Bayesian Decision Theory and the Representation of Beliefs

Bayesian Decision Theory and the Representation of
Beliefs

Edi Karni∗
Johns Hopkins University

March 29, 2006

Abstract
In this paper, I present an axiomatic choice theory of Bayesian decision makers
and define choice-based subjective probabilities that truly represent Bayesian decision
makers’ prior and posterior beliefs. I give an example that illustrates the potential
consequences of ascribing to a decision maker subjective probabilities that do not rep-
resent his beliefs. I argue that no equivalent results may be obtained in theories that
invoke Savage’s (1954) idea of a state space.
∗I am grateful to Itzhak Gilboa, Simon Grant, Robert Nau and, especially, Peter Wakker for their com-
ments and to the NSF for financial support under grant SES-0314249.

1

1 Introduction

In modern theories of decision making under uncertainty, a decision maker is characterized

by a preference relation on a set of alternatives, representing courses of action. Uncertainty

is present when, for some alternatives, the exact consequence is not determined solely by

the choice the alternative. With few exceptions, these theories are formulated using the

analytical framework of Savage (1954). This framework consists of a set of states, S; a set

of consequences, C; and the set, F, of all the mappings from the set of states to the set

of consequences. One element of S is the true state and elements of F, referred to as acts,

corresponding to the alternative courses of action. A decision maker is characterized by a

(prior) preference relation, < on F, whose structure, depicted axiomatically, is sufficient for

its representation by an expected utility functional,

Z (1)
u (f (s)) dπ (s) ,

S

where u is a utility (real-valued) function defined on the consequences, and π is a (subjective)

finitely additive, non-atomic, probability measure on S, and f is an act.

States resolve the uncertainty in the sense that, once the true state becomes known, the
unique consequence implied by each and every act becomes known. Subsets of S are events.
An event is said to obtain if the true state belongs to it. Presumably, implicit in the decision
maker’s preference relation are his prior beliefs, that is, a binary relation on the set of events
that have the interpretation “at least as likely to obtain as.” Clearly, any probability measure

2

on S gives rise to a definition of beliefs.1

In what follows I consider a decision maker who, upon receiving new information pertinent
to his assessment of the likely realization of alternative events, experiences a change of
preference that reflect his modified beliefs. That is to say, a decision maker whose change of
preferences is fully accounted for by the change in his beliefs. In this framework a decision
maker is considered Bayesian if (a) the representation of his prior preferences involve an
identifiable (unique) prior probability measure on the set S representing his prior beliefs;
(b) the representations of his posterior, information-dependent, preference relations involve
well-defined posterior probability measures on S representing his updated beliefs; and (c) the
representations of the posterior preference relations are obtained from the prior preference
relation by the updating the prior probabilities according to Bayes rule.

In Karni (1996, 2005) I argued that, in Savage’s (1954) theory and all other theories
that invoke Savage’s analytical framework, the definition of prior beliefs is predicated on the
convention that constant acts are constant utility acts. This convention is neither implied by
the axioms nor it is subject to refutation within Savage’s analytical framework. Without this
convention, however, it is impossible to disentangle the probabilities and marginal utilities;
consequently, there is an infinite set of prior probabilities consistent with a decision maker’s
prior preferences, and each such prior, when updated according to Bayes rule, gives rise to
posteriors that are consistent with the decision maker’s posterior preference relations. Hence

1 Let ºBbe a binary relation on the set of events define by E ºB E0 if and only if π (E) ≥ π (E0) , for all
events E and E0.

3

all theories that invoke Savage’s analytical framework, fail to identify a unique prior and, as

a result cannot be considered as model of Bayesian decision making. To see this, let γ be
Z

a strictly positive, bounded, real-valued function on S and let E (γ) = γ (s) dπ (s) . Then

S

the prior preference relation, depicted by the representation (1) is also represented by

Z (2)
uˆ (f (s), s) dπˆ (s) ,

S

where uˆ (·, s) = u (·) /γ (s) and πˆ (s) = π (s) γ (s) /E (γ) , for all s ∈ S.2

Consider next an experiment whose outcome is pertinent to the decision maker’s as-

sessment of the likely realization of events. Let X be the set of observations and denote by

q (x | s) the conditional probability of the observation x if the true state is s.3 Then, invoking

representation (1), the posterior preference relations {<x}x∈X of a Bayesian decision maker

are defined by: For all x ∈ X, and f, f 0 ∈ F, (3)
(4)
ZZ
f <x f 0 ⇔ u (f (s)) dπ (s | x) ≥ u (f 0(s)) dπ (s | x) ,

SS

where, for each x ∈ X,

π (· | x) = Z q (x | ·) π (·)
q (x | s0) dπ (s0)

S

is the posterior probability measure obtained by the application of Bayes theorem.

Similarly, invoking representation (2), the posterior preference relations, {<ˆ x}x∈X, of the

2This point has been recognized by Drèze (1987); Schervish, Seidenfeldt, and Kadane (1990); Karni

(1996), (2003); Karni and Schmeidler (1993), and Nau (1995).
3 In this case, s may be thought of as the parameters underlying the distribution of X, and {q (· | s)}s∈S

is a family of likelihood functions.

4

same Bayesian decision maker whose prior probability measure is πˆ (·) is defined by: For all

x ∈ X, and f, f 0 ∈ F,

ZZ (5)
f <ˆ xf 0 ⇔ uˆ (f (s)) dπˆ (s | x) ≥ uˆ (f 0(s)) dπˆ (s | x) , (6)

SS (7)

where

πˆ (· | x) = Z q (x | ·) πˆ (·) .
q (x | s0) dπˆ (s0)

S

However,

⎡Z ⎤

Z uˆ (f (s)) dπˆ (s | x) = ⎢⎢⎢⎢⎣ZS q (x | s0) dπ (s0) ⎥⎥⎥⎦⎥ Z u (f (s)) dπ (s | x) .
q (x | s0) dπˆ (s0)
S S

S

Hence, for all f, f 0 ∈ F and x ∈ X,

ZZ (8)
uˆ (f (s)) dπˆ (s | x) ≥ uˆ (f 0(s)) dπˆ (s | x) (9)

if and only if SS

ZZ
u (f (s)) dπ (s | x) ≥ u (f 0(s)) dπ (s | x) .

S S

Thus, <x= <ˆ x for all x ∈ X.

The fact that a decision maker updates his preferences using Bayes rule, does not imply
that either his prior or his posterior beliefs, as defined by the representing probabilities, are
unique. In other words, the mere fact that a decision maker is an expected utility maximizer
who uses Bayes rule to update his beliefs, does not imply that his beliefs are identifiable.

5

In Karni (2005), I presented an alternative framework for the analysis of decision making
under uncertainty. In this paper, I use the same analytical framework to develop a theory of
decision making under uncertainty in which the prior and posterior subjective probabilities
of Bayesian decision makers are identifiable, that is, unique prior and posterior subjective
probabilities that represent the beliefs of a Bayesian decision maker. Moreover, because the
definition of beliefs is choice based, this claim is testable (that is, subject to refutation)
within the realm of the revealed preference methodology.

The identifiability of the decision maker’s beliefs and the definition of subjective probabil-
ities is not an abstract philosophical issue. To illustrate the potential economic consequences
of ascribing subjective probabilities that do not represent the decision maker’s beliefs and
to motivate the ensuing theory, in the next section I discuss a moral hazard problem. The
analysis shows the potential pitfalls of ascribing the wrong subjective prior to the agent .
In the example, the principal knows the agent’s prior preference relation and, appropriately,
uses Bayes rule to update the agent’s preferences in the light of new information. But, be-
cause he ascribes the wrong prior to the agent, the principal fails to motivate the agent to
choose the action that the principal desires.

