences of their choices. Expected

utility theory provides a model of decision making under such uncertainty. This theory

deals with both objective and subjective uncertainty. It provides insights into actual

decisions and it may be used as a guide for decision making. The theory has been

extended to incorporate decisions made over time and the learning that results from

these decisions. It also provides the basis for the analysis of interacting decision makers

in a game.

1. Introduction

“The basic need for a special theory to explain behavior under conditions of

uncertainty”, noted Kenneth Arrow, “arises from two considerations: (1) subjective

feelings of imperfect knowledge when certain types of choices, typically involving

commitments over time, are made; (2) the existence of certain observed phenomena, of

which insurance is the most conspicuous example, which cannot be explained on the

assumption that individuals act with subjective certainty”. The literature is too vast for a

survey, and, in several directions lead to subtle issues of philosophy, economics and

probability theory. At one extreme are models that focus on a single decision-maker (an

investor, a central planner). At the other extreme are models - in the tradition of Walras

- with a large number of agents. In between are models - in the tradition of Cournot -

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FUNDAMENTAL ECONOMICS – Vol. I - Decision Making Under Uncertainty - David Easley and Mukul Majumdar

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CEOHAPLSTESRS2. A set P = {(p1,...,pn) ∈

Rn+with a small number of interacting agents.

The earliest treatments of decision making under uncertainty dealt with uncertain cash

flows and assumed that only the expected value mattered. The St. Petersburg paradox (a

random cash flow with infinite expected value that is clearly not worth more than a

finite amount) showed that this approach was unsatisfactory. In 1738, Daniel Bernoulli

proposed valuing uncertain cash flows according to the expected value of the utility of

money using a logarithmic utility function. Hence, both expected value and risk matters.

This approach was arbitrary, but it seemed more reasonable than assuming that decision

makers care only about expected values. (It does not, however, solve the St. Petersburg

paradox. Consider repeated tossing of a fair coin that pays exp(2n) if a head appears for

the first time on the nth toss.) In 1944, von Neumann and Morgenstern, in their analysis

of games, provided a set of axioms for decision makers preferences over uncertain

objects that lead to Bernoulli’s formulation with general utility functions over the

objects. This approach had the advantage that the reasonableness of the axioms would

be more easily judged than could the direct assumption of expected utility

maximization. von Neumann and Morgenstern’s formulation dealt only with objective

uncertainty. This is a limitation as often uncertainty is not objective, and can only be

subjectively accessed. In 1954, Leonard Savage extended the theory to deal with this

complication. His approach is elegant, but difficult. In this article we follow a simple

treatment.

2. Expected Utility

For models with a single agent, a basic agenda of research has been to cast the problem

of optimal choice under uncertainty in terms of maximization of “expected” utility. We

begin with the case in which the uncertainty the decision-maker faces is objectively

known. The basic ingredients of the single agent model of choice under uncertainty are:

1. A set X = {x1,...,xn} a finite set of prizes or consequences.

:∑pi = 1} of probabilities, or lotteries, on X.

t=1n3. Preferences ≥ defined on P.

Formally, preferences ≥ are a binary relation on P. That is, pairs of alternatives, in P are

ranked. If the decision-maker regards probability p to be “at least as good as”

probability q, then we write p ≥ q. These preferences reflect the decision-maker’s

valuation of prizes as well as his attitude toward risk.

2.1 Objective Expected Utility

The challenge has been to isolate axioms that enable one to impute to the decision-maker a utility function u on X, representing the decision-maker’s preferences. One

shows that, under some assumptions on preferences, the decision maker prefers one

probability p to another probability q if and only if the first probability yields a higher

expected utility, i.e. Ep(u(x)) > Eq(u(x)) where the expectation operation is taken with

respect to the probability distribution p or q on X.

