Subjective Expected Utility Theory with Costly Actions - Economics ...

Subjective Expected Utility Theory with Costly
Actions
Edi Karni ¤
Johns Hopkins University
November 13, 2003
Abstract
This paper explores alternative axiomatizations of subjective expected
utility theory for decision makers with direct preferences over actions;
including a general subjective expected utility representation with actiondependent
utility, and separately additive representations. In the context
of the state-space formulation of agency theory the results of this paper
constitute axiomatic foundations of the agent’s behavior.
1 Introduction
Agency theory admits alternative formulations, including the state-space formulation,
parameterized distribution formulation, and general distribution formulation.
1 In all these formulations the agent’s preferences are de…ned over action
- acts (that is, contingent payo¤) pairs. Despite the obvious interest in agency
theory and the huge volume of research that it inspired, almost no attention has
been paid to the development of axiomatic foundations of the agent’s decision
problem. This omission may re‡ect a belief that, because the agent is presumed
to be an expected utility maximizer, subjective expected utility theory provides
such a foundation. According to this belief actions are labels of the acts they
induce and do not enter the agent’s preferences directly. A careful examination
reveals that this belief, while containing an important element of truth, is not
fully justi…ed.
Situations requiring decision making under uncertainty in which, prior to
the resolution of uncertainty, the decision maker may take costly measures that
¤ My thinking on the issues discussed in this paper was inspired and stimulated by conversations
with Steven Matthews, for which I am deeply grateful. I also bene…ted from Steve’s
comments on an earlier version of this paper. I am particularly grateful to Larry Epstein and
Peter Wakker for their insightful comments and suggestions and to John Quiggin for calling
my attention to some of the references. Suggetions of Simon Grant, Matthew Jackson, Mark
Machina, and Zvi Safra help improve the exposition. Financial support by the NSF under
grant SES-0314249 is gratefully acknowleged.
1 See Hart and Holmstrom (1987) and Chambers and Quiggin (2000).
1

a¤ect the consequences associated with some states do not arise exclusively in
the context of principal-agent relationships. Indeed, such situations are common
in economic analysis and often arise even when the decision maker acts in
isolation. For instance, Robinson Crusoe may have to choose between building
his hut on a ‡ood plane or on high grounds and thereby, at a cost, a¤ect the
consequences of ‡ooding. Robinson Crusoe’s choice is not adequately captured
by traditional subjective expected utility theory because in that theory (e.g.,
Savage (1954)) the choice set includes all acts (that is, all assignments of consequences
to states) and, more importantly, acts do not enter the preferences
directly. This framework is inadequate to describe the choice facing Crusoe.
Building his hut on high grounds involves a larger expense of time and e¤ort.
Moreover, because the time and e¤ort to be spent must be determined in advance,
they cannot vary with the state of nature (e.g., they cannot depend on
the amount of precipitation). Put di¤erently, to …t into Savage’s framework,
the time and e¤ort spent building the hut must be incorporated into the description
of the consequences and the set of acts must include all assignments
of consequences to states. The problem is that the decision of where to locate
the hut must be made prior to the realization of the state yet acts, such as f
described below, must be contemplated:
States Flood No Flood
f
Hutlostina‡ood.
Little time and e¤ort spent
building hut on ‡ood plan
Hut stands.
Signi…cant amount of time and e¤ort spent
building hut on high ground
Put in more general terms, let X be the set of outcomes and denote by A the
set of actions (e.g., the hut damaged by a ‡ood is a consequence and time and
e¤ort spent is an action). If consequences are de…ned as action-outcome pairs
then Savage’s set of acts is given by (A £ X) S ; where S denotes the set of states
of nature. But, as the above example suggests, this set contains acts that are
incredible. To suppose that the decision maker is able to express meaningful
preferences as regards such acts creates a conceptual di¢culty with the theory. 2
This di¢culty is averted if, for instance, the only cost of actions is the
…nancial cost and the consequences represent …nancial rewards then actions are
indeed labels of the corresponding acts. To see this, let c (a) denote the …nancial
cost of implementing the action a and denote by x (s; a) the state-contingent
action-dependent monetary payo¤, then the state-contingent payo¤ function
xa (¢) =x (¢;a)¡c(a) is a Savage-type act and a is its label. In this example the
fact that the cost of action and the reward are additive is essential. However,
if the action involves spending time (e.g., while seeking employment) and the
state-contingent payo¤ is monetary (e.g., wages or unemployment insurance
bene…ts depending on the eventual state of employment) then, not being perfect
substitutes, actions must be separated from acts.
2 I am indebted to Steve Matthews for suggesting this discussion of Savage’s theory.
2