In section 3, I review the analytical framework and state, for expository convenience,
the main representation theorem of Karni (2005). In sections 4 and 5, I present alternative
formulations of Bayesian decision theory and prove that the prior and posterior subjective
probabilities are identifiable. Concluding remarks appear in section 6. The proofs appear in
the appendix.

6

2 A Moral Hazard Problem

2.1 The problem

The following moral hazard problem is formulated using the parametrized distribution ap-
proach. Let the agent be a risk-averse, Bayesian, expected utility—maximizing decision
maker. Suppose that the principal is not only aware of this but that he also knows the
agent’s prior preferences. The principal-agent relationship is governed by a contract specify-
ing the monetary payoff to the agent contingent on the outcome. If new pertinent information
becomes available before the terms of the original contract are fulfilled, the contract is subject
to renegotiation.

Even if it is state-independent, the agent’s prior preference relation admits infinitely
many representations, involving distinct probabilities and state-dependent utility functions
(see Karni [1996]). Among these, presumably, is a unique probability measure that is the
true representation of the agent’s beliefs. I intend to show that, even if the principal knows
the agent’s prior preferences but ascribes to him the wrong probabilities and, consequently,
the wrong utilities, it may happen that, if new information becomes available, the two parties
to the original contract agree to replace it with a contract that induces the agent to select
an action that is not in the principal’s best interest.

For the sake of concreteness I consider the following example. At the end of a regatta
monetary prizes will be awarded to the boats that finish first and second. Suppose that

7

there are three entrants, x, y and z, and the prizes satisfy m1 > m2 > m3 = 0, where mi
denotes the prize awarded to the boat in place i. According to the rules, if one of the boats
withdraws from the race, it is automatically assigned third place; if two boats withdraw, the
race is cancelled and no prizes are awarded. The relevant set of outcomes, Θ, consists of the
six permutations of the order in which the boats cross the finish line.4

The boat owners recruit skippers and put them in charge of training their crews. Suppose
that the owners cannot monitor the training and that the training entails costly effort on
the part of skippers and crews. To motivate the sailors, an owner (principal) may offer his
skipper (agent) an incentive contract.

In what follows, I describe the initial incentive contract and examine the implications
of renegotiating the contract of one of the remaining competitors - say the owner of x - if
one of the other entrants announces its withdrawal from the race. To simplify the analysis,
I assume that the set of actions is a doubleton, with a0 denoting training vigorously and
a training lightly. Let c : {a, a0} → R be a function depicting the disutility to the agent
associated with the actions and assume that c (a0) > c (a) = 0. Let Ei denote the event that
x finishes in position i (for example, E1 = {(x, y, z) , (x, z, y)}).

Recall that the skipper is assumed to be a risk-averse, Bayesian, expected utility-maximizer
and that the boat owner knows the skipper’s prior preferences and knows that the skipper is

4The outcomes of the race correspond to what I refer to later as effects. As this example illustrates, the
distinguishing characteristic of effects is that, unlike states, their likely realization may be influenced by the
decision maker.

8

Bayesian. Suppose that the boat owner ascribes to the skipper his own probabilities re-
garding the likely outcome, conditional on the level of training. This is the common prior
assumption, often invoked in the parametrized distribution formulation of principal-agent
problems. Under these assumptions, if the principal would like to induce the agent to train
vigorously, his problem may be stated as follows:

Program 1: Choose a contract w := (w (E1) , w (E2) , w (E3)) ∈ R3 so as to maximize

X3 (10)
u (mi − w (Ei)) π (Ei; a0)

i=1

subject to the individual rationality constraint

X3 (11)
v (w (Ei)) π (Ei; a0) − c (a0) ≥ v¯

i=1

and the incentive compatibility constraint

X3 X3 (12)
v (w (Ei)) π (Ei; a0) − c (a0) ≥ v (w (Ei)) π (Ei; a) ,

i=1 i=1

where u and {π (Ei, j) | i ∈ {1, 2, 3}, j ∈ {a, a0}} denote the utility functions and the

conditional (on the actions) subjective probabilities of the principal, and v and c denote,

respectively, the utility of money and the disutility associated with the actions ascribed by

the principal to the agent. By assumption, π (·, ·) also denote the conditional probabilities

ascribed by the principal to the agent.

By assumption, the principal employs a representation of the agent’s prior preferences
consistent with the agent’s choice behavior. This depiction of the principal’s problem is
consistent with the parametrized distribution formulation in the literature on agency theory.

9

2.2 Representation of the agent’s preferences

It is conceivable that, independently of his monetary reward, the agent cares about winning

but that the principal is unaware of this.5 Moreover, as the preceding discussion indicates, the

principal cannot detect the agent’s sentiments about winning from the agent’s choice behav-

ior. Suppose that, the agent’s true utility function is given by V (w, i, j) = Γ (j) v (w) /γ (i) ,

where 0 < γ (1) < γ (2) < γ (3) , and Γ (j) = P3 γ (i) π (Ei; j) , j ∈ {a, a0}. Note that the

i=1

presence of a nonadditive component, Γ (j) , in the agent’s utility of action, is also unde-

tectable by observing his choice behavior.6 Consistency of the agent’s prior preferences with

their representation in Program 1, requires that his beliefs be represented by the probabilities

πA (Ei; j) = γ (i) π (Ei; j), j ∈ {a, a0}, i = 1, 2, 3 (13)
Γ (j)

Hence the agent’s true objective function is given by: (14)

X3
Γ (j) γ−1 (i) v (w (Ei)) πA (Ei; j) − c (j) , j ∈ {a, a0}.

i=1

Program 1 may then be restated a follows:

Program 1’: Choose a contract, w ∈ R3, so as to maximize

X3 (15)
u (mi − w (Ei)) π (Ei; a0)

i=1

5This is, of course, a simplifying assumption. In general, it is sufficient that the prinicpal not know how

much the agent cares about winning.
6See Grossman and Hart (1983) for an example of principal-agent analysis in which the utility of action

has a multiplicative factor, as above.

10

subject to the individual rationality constraint (16)
(17)
X3
V (w (Ei) , i, a0) πA (Ei; a0) − c (a0) ≥ v¯

i=1

and the incentive compatibility constraint

X3 X3
V (w (Ei) , i, a0) πA (Ei; a0) − c (a0) ≥ V (w (Ei) , i, a) πA (Ei; a) .

i=1 i=1

Clearly, the solution, w∗, to the two programs is the same. Thus, insofar as the principal
is concerned, as long as the contract w∗ is in effect, the misspecification of the agent’s utilities
and probabilities is of no consequence.

More generally, this discussion shows that, from a substantive point of view, ascribing
to the agent the principal’s own beliefs is immaterial, provided that the utility functions are
adjusted so that the agent’s preferences are accurately represented. This implies that the
“common prior” assumption is harmless if the agent’s preferences are represented correctly
and no new information becomes available before the terms of the contract are fulfilled. I
revisit the “common prior” assumption at the end of this section.

2.3 New information and recontracting

Suppose that, after having signed a contract, the owner and skipper of boat x learn that
one of the entrants, say z, decides to withdraw from the race. This decision eliminates
the possibility of the two remaining boats finishing third. Equivalently, the event consist-
ing of all the outcomes in which z finishes first or second becomes null. Let E3 (z) de-

11

note the event consisting of outcomes in which z finishes third. Since both the principal
and the agent are Bayesian, the principal’s posterior probabilities are π (Ei; j | E3 (z)) =
π (E3 (z) ; j | Ei) π (Ei; j) /π (E3 (z) ; j) , and π (E3; j | E3 (z)) = 0, j ∈ {a, a0}. Moreover,
being aware that the agent is Bayesian, the principal updates the agent’s probabilities in
the same way. However - and here is the rub - the agent’s true posterior probabilities are
πA (E3; j | E3 (z)) = πA (E3 (z) ; j | Ei) πA (Ei; j) /πA (E3 (z) ; j) , and πA (E3; j | E3 (z)) = 0,
j ∈ {a, a0}. To simplify the notation, henceforth I denote the posteriors π (Ei; j | E3 (z)) and
πA (Ei; j | E3 (z)) by πˆ (Ei, j) and πˆA (Ei, j) , respectively.

