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Additive representations over actions and acts
Mathematical Social Sciences 48 (2004) 113 – 119 www.elsevier.com/locate/econbase
Additive representations over actions and acts Edi Karni * Department of Economics, Johns Hopkins University, 34th and Charles St. Baltimore, MD 21218-2685, USA Received 1 October 2003; accepted 1 March 2004 Available online 1 June 2004
Abstract This paper develops axiomatic foundations of rational decision making involving a choice among action – acts pairs. In the context of the state-space formulation of agency theory, this paper clarifies the assumptions underlying the additive representations that are often used to depict agents’ behavior in the analysis of principal – agent problems. The paper invokes the algebraic approach that renders the analysis general. D 2004 Elsevier B.V. All rights reserved. Keywords: Additive representation; Agency theory JEL classification: D81; D82
1. Introduction The state-space formulation of principal –agent relations in the presence of a moral hazard problem includes the following primitives: a state-space S, a set of actions A, a set of consequences X, and a technology t, represented by a mapping from A S to X. There is a true state of the world, sVaS, that, once known, resolves all the uncertainty. That is, if the true state is known, then the consequences of every action would be known. Thus, if the agent were to choose the action, a, the result would be the consequence t(a,sV)aX. In addition, it is assumed that the primitives are common knowledge, the action chosen is private information of the agent, and the true state of the world is not known at the time the action must be chosen. The principal and the agent have distinct interests in their relationship. The principal’s main interest are the consequences and he is concerned with the actions only insofar as they *Tel.: +1-410-516-7608; fax: +1-410-516-7600. E-mail address: [email protected] (E. Karni). 0165-4896/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2004.03.001
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affect the likely realization of the alternative consequences. By contrast, the agent is concerned with the consequences only to the extent that they affect his payoff under the contract, while his interest in the actions is direct. In other words, to the agent actions are costly and choosing an action has direct implications for his well-being. This difference of outlook requires distinct approaches to modeling the principal’s and the agent’s choice behavior. Assuming, as is customary in agency theory, that both the principal and the agent are expected utility maximizers, the principal’s preferences may be modeled using Savage’s (1954) theory. Specifically, the principal’s preference relation is a binary relation on (X R)S satisfying all of Savage’s (1954) postulates. The modeling of the agent’s preferences, however, must take into account his direct preferences on the actions. In particular, denote by F the set of all real valued functions on S, then agents are characterized by preference relations + on A F. (Note that, insofar as the agent is concerned, F constitutes the set of Savage-type acts). In Karni (2003), I explore the axiomatic underpinnings of alternative representations of agents’ preferences. In that work the choice set was M(A) H jAj, where M(A) is the set of probability measures on A, X is a compact interval, and H is the subset of X S containing all the simple acts (that is, acts that have finite range). In many applications of agency theory, however, it is assumed that the agent’s preferences are additively separable in income and actions (e.g., Holmstrom, 1979). In this paper, I develop an axiomatic model that dispenses with the use of probabilities as a primitive concept. In this model, the agent’s behavior is characterized by additive representation on A F. Moreover, in this paper, I invoke the algebraic approach and a representation result of Karni and Safra (1998) to obtain a model that is more general in the sense of not assuming a topological structure on the choice set.
2. Axioms and representations 2.1. Axioms Assume, henceforth, that the state-space is infinite and that the preference relation + on A F is a weak order (that is, + is complete and transitive). The strict preference relation, d, and the indifference relation, f , are the asymmetric and symmetric parts of +, respectively. Moreover, suppose that, given any action, the preference relations on the set of acts satisfy all the axioms of Savage (1954). Formally, for every given aaA, define a conditional preference relation +a on F by f +ag if and only if (a, f )+(a,g), for all f, gaF, then (A.0). For every given aaA, +a on F satisfies the postulates (P.1) – (P.7) of Savage (1954). The alternatives that the agent must choose from have two components, namely, action and act. The next axiom requires that the preference relation + displays independence of equal component. Formally: (A.1) (Independence). For all (a, f ),(aV, f V)aA F, (a, f )+(a, f V) implies (aV, f )+(a V, f V) and (a, f )+(a V, f ) implies (a, f V)+(aV, f V).