It seems, therefore, that an examination of the axiomatic foundations of subjective
expected utility theory for decision makers with direct preferences over
actions is warranted. As in other …elds, this study is intended to uncover and
allow clearer understanding of the behavioral premises underlying the theory,
thereby opening the way to a critical examination of its basic tenets.
In this paper I invoke the state-space formulation and pursue an approach in
which decision makers are supposed to have preference relations on M(A)£F jAj ;
where M (A) is the set of probability measure on A; and F is the subset of X S
containing all the simple acts: The resulting model …ts naturally situations in
which the eventual outcome is determined by the state of nature and stateindependent
actions taken by the decision maker, such as the decisions of the
agent in agency theory.
2 Preference Structures and Representations
2.1 Preliminaries
Let S be an in…nite set whose elements are states of nature and whose subsets
are events. Consequences, or outcomes, are real numbers representing monetary
payo¤s. Let the set of outcomes, X; be a compact interval in the real line.
Following Savage (1954), an act is a function from the set of states to the set
of outcomes. Let F denote the set of simple acts, namely, acts under which the
image of S is …nite. Assume that F is endowed with the product topology, then
it is a compact set. Denote by f a generic element of F and by x the constant
act f (s) =x for all s 2 S:
Let A = fa1; :::; ang; 2 · n

The next axiom requires that the evaluation of coordinates is independent
in the sense that preferences among alternatives of the form (e i ; (f;f¡i)); where
(h; f¡i) :=(f1;:::;fi¡1;h;fi+1; :::; fn) ; depends solely on the i¡th coordinate of
f. Formally,
(A.3) (Certainty Principle) For all f; g; h; l 2 F n ; (e i ; (f;f¡i)) < (e i ; (g; g¡i))
if and only if (e i ; (f;h¡i)) < (e i ; (g; l¡i)):
In view of the certainty principle, to simplify the notations, I shall henceforth
write (ai;f) < (aj;f 0 ) instead of (e i ; (f;f¡i)) < (e i ; (f 0 ; g¡j)):
To introduce the next axiom de…ne a partial mixture operation on D: Let
(®; f) and (¯;f) in D then (¸® +(1¡ ¸)¯;f) ;¸2 [0; 1] is an element of D such
that (¸® +(1¡ ¸)¯)i = ¸®i +(1¡ ¸)¯ i: This may be interpreted as a two-stage
lottery in which in the …rst stage the alternatives (®; f) and (¯;f) are obtained
with probabilities ¸ and 1 ¡ ¸; respectively. In the second stage the coordinate
of f is selected by the lottery, ® or ¯, that corresponds to the outcome of the …rst
stage. With this interpretation in mind assume that the decision maker prefers
(®; f) over (® 0 ; g) and (¯;f) over (¯ 0 ; g): Moreover, assume that if a decision
maker faces a choice between the alternatives d =(¸® +(1¡ ¸)¯;f) and d 0 =
¡ ¸® 0 +(1¡ ¸)¯ 0 ; g ¢ he reasons as follows: If the event whose probability is ¸ is
realized then he participates in the lottery (®; f) if he has chosen d andinthe
lottery (® 0 ; g) if he has chosen d 0 : Conditional on the realization of this event, he
is better o¤ with d: Bythesamelogichewouldalsopreferd over d 0 conditional
on the realization of the event whose probability is 1 ¡ ¸: Consequently, he
prefers d over d 0 unconditionally. Formally,
(A.4) (Constrained Independence) For all (®; f), (¯;f),(® 0 ; g), (¯ 0 ; g) in
D and ¸ 2 [0; 1) if (®; f) » (® 0 ; g) then (¯;f) < (¯ 0 ; g) if and only if
(¸® +(1¡ ¸)¯;f) < ¡ ¸® 0 +(1¡ ¸)¯ 0 ; g ¢ :
Constrained independence is weaker than the independence axiom of expected
utility theory (see discussion in Karni and Safra [2000]).
A real-valued function V on D is said to represent < if, for all (®; f) and
(¯;g) in D; (®; f) < (¯;g) if and only if V (®; f) ¸ V (¯;g) :
Theorem (Karni and Safra [2000]) Let < be a binary relation on D. Then
the following conditions are equivalent:
(a) < is a preference relation satisfying (A.1)-(A.4).
(b) There exist continuous non-constant functions V : D ! R and U (¢; ai) :
F ! R; ai2 A, such that V represents < and, for all (®; f) 2 D;
nX
V (®; f) = ®iU(fi; ai):
Moreover, if there are functions W : D ! R and W (¢; ai) :F ! R;ai 2 A;
such that W represents < and W (®; f) = Pn i=1 ®iW (fi; ai); then W (¢; ai) =
BU (¢; ai)+C; B > 0.
4
i=1