Consider next the issue of recontracting. The withdrawal of z from the race raises two
questions: First, are there perceived benefits to recontracting? Second, if the original con-
tract is replaced, is it necessarily true that the replacement contract induces the agent to
implement the action desired by the principal?

Before pursuing this analysis further, it is important to note that, in principle, the original
and the replacement contracts may be negotiated simultaneously or sequentially. In other
words, given that the announcement that one of the competitors has pulled out of the race is
made public before the beginning of training, the contract governing the principal-agent rela-
tionship in this event may be negotiated before or after the announcement. However, except
for the cost of contracting, there is no difference between the simultaneous and sequential
negotiations. If the contracts are negotiated simultaneously, the reservation utility of the
agent must be the same under both contracts. If the contracts are negotiated sequentially,
the first contract can be written to include a clause that makes it null and void if one of

12

the competitors withdraws from the race. The second contract is negotiated under the same
individual rationality and incentive compatibility constraints and is, therefore, no different
from the contract covering the same contingency that was negotiated simultaneously. In the
presence of contracting costs, the sequential negotiation is preferable on two counts: First,
if none of the competitors pulls out of the race, there is no need for a second contract, which
saves the contracting cost. Second, if one of the competitors does pull out, it is better to
incur the cost of recontracting later than earlier.

In what follows, I pursue the scenario of sequential contracting and analyze the case in
which the principal wants the agent to implement the rigorous training program. From the
principal’s point of view, the subsequent contract, w∗∗ = (w∗∗ (E1) , w∗∗ (E2)) , is the solution
to the following program:

Program 2: Choose w ∈ R2 so as to maximize (18)
(19)
X2 (20)
u (mi − w (Ei)) πˆ (Ei; a0)

i=1

subject to the individual rationality constraint

X2
v (w (Ei)) πˆ (Ei; a0) − c (a0) ≥ v¯

i=1

and the incentive compatibility constraint

X2 X2
v (w (Ei)) πˆ (Ei; a0) − c (a0) ≥ v (w (Ei)) πˆ (Ei; a) .

i=1 i=1

13

The solution, w∗∗, to Program 2, is given by the solution to the following equations:7

v (w (E1)) = v¯ + 1 − πˆ (E1; a) and v (w (E2)) = v¯ − πˆ πˆ (E1; a) a) . (21)
πˆ (E1; a0) − πˆ (E1; a) (E1; a0) − πˆ (E1;

Consider next the problem as viewed by the agent. The agent should agree to w∗∗

if, under the corresponding best action, it meets his reservation utility. Formally, w∗∗ is

acceptable to the agent if

X2 (22)
max V (w∗∗ (Ei) , i, j) πˆA (Ei; j) − c (j) ≥ v¯.

j∈{a,a0}
i=1

The second question, namely, whether it is necessarily true that w∗∗ induces the agent to

implement the action desired by the principal, may be restated as follows: Is the following

inequality necessarily true?

X2 X2 (23)
V (w∗∗ (Ei) , i, a0) πˆA (Ei; a0) − c (a0) ≥ V (w∗∗ (Ei) , i, a) πˆA (Ei; a)

i=1 i=1

I am interested in showing that it is possible that inequality (23) fails to hold. In this
instance, if z pulls out of the race, the original contract is null and void and the two parties
to the contract sign a new contract, w∗∗. But, contrary to the expectations and the interests
of the principal, the agent implements the light training program. In other words, because
he failed to ascribe to the agent the correct prior probabilities and utilities, the principal
misrepresents the agent’s posterior preference relation and, as a result, fails to provide the
agent with the incentives that would have induced him to act in the principal’s best interest.

7 See Salanie (1997)

14

2.4 An example

In order to simplify the calculations, assume that the principal is risk neutral. Let the agent’s
utility function be V (w, i, j) = Γ (j) √w/γ (i) , i ∈ {1, 2, 3}, j ∈ {a, a0}. Suppose that the
principal perceives the problem facing him as Program 1 above, with the prior probabilities

as follows:

⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞

Outcomes ⎜⎝⎜⎜⎜⎜⎜ x ⎟⎟⎟⎠⎟⎟⎟ ⎜⎜⎜⎝⎜⎜⎜ x ⎟⎟⎟⎟⎟⎠⎟ ⎜⎜⎝⎜⎜⎜⎜ y ⎟⎟⎟⎠⎟⎟⎟ ⎜⎜⎜⎜⎝⎜⎜ z ⎟⎟⎟⎠⎟⎟⎟ ⎜⎜⎜⎜⎜⎜⎝ y ⎟⎠⎟⎟⎟⎟⎟ ⎜⎜⎜⎝⎜⎜⎜ z ⎟⎟⎟⎟⎟⎟⎠
y z x x z y

z y z y x x (24)

π (·; a) 1.5 1.5 2 2 2.5 2.5
12 12 12 12 12 12

π (·; a0) 4 4 1 1 1 1
12 12 12 12 12 12

where, in describing the outcomes, the boats are ranked, from top to bottom, according to

their positions. Recall that the principal is the owner of boat x. Hence the principal’s prior

probabilities of the relevant events are:

π (E1; a) = 3 π (E2; a) = 4 π (E3; a) = 5
12 12 12
(25)

π (E1; a0) = 8 π (E2; a0) = 2 π (E3; a0) = 2
12 12 12

Suppose that the agent’s disutility associated with the rigorous training is c (a0) = 3 and

his prior reservation utility level is v¯ = 9.8333. Consider the contract

w∗= (w∗ (E1) = 225, w∗ (E2) = 144, w∗ (E3) = 25) . (26)

15

Insofar as the principal is concerned, this contract satisfies the agent’s perceived individual
rationality constraint

8√ 2√ 2√ (27)
12 225 + 12 144 + 12 25 − 3 = 9.8333 (28)
and the incentive compatibility constraint8

9.8333 = 3 √ + 4 √ + 5 √
225 144 25.
12 12 12

Moreover, the principal would rather the agent implement a0 under the contract w∗ than
a under the fixed-pay contract w¯ = 9.83332 = 96.694.

To see this, note that the value of the principal’s objective function under (w∗, a0) is

8 (1000 − 225) + 2 (500 − 144) − 2 25 = 571.83, (29)
12 12 12

far exceeding the value of his objective function under the fixed contract and a, which is

3 + 4 − 96.694 = 319.97. (30)
1000 500

12 12

Next consider the situation as seen by the agent. Suppose that the agent’s true prior

beliefs are represented by the action dependent probabilities as follows:

8The fact that, in this example, the incentive compatibility constraint is binding does not affect the
generality of the conclusion below.