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FUNDAMENTAL ECONOMICS – Vol. I - Decision Making Under Uncertainty - David Easley and Mukul Majumdar

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CEOHAPLSTESRS1The requirements for such a representation to exist are:

1. Completeness: for all p,q

∈ P either p

≥ q, q ≥ p or both.

2. Transitivity: for all p,q,r ∈ P if p

≥ q and q ≥ r, then p ≥ r.

3. Continuity: for all p,q,r ∈ P the sets {α ∈[0,1]: αp + (1-α)q ≥ r} and {α ∈[0,1]:r ≥

αp + (1-α)q} are closed.

4. Independence: for all p,q,r ∈ P and α ∈(0,1), p ≥ q if and only if αp + (1-α)r ≥ αq +

(1-α)r.

To interpret independence it is useful to break the probability

αp + (1-α)r into two

lotteries. Consider the (compound) lottery with probability α on “prize” p and

probability 1-α on “prize” r. The two lotteries αp + (1-α)r and αq + (1-α)r place

probability 1-α on the same prize r. With the remaining probability, α, the first gamble

gives p and the second gives q where p ≥ q. So it seems intuitive that p ≥ q if and only if

α p + (1-α) r ≥ αp + (1-α) r as long as the decision-maker cares only about the

consequences of gambling and not the process of gambling itself.

Theorem 1. A preference relation ≥ on P satisfies completeness, transitivity, continuity

and independence if and only if there exists a function u: X → R1 such that for any two

probabilities p and q on X, we have p / q if and only if Ep(u(x)) ≥ Eq(u(x)).

Clearly, the representation u(⋅) given in Theorem 1 is not unique. If u(⋅) is an expected

utility function for some preferences /, then so is V(x) = a + b u(x) for any numbers a

and b > 0.

Expected utility theory, which is developed here for the case of finite prize sets, extends

straightforwardly to continuous prizes. We focus on prizes x ∈

R1+; think of amounts of

money. The distribution on outcomes can be described by a cumulative distribution

function

F:

R+ → [0,1]. To tie this notation back to our earlier notation for discrete

xi

F(x) =

∑p(xi) where

p(xi) =

pi. For continuous

prizes,

P is the space of cumulative distribution functions on

R+1. If a decision-maker

has preferences on

P that satisfy the axioms above then there is utility function u:

1R+→R1 such that for any

F,

G ∈ P we have

F

≥

G if and only if

∫u(x)dF(x)≥∫u(x)dG(x).

2.2. Risk Aversion

A decision-maker who dislikes uncertainty prefers the expected value of any

distribution to the distribution itself. Such an individual is said to be

risk averse.

Definition: A decision-maker is risk averse if for any cumulative distribution function

F,

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FUNDAMENTAL ECONOMICS – Vol. I - Decision Making Under Uncertainty - David Easley and Mukul Majumdar

u∫xdF(x)≥∫u(x)dF(x).

()

This definition is equivalent to concavity of the utility function

u. The curvature of the

individual’s utility function provides a measure of his degree of risk aversion. This

curvature cannot be measured by

u”(Α) as the second derivative is not uniquely by

≥.

However;

u”(x)/u’(x) is invariant to the representation chosen and it can be used as a

measure of risk aversion.

Definition: The Arrow-Pratt coefficient of (absolute) risk aversion for an expected

utility function

u(x) is

λ(x,u) = -u”(x)/u’(x).

This measure is positive for all

x, for any risk averse decision maker. The measure is

increasing in the curvature of

u(⋅) and thus it is a reasonable measure of risk aversion.

Formally, if

u(x) =

f(v(x)), for all

x, for an increasing concave function

f(⋅) then

λ(x,u)

≥

λ(x,v) for all x.