2.2 The main result
For each a 2 A de…ne a conditional preference relation,

Theorem 1 Let < be a binary relation on D; then the following conditions are
equivalent:
(i) < is a preference relation satisfying (A.1) - (A.7).
(ii) There exist a nonatomic probability measure, ¼; on S, and continuous
real-valued functions, u on X and ª on A £ R; such that V represents <
and, for all (®; f) 2 D;
V (®; f) =
nX
i=1
®i
Z
S
ª(ai;u(f (s)) d¼ (s) ; (1)
where ª(a; ¢) is a monotonic increasing transformation. Moreover, ¼ is
unique with ¼(E) > 0 if and only if E is null, and u is unique up to
positive linear transformation.
Remark: In reality decision makers choose among action-act pairs. Theorem
1 implies that the restriction of the preference relation < to A £ F has the
following representation: For all (a; f) and (a 0 ;f 0 ) in A £ F;
(a; f) < (a 0 ;f 0 ) ,
Z
S
Z
ª(a; u (f (s)) d¼ (s) ¸
Hence the representation in (2) is applicable.
ª(a
S
0 ;u(f 0 (s))) d¼ (s) : (2)
Proof.
(i) ! (ii) :
The proof that (ii) ! (i) is straightforward. I shall prove that
Axiom (A.5) and Savage’s (1954) theorem imply that, for every given a 2 A;
there exist a nonatomic probability measure ¼ (¢; a) on S and a real-valued
function, u; on X such that,
Z
U(f; a) = u (f (s) ;a) d¼ (s; a) : (3)
S
where ¼ (¢; a) is unique and u (¢;a) is unique up to positive linear transformation.
Moreover, U(f; a) represents