16

⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞

Outcomes ⎜⎜⎜⎜⎜⎜⎝ x ⎟⎟⎠⎟⎟⎟⎟ ⎜⎜⎜⎜⎜⎜⎝ x ⎟⎠⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎝⎜⎜ y ⎟⎟⎟⎟⎟⎠⎟ ⎜⎜⎜⎝⎜⎜⎜ z ⎟⎟⎟⎟⎟⎟⎠ ⎝⎜⎜⎜⎜⎜⎜ y ⎟⎠⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜⎜⎝ z ⎠⎟⎟⎟⎟⎟⎟
y z x x z y

z y z y x x (31)

πA (·; a) 32,847 32,847 175,180 175,180 291,970 291,970

1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000

πA (·; a0) 15 15 15 15 20 20
100 100 100 100 100 100

This means that there are positive numbers γ (i), i = 1, 2, 3, satisfying γ (1) < γ (2) < γ (3) ,

and Γ (j) = P3 γ (i) π (Ei; j) , j ∈ {a, a0}, such that πA (Ei; j) = γ (i) π (Ei; j) /Γ (j) .

i=1

The solution to γ (i) , i = 1, 2, 3, and Γ (j) , j ∈ {a, a0}, is γ (1) = 6.0336, γ (2) = 24.134,

γ (3) = 32.179, Γ (a0) = 13.408, and Γ (a) = 22.961.9

9 The calculation of {γ (i) | i = 1, 2, 3} and Γ(j), j ∈ (a, a0) is based on the following equations:

= 0.38

12
γ(1)

8 γ (1)+ 3 γ(2)+ 2 γ (3)
12 12 12

= 0.32

12
γ(2)

8 γ (1)+ 3 γ(2)+ 2 γ (3)
12 12 12

= 0.42

12
γ(3)

8 γ (1)+ 3 γ(2)+ 2 γ (3)
12 12 12

the solution to which is

γ (1) = 6.0336, γ (2) = 24.134, γ (3) = 32.179

Thus Γ (a) = 3 6.0336 + 4 24.134 + 5 32.179 = 22.961
and 12 12 12

Γ (a0) = 8 2 2
6.0336 + 24.134 + 32.179 = 13.408.
12 12 12

17

Hence his probabilities of the relevant events are10

πA (E1; a) = 0.065694 πA (E2; a) = 0.35036 πA (E3; a) = 0.58394 (32)

πA (E1; a0) = 0.3 πA (E2; a0) = 0.3 πA (E3; a0) = 0.4

Because w∗ is the solution of Program 1, it satisfies the individual rationality and in-
centive compatibility constraints. Thus the agent accepts contract w∗ and, under its terms,
implements the rigorous training program, as expected by the principal.

Consider next the effect of z pulling out of the race. The principal’s posterior probabili-
ties, which are also the posterior probabilities he ascribes to the agent, are obtained by the
application of Bayes rule. They are given by:

πˆ (E1; a) = 15 πˆ (E2; a) = 20
35 35
(33)

πˆ (E1; a0) = 4 πˆ (E2; a0) = 1
5 5

The principal considers offering the agent the contract, w∗∗ = (w∗∗ (E1) , w∗∗ (E2)) , that
would motivate him to implement a0. This contract is obtained by the solution to equations

(21) and is given by w∗∗ (E1) = 129.32, and w∗∗ (E2) = 75.333.11

10 The conditional probabilities {πA (Ei; a) | i = 1, 2, 3} are the solution to the equations

3 6.0336 = πA (E1; a)
12
3 4 5
12 6.0336+ 12 24.134+ 12 32.179

4 24.134 = πA (E2; a)
12
3 4 5
12 6.0336+ 12 24.134+ 12 32.179

5 32.179 = πA (E3; a)
12
3 4 5
12 6.0336+ 12 24.134+ 12 32.179

11 Specifically, w∗∗ is given by the equations

pp
w∗∗ (E1) = 9.8333 + 20 , and w∗∗ (E2) = 9.8333 − 15 .
13 13

18

The principal’s payoff under (w∗∗, a0), is12

41 (34)
5 (1000 − 129.32) + 5 (500 − 75.333) = 781.48.
His payoff under (w¯ , a) , is

151000 + 20500 − 96.694 = 617.59. (35)
35 35

Clearly, the principal is better of with (w∗∗, a0) than with (w¯ , a) . In other words, the principal

is better off offering to replace contract w∗ by contract w∗∗, expecting the agent to implement

the rigorous training program.

Next I show that the agent accepts w∗∗ and, contrary to the anticipations of the principal,
chooses action a. To begin with, note that the agent’s posterior beliefs are represented by

πˆA (E1; a) = 32,847 = 0.1579 πˆA (E2; a) = 175,180 = 0.8421
32,847+175,180 32,847+175,180
(36)

πˆA (E1; a0) = 0.5 πˆA (E2; a0) = 0.5

If the agent accepts w∗∗ and implements a, his posterior expected utility is

Ã√ √!
22.961 0.1579 129.32 + 0.8421 75.333 = 13.787,
6.0336 24.134 (37)

12 The principal believes that if the agent acepts w∗∗, it will induce him to select the action a0.

19

which exceeds his reservation utility. Thus the agent accepts w∗∗. The agent’s payoff under
(w∗∗, a0) is

Ã√ √!
0.5 129.32 0.5 75.333
13.408 6.0336 + 24.134 − 3 = 12.046. (38)

Comparing the agent’s payoffs corresponding to w∗∗ under the two actions, it is clear
that, against the principal’s best interest, the agent will implement the action a.

2.5 Conclusion and explanation

This example illustrates the cost to the principal of misrepresenting the agent’s prior prob-
abilities and utilities, even if the representation of the agent’s prior preferences is accurate.
To avoid this pitfall, the principal must know the agent’s prior and posterior preferences. As
the theory developed in this paper shows, this requires that the principal know the unique
subjective probabilities that truly represent the agent’s beliefs.

This conclusion undermines the use of the common prior assumption, which pervades
agency theory. As argued earlier, if the principal knows the agent’s prior preferences and
no new information becomes available, the common prior assumption is harmless and con-
venient. However, as the example shows, in situations in which recontracting under new
information is possible, the use of the common prior assumption is no longer harmless or
automatically justified. If the posterior contracts are to implement the second-best solutions,

20

the agent’s true prior needs to be identified and it may or may not agree with the principal’s.

The result in this example reflects the fact that the agent believes that his choice of the
more costly action shifts relatively large probability mass from the worst outcome, E3, to the
best outcome, E1, but little from E2 to E1. Hence the agent readily responds to incentives
when E3 is feasible. However, once E3 has been eliminated, the agent’s responsiveness
diminishes. Because he ascribed to the agent the wrong probabilities, the principal fails to
appreciate this lack of responsiveness and offers the agent the “wrong” replacement contract.

3 The Theory

3.1 The analytical framework

Let Θ be a finite set of effects and A a set of actions. Actions represent initiatives that
decision makers may take to influence the likely realization of effects. A bet, b, is a mapping
from Θ into R, the set of real numbers. Bets have the interpretation of monetary payoffs
contingent on the effects. Let B := RΘ denote the set of all bets and assume that it is
endowed with the R|Θ| topology. Denote by (b−θr) the bet obtained from b ∈ B by replacing
the θ−coordinate of b, that is, b (θ) , with r. In this framework, effects are analogous to
states, in the sense that they resolve the uncertainty associated with bets. However, unlike
states, decision makers believe that they may influence the likely realization of effects by
their actions. The outcomes of the regatta described in the preceding section are examples

21

of effects.

Decision makers are supposed to choose actions and, at the same time, place bets on the
effects. For example, a decision maker may adopt an exercise and diet regimen to reduce
the risk of heart attack and at the same time take out health insurance and life insurance
policies. The diet and exercise regimens correspond to actions, the states health are effects,
and the financial terms of an insurance policy constitute a bet. Formally, the choice set, C,
consists of all the action-bet pairs (that is, C = A × B). A choice of an action a and a bet
b results, ultimately, in an effect-payoff pair, or consequence, (θ, b (θ)). Let C denote the set
of all consequences (that is, C = Θ × R).

A decision maker is characterized by prior binary relations, < on C, that have the usual
interpretation of a preference relation. The strict preference relation, Â, and the indifference
relation, ∼, are the asymmetric and symmetric parts of <, respectively. For each a ∈ A, the
preference relation < on C induces a conditional preference relation on B defined as follows:
For all b, b0 ∈ B, b <a b0 if and only if (a, b) < (a, b0) .