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Independence implies that +a = +aV, for all a,aVaA. To emphasize this fact and to simplify the notations, I denote the induced relation +a on F by +F. Similarly, define an induced binary relation +A on A by a+Aa V if there exist faF such that (a, f )+(a V, f ). Again, independence implies that this relation is well-defined. Actions and acts are essential if the asymmetric parts of the induced relations +A and +F, A and F, respectively, are nonempty. A sequence {an}aA is an increasing standard sequence if there exist f, f Va F such that f VdF f and, for all n z 1, (an, f V) f (an + 1, f ). A decreasing standard sequence is defined analogously. A standard sequence is bounded if there exist a,a VaA such that, for all n, a+Aan+Aa V. The standard (bounded) sequence in F is defined similarly. Definition. The relation + is Archimedean if every bounded standard sequence is finite. The relation + satisfies solvability if, whenever a,a V,aWaA and f, f Va F for which (aV, f V)+(a, f )+(aW, f V), then there is a¯aA such that (a¯, f V) f (a, f ) and a similar condition holds with A and F interchanged. (A.2) (Solvability). The relation + satisfies solvability. Because there are at most two essential components, independence is not sufficient for the sought-after additive representation. The next well-known hexagon condition is also required. (A.3) (Hexagon condition). For all a,a V,aWaA and f, f V, f Wa F, if (a,f V) f (a V, f ) and (aW, f ) f (aV, f V) f (a, f W), then (a V, f W) f (aW, f V). 2.2. Additively separable representation An array of real-valued functions (Vj)jn= 1 is said to be a jointly cardinal representation n of Pn + on a product set D ¼ jj¼1Ds if, for all d,dVaD, d+d Vif and only if Pna binary relation j¼1 Vj ðdj Þz j¼1 Vj ðdj VÞ , and the class of all functions that constitute an additive representation of + consists of those arrays of functions, (Wj)jn= 1, that satisfy Wj = bVj + aj, b>0, j = 1,. . .,n. Theorem 1 . Let + be a binary relation on A F and suppose that both actions and acts are essential. Then the following conditions are equivalent: (i) + is an Archimedean weak order satisfying (A.0) (A.1) (A.2) (A.3). (ii) There exist a nonatomic probability measure p on S, a real-valued function, u, on R, a real-valued function, v, on A, and a strictly monotonic increasing, function U: R ! R, such that, for all (a,f ) and (aV,f V) in A F, Z Z ða; f Þ+ðaV; f VÞZU uðf ðsÞÞdpðsÞ þ vðaÞzU uðf VðsÞÞdpðsÞ þ vðaVÞ: S
S
ð1Þ Moreover, p is unique, U and v are jointly cardinal representation of +, and u is unique up to positive affine transformation.
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Proof. (i) ! (ii). Since both actions and acts are essential, Karni and Safra (1998) imply that the binary relation + on A F is an Archimedean weak order satisfying (A.1), (A.2), and (A.3) if and only if there exist real-valued functions v on A and V on F such that, for all (a, f ) and (aV, f V) in A F, ða; f Þ+ðaV; f VÞZV ð f Þ þ vðaÞzV ð f VÞ þ vðaVÞ:
ð2Þ
Moreover, V and v are jointly cardinal representation of +. By (A.0) and Savage’s (1954) theorem, there exist a unique, nonatomic, probability measure, p, on S and a real-valued function, u, on R, unique up to positive linear transformation, such that, for every f,gaF, Z
uðf ðsÞÞdpðsÞ z
f +F g Z S
Z uðgðsÞÞdpðsÞ;
ð3Þ
S
However, by Eq. (2), f +Fg if and only if V( f ) z V ( g). Hence, Z V ðf Þ ¼ U uðf ðsÞÞdpðsÞ ;
ð4Þ
S
where U is strictly monotonic increasing function. (ii) ! (i). The proof that Eq. (2) implies that + is an Archimedean weak order satisfying (A.1), (A.2), and (A.3) is straightforward. The proof that Eq. (4) implies (A.0) is an implication of Savage’s (1954) theorem. 5 2.3. Subjective expected utility representation In many applications, it is assumed that U is a positive affine function. This specification requires additional restrictions on the preference relation. One such restriction is given in the following condition whose statement requires the following additional notation: For all ya R, denote by y the constant function faF, such that f(s) = y for all saS. For every given EoS and x, yaR, denote by xEy the act faF satisfying f(s) = x for all saE and f (s) = y otherwise. (A.4) (Uniform utility differences). Let a,aVaA and y,y VaR satisfy (a, f )d(aV, f ) for all faF, (a,y)d(a,y V) for all aaA, and (a,y V) f (aV,y). Then, for all x, x VaR, (a, x V) f (aV, x) if and only if (a, x VEy) f (a, xEy V) for all EoS such that (a,wE z) f (a, zEw) for all z,waR and aaA. Axiom (A.4) looks complicated, but it is easy to understand when dissected. To begin with the last part of (A.4), namely, the requirement that(a,wEz) f (a,zEw) for all z,waR and aaA, implies that E is an event whose (subjective) probability is one-half. In other words, E and its complement in S are ethically neutral events in the sense of Ramsey (1931). (Notice that the nonatomicity of the subjective probability measure in Savage
E. Karni / Mathematical Social Sciences 48 (2004) 113–119
117