where, for each E µ S; ¼ (E;a) = R
E
u (x0 ;a) > 0 if and only if u (x; a0 ) ¡ u (x0 ;a0 ) > 0: Hence
¼ (s; a) ds: But, by (A.6), u (x; a) ¡
¼ (E;a) ¡ ¼ (G; a) ¸ 0 , ¼ (E;a 0 ) ¡ ¼ (G; a 0 ) ¸ 0: (7)
But equation (7) implies that, for every E µ S and all a; a 0 2 A; ¼ (E;a) =
¼ (E;a 0 ) = ¼ (E) : To prove the last assertion suppose, by way of negation,
that for some E ½ S and a; a 0 2 A; ¼ (E;a) >¼(E;a 0 ) : Because the measures
¼ (¢;a) ;a 2 A are nonatomic, by Liapunov’s theorem 3 there exist K ½ S such
that ¼ (E;a) >¼(K; a) =¼ (K; a 0 ) >¼(E;a 0 ) : (To see this observe that the
point (¼ (E;a) ;¼(E;a 0 )) is below the diagonal of the unit square. But, by
Liapunov’s theorem, the set f(¼ (T;a) ;¼(T;a 0 )) 2 [0; 1] 2 j T 2 2 S g is closed,
convex and contains the diagonal. Let r satisfy ¼ (E;a) >r>¼(E;a 0 ) and K
be an event whose image under (¼ (¢;a) ;¼(¢;a 0 )) is on the diagonal and satis…es
¼ (K; a) =¼ (K; a 0 )=r:) Then Substituting K for G in equation (7) leads to a
contradiction. Hence ¼ (¢; a) =¼ (¢) for all a 2 A: Moreover, it is easy to verify
that ¼ (E) > 0 if and only if E is null.
Equation (3) together with the Theorem of Karni and Safra (2000) imply
that
V (®; f) =
nX
i=1
®i
Z
u(fi (s) ;ai)d¼ (s) ;
S
where V and u are continuous functions. But, by (A.6), u(¢;a) is ordinally equivalent
to u (¢;a 0 ) for all a; a 0 2 A: Hence, for all ai 2 A; u(¢;ai) =ª(ai;u(¢;a1))
where ª(ai; ¢) is monotonic increasing continuous transformation. Let u(¢) =
u(¢;a1) to obtain the representation in (ii) :
The uniqueness follows from Savage’s theorem.
2.3 Action-a¢ne utility representation
Insofar as the dependence of the utility function on the action is concerned,
the result in Theorem 1 is quite general. In some applications the e¤ect of
the choice of action on the decision maker’s well-being calls for a more speci…c
functional form of the utility function. For example, when actions correspond to
levels of e¤ort, it is commonplace to assume some form of separability between
the utility of money and the utility of e¤ort. I explore next the axiomatic
underpinnings of two alternative, more speci…c, representations of the decision
maker’s preferences. The …rst, action-a¢ne utility representation, was used in
Grossman and Hart (1983).
The next axiom requires that, for any given event, E; the “intensity of preferences”
between consequences on the complementary event S ¡E be independent
of the action. The formal statement of this idea is an adaptation of Wakker’s
(1987) cardinal consistency axiom. 4
3 See Hildenbrand (1974).
4 See Karni (2003, 2003a) for di¤erent adaptations of the same idea.
7

(A.8) (Action-independent intensity of preferences) For all f;g;h;l 2
F , a; a0 2 A, E ½ S; and w; x; y; z; 2 X, iffEw