An effect θ is said to be nonnull given the action a if (a, (b−θr)) Â (a, (b−θr0)) , for some
b ∈ B and r, r0 ∈ R, and it is null given the action a otherwise. I assume that every effect
is nonnull for some action a. Given a preference relation <, denote by Θ (a; <) the subset of
effects that are nonnull given a according to <. To simplify the notation, when there is no
risk of confusion, I write Θ (a) instead of Θ (a; <).

Two effects, θ and θ0, are said to be elementarily linked if there are actions a, a0 ∈ A

22

such that θ, θ0 ∈ Θ (a) ∩ Θ (a0) . Two effects are said to be linked if there exists a sequence of
effects θ = θ0, ..., θn = θ0 such that every θj is elementarily linked with θj+1. This condition
will be used to link a utility scale under a with that under a0.

Actions may affect the decision maker’s well-being directly. For example, implementing

a diet regimen may require that the decision maker avoids eating food he likes. With this

in mind, I refer to a bet as constant valuation if, once accepted, it leaves the decision maker

indifferent among actions whose direct utility cost are just compensated for by the improved

chances of winning a better outcome they afford.13 I assume that the set of actions is

sufficiently rich so that the variations in the likely realization of alternative effects induced

by elements of A, and taking into account the associated direct utility implications, there

are no two bets that are equally desirable under all actions. That is, for every b 6= ¯b, there

exist a ∈ A, such that “not (a, b) ∼ ¡ ¯b¢ ”. Formally, let I (a; b) = {b0 ∈ B | (a, b0) ∼ (a, b)}
a,

then the idea of constant valuation bets is formalized as follows:

Definition 1: A bet ¯b is said to be a constant-valuation bet if ¡ ¯b¢ ∼ ¡ ¯b¢ for all
a, a0,

a, a0 ∈ A, and ∩a∈AI ¡ ¯b¢ = {¯b}.
a;

The last requirement implies that if ¯b is a constant valuation bet then no other b ∈ B

satisfies (a, b) ∼ (a0, b) for all a, a0 ∈ A and (a, b) ∼ ¡ ¯b¢ for some a ∈ A.14 The set
a,

13 Because of the possible direct impact of the actions on the decision maker’s well-being, constant valaution

bets are different from Drèze’s (1987) notion of “omnipotent” acts and the idea of constant valaution acts

in Karni (1993, 2005a) and Skiadas (1997).
14 In fact, I need assume only that there exists a subset, Aˆ, of A such that the conditions in definition 1

23

of all constant valuation bets is denoted by Bcv. If b∗∗ and b∗ are constant valuation bets
satisfying (a0, b∗∗) Â (a0, b∗) then transitivity of < implies (a, b∗∗) Â (a, b∗) for all a ∈ A.
Since transitivity will be assumed, I write b∗∗ Â b∗ instead of (a, b∗∗) Â (a, b∗) .

The following richness assumption is maintained throughout.

(A.0) Every pair of effects is linked, there exist constant-valuation bets b∗∗ and b∗ such that
b∗∗ Â b∗ and, for every (a, b) ∈ C, there is ¯b ∈ Bcv satisfying (a, b) ∼ ¯b.

3.2 Axioms

The first two axioms are standard.

(A.1) (Weak order) < on C is a complete and transitive binary relation.
(A.2) (Continuity) For all (a, b) ∈ C the sets {(a, b0) ∈ C | (a, b0) < (a, b)} and {(a, b0) ∈

C | (a, b) < (a, b0)} are closed.

The third axiom requires that the “intensity of preferences” for monetary payoffs contin-
gent on any given effect be independent of the action that resulted in that effect.

(A.3) (Action-independent betting preferences) For all a, a0 ∈ A, b, b0, b00, b000 ∈ B, θ ∈

¡ ¢¡ ¢
Θ (a) ∩ Θ (a0) , and r, r0, r00, r000 ∈ R, if (a, b−θr) < a, b0−θr0 , a, b−0 θr00 < (a, b−θr000) ,

¡ ¢¡ ¢¡ ¢¡ ¢
and a0, b−00 θr0 < a0, b−000θr then a0, b0−0 θr00 < a0, b0−00θr000 .

hold for all a, a0 ∈ A. Karni (2005) provides more details.

24

To grasp the meaning of action-independent betting preferences, think of the preferences

¡ ¢¡ ¢
(a, b−θr) < a, b0−θr0 and a, b0−θr00 < (a, b−θr000) as indicating that, given action a and effect

θ, the intensity of the preferences of r00 over r000 is sufficiently larger than that of r over r0

as to reverse the preference ordering of the effect-contingent payoffs b−θ and b0−θ. The axiom
requires that these intensities not be contradicted when the action is a0 instead of a and bets

are b00 and b000.15

3.3 Representation

It was shown in Karni (2005) that a preference relation on C has the structure described
by axioms (A.1)—(A.3) if and only if there is a continuous utility function, u, on the set of
consequences, a family of action-dependent probability measures, {π (·; a) | a ∈ A}, on the
set of effects, and a family of continuous strictly increasing functions {fa : R → R | a ∈ A},

¡P ¢
such that the assignment (a, b) → fa θ∈Θ u (b (θ) ; θ) π (θ; a) represents the preference
relation. Furthermore, for each a ∈ A, the probability measure π (·; a) is unique, satisfying
π (θ; a) = 0 if and only if θ is null given a.

Theorem 1 (Karni [2005]) Suppose that assumption (A.0) is satisfied, and | Θ(a) |≥ 2 for
all a ∈ A, then:

15 Action-independent betting preferences invokes Wakker’s (1987) idea of cardinal consistency. A more
elaborate discussion of this axiom is provided in Karni (2005). There it is also shown that, for every given
action, this axiom embodies the Sure Thing Principle and, in the case of two effects, the Hexagon Condition.

25

(a) The conditions (a.i) and (a.ii) below are equivalent:

(a.i) The preference relation < on C satisfies (A.1) — (A.3).

(a.ii) There exists a continuous function u : C → R, a family of probability measures

{π (·; a)}a∈A on Θ, and a family of continuous increasing functions {fa : R → R}a∈A

such that, for all (a, b) , (a0, b0) ∈ C,

à !à !
XX
(a, b) < (a0, b0) ⇔ fa u (b (θ) ; θ) π (θ; a) ≥ fa0 u (b0 (θ) ; θ) π (θ; a0) .

θ∈Θ θ∈Θ

(b) u is unique and {fa}a∈A are unique up to common, strictly increasing, transformation.

(c) For each a ∈ A, π (·; a) is unique and π (θ; a) = 0 if and only if θ is null given a.

The uniqueness of the probabilities in Theorem 1 is predicated on a normalization of the

functions u and {fa}a∈A.16 This normalization is not implied by the axioms. Consequently, it
is subject to the objection expressed against Savage’s model in the introduction, namely, that

the arbitrary choice of the normalization, renders the prior probabilities meaningless. To see

this, consider again the representation of the prior preference relation, <, in Theorem 1, fix
P

a function γ : Θ → R++ and let Γ (a) = θ∈Θ γ (θ) π (θ; a) , uˆ (b (θ) ; θ) = u (b (θ) ; θ) /γ (θ) ,

πˆ (θ; a) = γ (θ) π (θ; a) /Γ (a) , and fˆa = fa ◦ Γ (a) . Then the prior preference relation < is

represented by

à !à !
Γ (a) X u (b (θ) ; θ) γ (θ) π (θ; a) X
fa γ (θ) Γ (a) = fˆa uˆ (b (θ) ; θ) πˆ (θ; a) . (39)

θ∈Θ θ∈Θ

16 The normalization constitutes of assigning b∗∗ and b∗ utilities as follows: u (b∗ (θ) , θ) = 0 for all θ ∈ Θ,

and P u (b∗∗ (θ) , θ) π (θ, a) = 1, a ∈ A. Then, for all a ∈ A, fa (1) = 1 and fa (0) = 0.