(1954) model assures that such events exist.) According to this interpretation of E, the condition that, for all x,x VaR, (a, x V) f (aV, x) if and only if (a, x VEy) f (a, x Ey V) implies that x and xVhave the same von Neumann – Morgenstern (that is, u) utility difference as y and y V. The equivalences (a, yV) f (a V,y) and (a, x V)f (a V,x) imply that x and x Vhave the same utility differences as y and yV in terms of the transformation U of the subjective expected utility. The addition of Axiom (A.4) implies that the transformation U must be positive linear. Formally, Theorem 2. Let + be a binary relation on A F and suppose that both actions and acts are essential. Then the following conditions are equivalent: (i) + is an Archimedean weak order satisfying (A.0) (A.1) (A.2) (A.3) (A.4). (ii) There exist a unique, nonatomic, probability measure p on S, a real-valued function, u, on R, and a real-valued function, v, on A such that, for all (a,f ) and (aV,f V) in A F, ða; f Þ+ðaV; f VÞZ
Z
uð f ðsÞÞdpðsÞ þ vðaÞz
S
Z
uð f VðsÞÞdpðsÞ þ vðaVÞ:
ð5Þ
S
Moreover, u and v are jointly cardinal representations of +. Proof. From Theorem 1, if E satisfies (a,wEz) f (a, zEw) for all z,waR and aaA then p(E) = 1/2. Moreover, (a,x V) f (aV, x) if and only if U(u(x V)) U(u(x)) = v(aV) v(a) and (a, x VE y) f (a,xEy V) if and only if u(x V) u(x) = u( y V) u( y) (where used has been made of the fact that p(E ) = 1 p (E )). Now, (a,y V) f (a V,y) implies that u( y V) u( y) = v(aV) v(a). Hence, by (A.4), for all x,x VaR, Uðuðx VÞÞ UðuðxÞÞ ¼ uðx VÞ uðxÞ: Thus, U is a positive affine transformation and the representation follows.
5
3. Discussion The study of the designs of incentive contracts that mitigate the welfare loss associated with moral-hazard problems has been the focal point of economic analysis in the last three decades. The modeling of principal – agent relationships in this context admits alternative formulations, including the state-space formulation and the parameterized distribution formulation (see Hart and Holmstrom, 1987; Chambers and Quiggin, 2000). Either formulation requires the assignment of objective functions to the parties involved, typically assuming that both the principal and the agent are expected utility maximizers. Despite the widespread interest in the problem little attention was devoted to the study of the axiomatic foundations of agency theory. This lack of interest may be due to the belief that the axiomatic underpinnings of the behavior of the contracting
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parties already exists in the theory of Savage (1954). In the case of the parameterized distribution formulation, this belief is utterly misguided (see Karni, 2003). In the case of the state-space formulation, this paper demonstrates clearly, that the traditional subjective expected utility theory does not capture some essential aspects of the agent’s behavior. Even if the agent is a subjective expected utility maximizer, the axiomatic structure of expected utility theory must be amended to accommodate the direct effect of the actions on the agent’s preferences. In a typical principal –agent problem, both parties are supposed to have preference relations over actions and contracts. The preference relations axiomatized in this paper are over action –acts pairs. To translate the model and results of this paper into the terms used in the literature dealing with contract theory, let a contract, w, be a function from X to R, representing the monetary payoff of the agent contingent on the realized consequence. Given a contract, w, the agent’s action-dependent contingent monetary payoff is given by (wBt)(a, ). In other words, each action– contract pair, (a,w), induces an act in F given by (wBt)(a, ): S ! R. The preference relation + on A F induces a preference relation +ˆ on ˆ A W as follows: (a,w) +ˆ (aVwV) if and only if (a,(wBt)(a, ))+(aV,(wBt)(aV, )). Let +P and + denote the preference relations of the principal and the agent, respectively, on A W. Then the principal’s problem may be stated as follows: Choose (a*,w*) such that: ða*; ðw*BtÞða*; ÞÞ+P ða; ðwBtÞða; ÞÞ; for all ða; wÞa A W subject to the incentive compatibility constraints: ˆ ða*; ðw*BtÞða*; ÞÞ+ða; ðw*BtÞða; ÞÞ; for all aa A and the participation constraint ˆ a; wÞ ¯ ða*; ðw*BtÞða*; ÞÞ+ð¯ where (¯a ,w¯) represents the action – contract pair that is equivalent to the agent’s best alternative course of action should he reject the contract. Assume next that the agent’s preferences satisfy (A.0) (A.1) (A.2) (A.3) (A.4) and that the principal’s preferences on (X R)S satisfy Savage’s (1954) postulates. Let P(x;a) = pP(t 1(a, x)) for every xaX, where pP is the subjective probability measure of the principal on S. Let U: X R ! R denote the principal’s von Neumann –Morgenstern utility function. Similarly, let p(x;a) = p(t 1(a,x)) for every xaX, where p is the subjective probability measure of the agent on S. Let u: R ! R denote the agent’s von Neumann– Morgenstern utility function. Then the principal’s problem may be restated as follows: Choose (a*,w*) so as to maximize Z U ðx; wðxÞÞdPðx ; aÞ X
subject to the incentive compatibility constraints: Z Z uðw*ðxÞÞdpðx ; a*Þ þ vða*Þz uðw*ðxÞÞdpðx ; aÞ þ vðaÞ; for all aaA; X
X
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and the participation constraint Z uðw*ðxÞÞdpðx ; aÞ þ vða*Þz uo X
where u0 is the reservation utility level of the agent.