Proof. (i) ) (ii) : Fix an event E ½ S such that both E and S ¡ E are
nonnull and a 2 A; then, by Theorem 1, there exist f;g;2F andsuchthat
Z
[ª (a; u(f (s))) ¡ ª(a; u(g (s)))] d¼ (s) =³>0: (10)
E
Given a0 2 A; by Theorem 1, there exist acts h; l 2 F that satisfy
Z
[ª (a
E
0 ;u(h (s))) ¡ ª(a 0 ;u(l (s)))] d¼ (s) =">0. (11)
By continuity of the functions ª(a; ¢) and u; for every ^ ³ 2 [¡³;³]; ^" 2 [¡"; "];
there exist f 0 ;g 0 ;h 0 ;l 0 2F and such that
and
Z
Z
[ª (a; u(f
E
0 (s))) ¡ ª(a; u(g 0 (s)))] d¼ (s) = ^ ³ (12)
[ª (a
E
0 ;u(h 0 (s))) ¡ ª(a 0 ;u(l 0 (s)))] d¼ (s) =^". (13)
De…ne Á a 0 by ª(a 0 ;u(¢)) = Á a 0 ± ª(a; u(¢)) for all a 0 2 A: Then Á a 0 is
a continuous function. To show that Á a 0 is positive a¢ne or constant take
®; ¯; °; ± 2 R such that ¡³ · ® ¡ ¯ = ° ¡ ± · ³ and ¡" · Á a 0 (¯) ¡ Á a 0 (®) · ":
Let w; x; y; z 2 R satisfy ª(a; u(x)) ¼ (S ¡ E) =®; ª(a; u(w)) ¼ (S ¡ E) =¯;
u (z; a) ¼ (S ¡ E) =° and ª(a; u(y)) ¼ (S ¡ E) =±: Take ^ f;^g 2 F such that
Z
E
h ³
ª a; u( ^ ´
i
f (s)) ¡ ª(a; u(^g (s))) d¼ (s) =® ¡ ¯: (14)
Thus ^ fEw »a ^gEx and ^ fEy »a ^gEz:
Take ^ h; ^ l2F such that,
Z
E
h ³
ª a 0 ;u( ^ ´ ³
h (s)) ¡ ª a 0 ;u( ^ ´i
l (s)) d¼ (s) =Áa0 (®) ¡ Áa0 (¯) : (15)
Since ª(a 0 ;u(¢)) = Á a 0 ± ª(a; u(¢)) this implies ^ hEw »a 0 ^ lEx: Applying (A.8)
twice yields ^ hEy »a0 ^lEz: Thus
Z
Á a 0 (±) ¡ Á a 0 (°) =
E
h ³
ª a 0 ;u( ^ ´ ³
h (s)) ¡ ª a 0 ;u( ^ ´i
l (s)) d¼ (s) =Áa0 (¯) ¡ Áa0 (®) :
(16)
By Wakker (1987) Lemma 4.4 this implies that Á a 0 is a¢ne. But Á a 0 is nondecreasing.
(To see this, let fEw

(ii) ) (i) : Assume that there exist positive a¢ne or constant transformations
Á a 0 such that ª(a 0 ;u(¢)) = Á a 0 ± ª(a; u(¢)) for all a 0 2 A. Fix an event
E ½ S such that E and S ¡ E c are nonnull and suppose that fEw

Two actions, a; a 0 2 A are elementarily linked if there are x; x 0 ;y;y 0 2 X
such that (a; x) Â (a; y) ; (a; x) » (a 0 ;x 0 ) and (a; y) » (a 0 ;y 0 ) : Two actions, a
and a 0 are said to be linked if there exists a …nite sequence a = a0;a1; :::; an = a 0
such that every aj is elementarily linked with aj+1: The axiomatic underpinning
of the separately additive speci…cation consists of the preceding axioms and, in
addition, the following independence condition,
(A.9) (Independence) For all E ½ S; a; a 0 2 A and x; x 0 ;y;y 0 2 X; (a; xEy) <
(a 0 ;x 0 E y) if and only if (a; xEy 0 ) < (a 0 ;x 0 E y0 ) :
The next theorem characterizes the separately additive representation.
Theorem 4 Let < be a binary relation on D and suppose that every pair of
actions a; a 0 2 A is linked. Then the following conditions are equivalent:
(i) < is a preference relation satisfying (A.1)-(A.9).
(ii) There exist a nonatomic probability measure ¼ on S, real-valued continuous
functions, u on X and ¹ on A, and a continuous function V representing