θ∈Θ

26

But πˆ (θ; a) =6 π (θ; a) for some a and θ. Therefore the prior subjective probabilities are not
unique.

4 Bayesian Preferences and Subjective Probabilities

4.1 Bayesian decision makers

A Bayesian decision maker is characterized by a prior preference relation, a set of information-
dependent posterior preferences, and the condition that the posterior preferences are obtained
from the prior preference solely by the updating his subjective probabilities using Bayes’
rule (that is, leaving intact the other functions that figure in the representation). To model
the behavior of Bayesian decision makers it is necessary to formally incorporate into the
analytical framework new information.

Let X be a finite set of possible observations representing new information. Depending
on the context, elements of X represent experimental results, a study of existing evidence,
expert opinions, or any other information that might influence the decision maker’s beliefs
regarding the likely realization of the alternative effects.17 Note that Θ ⊂ X, that is,
because, ultimately, the uncertainty regarding the payoffs of the different bets is resolved,
the effects are observations. For every x ∈ X, let <x on C be a (posterior) preference relation
characterizing the decision maker after he has been informed of the observation x.

17 Inn the example in Section 2, the withdrawal of z from the regatta is an observation.

27

Consider a decision maker who, prior to choosing an action-bet pair, receives information
pertinent to his assessment of the likely realization of alternative effects, conditional on his
choice of action. Suppose that the sole relevance of new information is to render null, effects
that were nonnull according to the decision maker’s prior preferences. For instance, in the
example of Section 2, the withdrawal of one of the original entrants from the race renders
the event that one of the remaining entrants finishes third null. To formalize this idea and to
define a Bayesian decision maker, I use the following notation: For each observation x ∈ X
and action a ∈ A, let Θ (a, x) denote the set of effects that is nonnull given a and x according
to < . This notation makes it clear that the impact of the same observation on the decision
maker’s beliefs is subjective.

Definition 2: A decision maker whose prior preference relation < on C satisfies (A.1)—(A.3)

¡P ¢
is Bayesian if his prior preference relation, <, is represented by fa θ∈Θ u (b (θ) ; θ) π (θ; a)

and his posterior preferences {<x}x∈X are represented by fa ¡P u (b (θ) ; θ) π (θ | x, ¢ ,
a)
θ∈E

where, for all x ∈ X, and a ∈ A, π (θ | x, a) = π (θ; a) / P π (θ0; a) if θ ∈ Θ (a, x)

θ0∈Θ(x,a)

and π (θ | x, a) = 0 otherwise.

4.2 The uniqueness of the priors

The next theorem asserts that if a decision maker is Bayesian in the sense of definition 2 then
the prior, action-dependent, subjective probabilities representing his beliefs is identifiable.

28

Theorem 2 Suppose that there are at least two effects, and the prior preference relation, <
on C, satisfies (A.0)—(A.3). If the decision maker is Bayesian then the representation of his
prior preferences admits a unique set of action-dependent probability measures.

The proof is given in the Appendix.

4.3 An extension

Next, I generalize the definition of Bayesian decision makers to allow for more general infor-
mation structure.

A experiment is a pair {X, {q (· | θ, a)}a∈A,θ∈Θ}, where X is a finite set of observations
P

and q (· | θ, a) is a likelihood function on X (that is, q (· | θ, a) ≥ 0 and x∈X q (· | θ, a) = 1).

Consider the following definition:

Definition 3: A decision maker whose prior preference relation, < on C, is represented

by ¡P ¢ is Bayesian if his posterior preference relations, {<x}x∈X ,
fa θ∈Θ u (b (θ) ; θ) π (θ; a)

¡P ¢
are represented by fa θ∈E u (b (θ) ; θ) π (θ | x, a) , where, for all x ∈ X, θ ∈ Θ, and a ∈ A,

π (θ | x, a) = P q (x | θ, a) π (θ; a) a) . (40)
θ0∈Θ q (x | θ0, a) π (θ0;

It can be shown that, under the hypothesis of Theorem 2, if a decision maker is Bayesian
in the sense of definition 3, then the representation of his prior preferences admits a unique
(identifiable) set of action-dependent probability measures.18

18 The proof is analogous to that of Theorem 2 and is omitted.

29

5 Implicit State-Space Models of Bayesian Decision

Making

5.1 A state-space formulation

For reasons that are discussed at length in Karni (2005), I avoid the use of a state-space as an
ingredient of the analytical framework. This does not precludes the formulation of a model of
Bayesian decision making using the concept of a state-space. In fact, as the discussion below
shows, the insights and results of the preceding section may be extended to a model that
includes a state space, the interpretation of which depends on the context. It may represent
an image of the world, invoked by the decision maker, or an analytical framework within
which decision making is modeled. In either case, however, the state-space and associated
concepts are tacit constructs that lack well-defined choice-based foundations. The merit of
introducing a state-space, is the theoretical clarity that is affords. This is demonstrated by
the following models.

Let Ω be an abstract set of states, F an algebra on Ω, and P a probability measure on
F. A technology is a function t : Ω × A → Θ such that, for each a ∈ A, t (·, a) : Ω → Θ
is a F-measurable function. The choice of an action a together with a state ω produce the
effect t (ω, a).19 The probability space (Ω, F, P ) together with a technology t constitute the

19 It may seem that t (·, a) is a Savage act. This interpretation, however, is unwarranted because the direct
utility implication of the actions are state-contingent.

30

decision maker’s tacit perception of the world and/or its modeler’s depiction.

The decision maker’s priors on Θ are given by: π (·, a) = P (t−1 (·, a)) for all a ∈ A.
Moreover, for every observation x ∈ X, let Ωx denote the smallest subset of Ω such that
P (Ωx | x) = 1.20 Then a decision maker is Bayesian if, for all θ ∈ Θ, a ∈ A, and x ∈ X,

π (θ | a, x) = P (t−1 (θ, a) ∩ Ωx) (41)
P (Ωx) .

Note that in this formulation, θ is null given a and x if P (t−1 (θ, a) ∩ Ωx) = 0. Denote

by Θ (a, x) the set of effects that are nonnull given a and x. Then, the updating of the

probabilities by a decision maker who is Bayesian according to definition 2 may be written

as,

π (θ | a, x) = ¡ P (t−1 (θ, a) ∩ Ωx) ¢. (42)
P ∪θ∈Θ(a,x) (t−1 (θ, a) ∩ Ωx)

In many instances, the application of Bayes’ rule is based on the assumption that the
distribution of outcomes is governed by some parameters, and the updating pertains to the
distribution on the parameters. I propose below a formulation of the behavior of Bayesian
decision makers that corresponds better to this practice. The approach I take is that the
probabilities that figure in the representation of the decision maker’s prior preferences are
derived from underlying beliefs and a more sophisticated perception of the world that is
given by a probability space (Ω, F, P ) and a technology t.

The belief structure presumes the existence of a probability space of parameters, (Ψ, S, η) ;

20 Let {Ωα ⊂ Ω | P (Ωa | x) = 1}, then Ωx = ∩αΩa. One may also identify x with Ωx.

31

and a family, {µ (· | ψ)}ψ∈Ψ, of conditional probability distributions on Ω such that, for all
P

E ∈ F, P (E) = ψ∈Ψ µ (E | ψ) η (ψ) . In this instance, the prior probability of θ ∈ Θ given

a ∈ A is:

π (θ; a) = X µ ¡t−1 (θ, a) | ¢ η (ψ) . (43)
ψ

ψ∈Ψ

It is noteworthy that the probability space and the technology may depict the decision
maker’s actual view of the world or that attributed to him by the modeler. In either case,

insofar as the analysis of decision making is concerned, whether they depict correctly the

connections between states and effects and the parameters to the distributions, is beside the

point.