A typical application of this theory is to the choice of insurance. Suppose that an

individual begins with wealth

w > 0. With probability

p1 he will lose

L1, with

probability

p2 he will lose

L2 and with probability 1 -

p1 -

p2 he will retain his initial

wealth. He is offered a menu of insurance policies that pay

πi in the event of loss

Li with

cost or premium

C =

α(p1π1+p2π2). The individual can choose any level

πi

≤

Li, and he

pays a premium determined by

C. If

α = 1 then this actuarially fair insurance. Suppose

that the individual is risk averse with utility function on money given by

u(⋅). Then an

optimal insurance contract maximizes expected utility

p1u(w-C-L1+π1) +

p2u(w-C-L2+π2) + (1-p1-p2)u(w-C) over feasible payoffs.

For actuarially fair insurance it is immediate from the first order conditions for this

maximization problem that

πi = Li for all

i. That is, the individual fully insures and his

wealth will be

w -

C. For

α > 1, the solution involves a deductible D. The optimal policy

is characterized by

Li-πi=D > 0 for all i, where the optimal deductible depends on how

risk averse the individual is and on how unfair the insurance is.

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Bibliography

Allais M. (1953). Le comportement de l’homme rationnel devant le risque, critique des postulats et

axiomes de l’école Américaine. Econometrica 21, 503-546. [A paradox that challenges expected utility

©Encyclopedia of Life Support Systems (EOLSS)

FUNDAMENTAL ECONOMICS – Vol. I - Decision Making Under Uncertainty - David Easley and Mukul Majumdar

theory]

Anscombe F. and Aumann R. (1963). A definition of subjective probability. Annals of Mathematical

Statistics 34, 199-205. [A modern treatment of subjective expected utility theory]

Arrow K. (1971). Essays in the Theory of Risk Bearing. Chicago: Markham. [A collection of essays on

choice under uncertainty, some of which are landmarks in the progress of economic theory]

Bernoulli D. (1738). Specimen theoriae novae de mensura sortis, in Commentarii Academe Scientiarum

Imperialis Petrogolitanae, 5, 175-192. [An early article arguing that expected utility of prizes rather than

the expected value of prizes is relevant for decision making]

Berry, D.A. and Friestedt, B. (1985). Bandit Problems, Chapmen and Hall: London [A monograph

dealing with bandit problems has an extended list of references]

Blackwell, D. (1965). Discounted dynamic programming. Annals of Mathematical Statistics 36 226-235

Fudenberg D. and Tirole J. (1991). Game Theory. Cambridge: MIT Press. [A standard game theory

textbook]

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Biographical Sketches

©Encyclopedia of Life Support Systems (EOLSS)

Harsanyi J. (1967-68). Games with incomplete information played by Bayesian players. Managment

Science 14, 159-182, 320-334, 486-502. [A series of articles on decision making and games with

incomplete information]

Majumdar M. (1998). Organizations with Incomplete Information. (ed. Mukul Majumdar) Cambridge:

Cambridge University Press [A collection of essays and review articles dealing with decision making

with incomplete information]

Mas-Colell, A.,Whinston, M.D. and Green, J.R. (1995). Microeconomic Theory, Oxford University Press:

New York [Chapter 6 provides a useful exposition of choice under uncertainty]

Myerson R. (1991). Game Theory: Analysis of Conflict. Cambridge: Harvard University Press. [A

standard game theory textbook]

Savage L. (1954). The Foundations of Statistics. New York: Wiley [The pioneering work in subjective

probability]

Simon, H.A. (1972). Theories of bounded rationality. In Decision and Organizations (eds. McGuire, C.B.

and Radner, R.), North Holland: Amsterdam 161-176

Von Neumann J. and Morgenstern O. (1944). Theory of Games and Economic Behavior. Princeton:

Princeton University Press. [The original axiomatic development of expected utility theory]

David Easley is H. Scarborough Professor of Economics at Cornell University. A Fellow of the

Econometric Society, he is a leading contributor to the literature on decision making under uncertainty.

Mukul Majumdar is H.T. and R.I. Warshow Professor of Economics at Cornell University and has made

wide-ranging contributions to economic theory.

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