3 Discussion
To begin with consider the application of the models developed in the preceding
section to agency theory. A technology, t; is a mapping from the set A£S to the
set of consequences. The agent’s choice of action a 2 A and nature’s “choice” of
astates 2 S jointly determine the outcome t(a; s) 2 R: Thus given a technology,
t; andanaction,a; t (¢; a) is an act in F . In other words, ta := t (a; ¢) 2 F is
a random variable whose value ta (s) is the monetary payo¤s associated with
the action a in the state s: The full support assumption often invoked in the
formulation of agency problem requires that the range of possible payo¤s be
uniform across actions, that is, T = ta (S) for all a 2 A: 5
An incentive contract is a function w : T ! R, wherew (x) is the payo¤ to
the agent as a function of the realized outcome x: Let W denote the set of all
incentive contracts. Note that, given a technology t; each action a and incentive
contract w correspond to the act ^w (a;w) 2 F de…ned by ^w (a;w)(s) =w (ta(s))
for all s 2 S: Similarly, the residual payo¤ ta ¡ ^w (a;w) is also an act in F:
Suppose that the principal’s preference relation, < P ; on F satis…es Savage’s
(1954) axioms and that the agent has a preference relation, < A ; on M (A)£F jAj
satisfying the axioms in either Theorems 1, 2, or 3.
Assume that the production technology is common knowledge, the level of
output is veri…able, and the choice of action is private information of the agent.
Under these assumptions the information regarding the state becomes a focal
issue in the formulation of the agency problem. In decision theory it is generally
assumed that the true state of nature becomes known ex post. Presumably in
order to verify whether he is correctly rewarded according to the act that he
chose, the decision maker must be able to observe the state. However, if the state
and the consequences are observable and the technology is known, each action
the agent may choose can be inferred on some events (subset of states). Hence
the principal is in a position to implement the …rst-best solution by imposing
a su¢ciently large penalty on all the events-output combinations that, under
the known technology, indicate that the agent chose an action other than that
required to attain the …rst-best solution. Thus for the agency problem to be
nontrivial it is necessary to assume that the states are not directly observable. 6
Within this framework the principal’s problem may be stated as follows:
Choose an action-contract pair (a ¤ ;w ¤ ) such that
¡ ta ¤ ¡ ^w (a ¤ ;w ¤ )
¢ P
< ¡ ¢
ta ¡ ^w (a;w) for all (a; w) 2 A £ W (28)
subject to the incentive compatibility constraints
¡ ¢ ¤ A
a ; ^w(a ¤ ;w ¤ ) < ¡ ¢
a; ^w (a;w ¤ ) for all a 2 A (29)
and the participation constraint
¡ ¢ ¤ A
a ; ^w(a ¤ ;w ¤ ) < (¹a; ¹w) ; (30)
5 See Shavel (1979).
6 See, for example, Quiggin and Chambers (1998).
12

where (¹a; ¹w) denotes the “outside option,” that is, the best course of action
available to the agent in case he declines the contract. This outside option is
the best action-contract pair attainable by the agent.
Typically the principal-agent problem is stated in terms of the utility representation
of the preferences of the two parties and not in terms of their preference
relations. To obtain such a representation let S (x; a) =fs 2 S j ta (s) · xg for
every x 2 T and a 2 A. For every a de…ne j 0 s subjective cumulative distribution
function, ¦ j (¢; a) on T by
¦ j (x; a) =¼ j (S (x; a)) ; j 2fP; Ag; (31)
where ¼j is j0s subjective probability derived from the corresponding preference
relation. Using these notations and Theorem 3, the principal-agent problem
(28) - (30) may be restated in its conventional form as follows: Choose an
action-contract pair (a ¤ ;w ¤ ) such that
(a ¤ ;w ¤ ) 2 arg max
A£W
subject to the incentive compatibility constraints
Z
Z
u
T
A (x ¡ w ¤ (x)) d¦ A (x; a ¤ )+v (a ¤ ) ¸
and the participation constraint
Z
Z
u
T
P (x ¡ w ¤ (x)) d¦ P (x; a ¤ ) (32)
u
T
A (x ¡ w ¤ (x)) d¦ A (x; a)+v (a) ; 8a 2 A
(33)
u
T
A (x ¡ w ¤ (x)) d¦ A (x; a ¤ ) ¸ u0; (34)
where u0 is the reservation utility level associated with the outside option.
Note that the similar representations apply under the conditions of Theorems
1 and 2 with the corresponding utility functions.
An alternative formulation of agency theory, the parameterized distribution
formulation, pioneered by Mirrlees (1974, 1976), avoids the use of a state space
as a primitive concept. In a recent paper Karni (2003b) I developed an axiomatic
foundation of the parameterized distribution formulation of agency theory.
13

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