Let {p (· | ψ)}ψ∈Ψ be a family of likelihood functions on X, the set of observations. Then

by Bayes’ rule, for all x ∈ X,

π (θ | a, x) = X µ ¡t−1 (θ, a) | ¢ P p (x | ψ) η (ψ) (44)
ψ ψ0∈Ψ p (x | ψ0) η (ψ0) .
ψ∈Ψ

The following examples provide two interpretations of this model.

Example 3 A decision maker needs to undergo an operation whose outcomes (effects) are
success, θ, or failure, θ0. Let Ψ denote the set of possible parameter values underlying the
binomial distribution of success. The operation may be performed in one of two hospitals,
H1 or H2. Let a1 be the action of undergoing surgery at H1, and a2 is the action of un-
dergoing surgery at H2. An observation is the outcome record of the operation at the two
hospitals. Then, p (x | ψ) is the probability of observing x conditional on the success rate

32

being governed by a binomial process whose parameter is ψ. In this instance π (θ; ai) and
{π (θ | ai, x)}x∈X represent, respectively, the prior and posterior subjective probabilities of
success if the operation is performed at Hi.

Example 4 A decision maker considers adopting a diet and exercise regimen to reduce the
risk of heart disease. In this instance, effects are the decision maker’s cardiac conditions,
and actions correspond to distinct dietary and exercise regimens. Supposing that different
individuals have distinct genetic predispositions to heart disease, let Ψ denote the set of
parameters representing the relevant genetic information, and η the distribution of these
parameters. A medical test designed to identify the existence of a predisposition to heart
disease is available. Denote by X the set of possible results of the test and let p (x | ψ) denote
the probability of observing x conditional on the ψ being the true genetic information. It is
conceivable that the decision maker believes that, whatever might be his genetic constitution,
he can reduce the likelihood of heart disease by exercising regularly and maintaining a low-
fat diet. This belief is captured by the technology, where a ∈ A corresponds to alternative
diet and exercise regimens. The family {µ (t−1 (·, a) | ψ)}ψ∈Ψ of likelihood functions on Θ
represent this belief.

As before, let <x denote the posterior preference relation on C, characterizing the decision
maker after he has been informed of the observation x. Choice-based definitions of prior and
posterior subjective probabilities apply to {π (·; a)}a∈A and {π (· | x, a)}(x,a)∈X×A, only. The
implicit structure, consisting of the probability space and technology, serves as model that

33

links the prior and posterior subjective probabilities. The theory does not require that it
be well-defined. All that is needed is that the prior and posterior probabilities, themselves
observable in principle, be linked by the application of Bayes’ rule to update the distribution
on the parameter space. Accordingly, Bayesian decision maker may be defined, in this
framework, as follows:

Definition 4: A decision maker is Bayesian if there exists a probability space (Ω, F, P ) ; a

technology t; a parameter probability space (Ψ, S, η) ; a family, {µ (· | ψ)}ψ∈Ψ, of con-

ditional probability distributions on Ω, and a set of likelihood functions {p (· | ψ)}ψ∈Ψ;

¡P ¢
such that his prior preference relation, <, is represented by fa θ∈Θ u (b (θ) ; θ) π (θ; a)

and his posterior preferences {<x}x∈X are represented by fa ¡P u (b (θ) ; θ) π (θ | x, ¢ ,
a)
θ∈E

where, π (θ | a) , and {π (θ | x, a)}x∈X are given by equations (43) and (44), respectively.

5.2 Representation of the decision maker’s beliefs

Similarly to Theorem 2, the next theorem establishes that, if a decision maker is Bayesian in
the sense of definition 4, then the prior, action-dependent, subjective probability measures
representing his beliefs are identifiable.

Theorem 5 Suppose that there are at least two effects and the prior preference relation,
< on C, satisfies (A.0)—(A.3). If the decision maker is Bayesian in the sense of definition
4, then the representation of his prior preferences admits a unique set of action-dependent

34

probability measures.

The proof is outlined in the Appendix.

Remark: It is worth noting that it is possible to define Bayesian decision makers using
the state-space cum technology model without assuming the parametric structure. More
specifically, consider the following variation on the definition of Bayesian decision maker:

Definition 5: A decision maker is Bayesian if there exists a probability space (Ω, F, P )

and a technology t, such that his prior preference relation, < on C, is represented

by ¡P ¢ and his posterior preference relations, {<x}x∈X , are
fa θ∈Θ u (b (θ) ; θ) π (θ; a)

¡P ¢
represented by fa θ∈E u (b (θ) ; θ) π (θ | x, a) , where, π (θ | a) , and {π (θ | x, a)}x∈X

are given by equations (41) and (42), respectively.

By an argument analogous to the one used in the proof of Theorem 3, it can be shown
that, under the conditions in the hypothesis of Theorem 3, if a decision maker is Bayesian
in the sense of definition 5 the representation of his prior preferences admits a unique set of
action-dependent probability measures.

The implicit probability space cum technology structure is not subject to verification by
the observation of choice behavior, and is not necessarily unique. Distinct probability spaces
and technologies may be consistent with the same prior and posterior probability measures
characterizing a Bayesian decision maker’s beliefs. Therefore, from a choice-theoretic view-
point, the specification of the underlying probability space and technology is more a matter

35

of convenience than of substance.

6 Concluding Remarks

The traditional theory of decision making under uncertainty does not imply that the prior
and posterior beliefs of a Bayesian decision maker are identifiable. In this paper, I advance
a Bayesian decision theory that yields unique prior and posterior beliefs. The failure of
the traditional theory stems from the fact that it does not permit decision makers to take
actions that may influence the likely realization of events in the state-space. In the theory
of this paper, the ability to influence the likely realization of effects by their choice of action,
in conjunction with their expressed willingness to bet, and the changes thereof when new
information becomes available, yields the additional data needed to identify the subjective
probability that represent the decision maker’s beliefs. Using these probabilities, it is possible
to define a binary relation D on A × 2Θ, by (a, E) D (a0, E0) if π (E, a) ≥ π (E0, a0), for all E,
E0 ∈ E and a, a0 ∈ A. This binary relation is well-defined, choice based, and has the following
interpretation: The decision maker believes that E is more likely to obtain under the action
a that E0 is under a0.

36

References

[1] Anscombe, F., Aumann, R. J. 1963. A Definition of Subjective-Probability. Annals of
Mathematical Statistics 34, 199-205.

[2] Drèze, J. H. 1987. Decision Theory with Moral Hazard and State-Dependent Prefer-
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38

APPENDIX

A. Proof of Theorem 2. Let < on C satisfy (A.0)—(A.3). Then the prior preference
relation is represented as in Theorem 1.

Suppose that the decision maker is Bayesian and, for each x ∈ X, let <x on C represent

his posterior preferences conditional on the observation x. If the representation of the prior

preference relation < on C is given by à !
X
(a, b) → fa u (b (θ) ; θ) π (θ; a) , (45)

θ∈Θ

then, for every x ∈ X , the posterior preference relation is represented by

à !
X
(a, b) →7 fa u (b (θ) ; θ) π (θ | x, a) , (46)

θ∈Θ

where π (θ | x, a) = π (θ; a) / P π (θ0 ; a) if θ ∈ Θ (x, a) and π (θ | x, a) = 0 if θ ∈

θ0∈Θ(x,a)

Θ − Θ (x, a) .

Given a non-constant function γ : Θ → R++ let Γ (a) = P (θ) π (θ; a) .21 Then the
θ∈Θ γ

representation of the prior preferences < on C is given by
Ã!
X
(a, b) → fˆa uˆ (b (θ) ; θ) πˆ (θ; a) , (47)

θ∈Θ

where uˆ (b (θ) ; θ) = u (b (θ) ; θ) /γ (θ) , πˆ (θ; a) = γ (θ) π (θ; a) /Γ (a) , and fˆa = fa ◦ Γ (a) . For

every x ∈ X , the corresponding posterior preference relation is represented by:

Ã!
X
(a, b) →7 fˆa uˆ (b (θ) ; θ) πˆ (θ | x, a) , (48)

θ∈Θ

21 Notice, since γ is not constant, Γ (γ, a) 6= 1 for some a ∈ A.

39

where πˆ (θ | x, a) = πˆ (θ; a) / P πˆ (θ0 ; a) if θ ∈ Θ (x, a) and π (θ | x, a) = 0 if θ ∈

θ0∈Θ(x,a)

Θ − Θ (x, a) . But

πˆ (θ | x, a) = P γ (θ) π (θ; a) ; a) . (49)
∈Θ(x,a) γ (θ0) π (θ0 (50)
θ0

Thus, by definition,

X (θ | x, a) = PθP0∈Θθ0(∈xΘ,a()xγ,a)(θπ0)(θπ0 ; a) a) X (θ | x, a) .
uˆ (b (θ) , θ) πˆ (θ0; u (b (θ) ; θ) π

θ∈Θ θ∈Θ

Hence, in general,

XX (51)
uˆ (b (θ) , θ) πˆ (θ | x, a) =6 u (b (θ) , θ) π (θ | x, a) .

θ∈Θ θ∈Θ

Moreover, by definition,

fˆa à uˆ (b (θ) , θ) πˆ (θ | x, ! = fa ÃP π (θ0; a) X u (b (θ) ; θ) π (θ | x, ! . (52)
X a) Pθ0∈Θ(x,a) πˆ (θ0; a) a)
θ∈Θ
θ∈Θ θ0∈Θ(x,a)

To show that, for every non-constant function γ : Θ → R++ and prior probabilities

{π (·; a)}a∈A there exist observation x ∈ X such that <x=6 <ˆ x, fix {π (·; a)}a∈A and non-

constant γ. Let ¯b, ¯b0 ∈ B be such that u ¡¯b (θ) , ¢ = u¯ and u ¡¯b0 (θ) , ¢ = u¯0 for all θ ∈ Θ
θ θ

and fa (u¯) = fa0 (u¯0) . By Theorem 1, ¡ ¯b¢ ∼ ¡ ¯b0 ¢ . Moreover, by the representation in
a, a0,

equation (46), ¡ ¯b¢ ∼x ¡ , ¯b0¢, for all x ∈ X.
a, a0

Let x ∈ X be an observation such that

P π (θ0; a) =6 P ∈Θ(x,a0) π (θ0 ; a0 ) . (53)
Pθ0∈Θ(x,a) πˆ (θ0; a) Pθ0 ∈Θ(x,a0) πˆ (θ0 ; a0 )

θ0∈Θ(x,a) θ0

(To see that such an observation exists, note that the non-constancy of γ together with the

richness of the set of actions that accounts for the constant valuation bets being well-defined,

40

imply that Γ (a) 6= Γ (a0) for some a, a0 ∈ A. Thus there are θ ∈ Θ such that

π (θ; a) π (θ; a0) (54)
πˆ (θ; a) =6 πˆ (θ; a0) .

¡¢ ¡ ¢
Consider the observation xθ = θ, then Θ xθ, a = Θ xθ, a0 = {θ}, and inequality (53) is

reduced to (54)).

Observe next that fa (u¯) = fa0 (u¯0) together with equation (53) imply that

fa ÃP π (θ0; ! 6= fa0 ÃP π (θ0; ! . (55)
Pθ0∈Θ(x,a) πˆ (θ0; a) Pθ0∈Θ(x,a0) πˆ (θ0; a0) (56)

θ0∈Θ(x,a) u¯ θ0∈Θ(x,a0) u¯0
a) a0)

Thus, by equation (52),

à !à X !
X ¡¯b ¢ ¡¯b0 ¢
fˆa uˆ (θ) , θ πˆ (θ | x, a) =6 fˆa uˆ (θ) , θ πˆ (θ | x, a) .

θ∈Θ θ∈Θ

Hence, by definition, “not ¡ ¯b¢ ∼ˆ x ¡ ¯b0¢”. Hence <x6= <ˆ x.
a, a0,

But <x6= <ˆ x implies that {fa, π (· | x, a)}a∈A and u and {fˆa, πˆ (· | x, a)}a∈A and uˆ do
not represent the same posterior preferences and, therefore, cannot both be true. Thus

there is a unique representation {fa, π (·; a)}a∈A and u of the prior preference relations of a
Bayesian decision maker that induces that correct posterior representations. The probabili-

ties {π (·; a)}a∈A in this representation are unique. ¥

B. Proof of Theorem 5. The proof is similar to that of Theorem 2, and is only outlined
below.

41

Suppose that the decision maker is Bayesian in the sense of definition 4. Invoking Theo-

rem 1, let his prior preference relation, < on C, be represented by

à !
X
(a, b) → fa u (b (θ) ; θ) π (θ; a) , (57)

θ∈Θ

where π (θ; a) is given in equation (43). For every x ∈ X , let the posterior preference relation

be represented by à !
X
(a, b) 7→ fa u (b (θ) ; θ) π (θ | x, a) , (58)

θ∈Θ

where π (θ | x, a) is given in equation (44).

P
Fix a non-constant function γ : Θ → R++ let Γ (a) = θ∈Θ γ (θ) π (θ; a) . Then, by equa-

tion (43) Γ (a) = P γ (θ) P µ (ta−1 (θ) | ψ) η (ψ) . Define uˆ (b (θ) ; θ) = u (b (θ) ; θ) /γ (θ) ,

θ∈Θ ψ∈Ψ

πˆ (θ; a) = γ (θ) π (θ; a) /Γ (a) , and fˆa = fa ◦ Γ (a) . Then the prior preferences, < on C, is

represented by Ã!

(a, b) → fˆa X uˆ (b (θ) ; θ) πˆ (θ; a) . (59)
(60)
θ∈Θ

Moreover, for every x ∈ X , the posterior preference relation <x is represented by:

Ã!
X
(a, b) →7 fˆa uˆ (b (θ) ; θ) πˆ (θ | x, a) ,

θ∈Θ

where

γ (θ) P µ (t−1 (θ, a) | ψ) S η(ψ)p(x|ψ) ) γ (θ) π (θ | x, a),
(θ0, a) ∈Ψ η(ψ0)p(x|ψ0 Γ (x, a)
πˆ (θ | x, a) = ψ∈Ψ ψ0 = (61)
P P
γ (θ0) µ (t−1 | ψ) S η(ψ)p(x|ψ)
θ0∈Θ ψ∈Ψ ψ0 ∈Ψ η(ψ0)p(x|ψ0)

and Γ (x, a) = P γ (θ0) π (θ0 | x, a) .

θ0∈Θ

Thus

X 1X
uˆ (b (θ) , θ) πˆ (θ | x, a) = u (b (θ) ; θ) π (θ | x, a) , (62)
Γ (x, a)
θ∈Θ θ∈Θ

42

and, for all a ∈ A,

à !à !
X Γ (a) X
fˆa uˆ (b (θ) , θ) πˆ (θ | x, a) = fa Γ (x, a) u (b (θ) ; θ) π (θ | x, a) . (63)

θ∈Θ θ∈Θ

As in the proof of Theorem 2, if the set of observations is sufficiently rich then, for every

non-constant bounded function γ : Θ → R++ and prior probabilities {π (·; a)}a∈A, there is
an observation x ∈ X such that <x=6 <ˆ x.22 The conclusion follows from by the argument

used in Theorem 2. ¥

22 For this purpose X is rich if, for every ψ ∈ Ψ there is x ∈ X such that p(x | ψ) > 0 and p(x | ψ0) = 0 for
all ψ0 ∈ Ψ − {ψ}.

43