Acknowledgements Financial support by the NSF under grant SES-0314249 is gratefully acknowledged.
References Chambers, R.J., Quiggin, J., 2000. Uncertainty, Production, Choice, and Agency Cambridge Univ. Press, Cambridge. Hart, O., Holmstrom, B., 1987. The theory of contracts. In: Bewley, T. (Ed.), Advances in Economic Theory. Cambridge Univ. Press, Cambridge, pp. 71 – 155. Holmstrom, B., 1979. Moral hazard and observability. The Bell Journal of Economics 10, 74 – 91. Karni, E., 2003. Subjective expected utility theory with costly actions. Games and Economic Behavior (in press). Karni, E., Safra, Z., 1998. The hexagon condition and additive representation for two dimensions: an algebraic approach. Journal of Mathematical Psychology 42, 393 – 399. Ramsey, F.P., 1931. Truth and probability. In: Paul, K. (Ed.), The Foundations of Mathematics and Other Logical Essays. Trench, Truber and Co, London, pp. 156 – 198. Savage, L.J., 1954. The Foundations of Statistics. Wiley, New York.
Additive representations over actions and acts Edi Karni * Department of Economics, Johns Hopkins University, 34th and Charles St. Baltimore, MD 21218-2685, USA Received 1 October 2003; accepted 1 March 2004 Available online 1 June 2004
Abstract This paper develops axiomatic foundations of rational decision making involving a choice among action – acts pairs. In the context of the state-space formulation of agency theory, this paper clarifies the assumptions underlying the additive representations that are often used to depict agents’ behavior in the analysis of principal – agent problems. The paper invokes the algebraic approach that renders the analysis general. D 2004 Elsevier B.V. All rights reserved. Keywords: Additive representation; Agency theory JEL classification: D81; D82
1. Introduction The state-space formulation of principal –agent relations in the presence of a moral hazard problem includes the following primitives: a state-space S, a set of actions A, a set of consequences X, and a technology t, represented by a mapping from A S to X. There is a true state of the world, sVaS, that, once known, resolves all the uncertainty. That is, if the true state is known, then the consequences of every action would be known. Thus, if the agent were to choose the action, a, the result would be the consequence t(a,sV)aX. In addition, it is assumed that the primitives are common knowledge, the action chosen is private information of the agent, and the true state of the world is not known at the time the action must be chosen. The principal and the agent have distinct interests in their relationship. The principal’s main interest are the consequences and he is concerned with the actions only insofar as they *Tel.: +1-410-516-7608; fax: +1-410-516-7600. E-mail address: [email protected] (E. Karni). 0165-4896/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2004.03.001
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E. Karni / Mathematical Social Sciences 48 (2004) 113–119
affect the likely realization of the alternative consequences. By contrast, the agent is concerned with the consequences only to the extent that they affect his payoff under the contract, while his interest in the actions is direct. In other words, to the agent actions are costly and choosing an action has direct implications for his well-being. This difference of outlook requires distinct approaches to modeling the principal’s and the agent’s choice behavior. Assuming, as is customary in agency theory, that both the principal and the agent are expected utility maximizers, the principal’s preferences may be modeled using Savage’s (1954) theory. Specifically, the principal’s preference relation is a binary relation on (X R)S satisfying all of Savage’s (1954) postulates. The modeling of the agent’s preferences, however, must take into account his direct preferences on the actions. In particular, denote by F the set of all real valued functions on S, then agents are characterized by preference relations + on A F. (Note that, insofar as the agent is concerned, F constitutes the set of Savage-type acts). In Karni (2003), I explore the axiomatic underpinnings of alternative representations of agents’ preferences. In that work the choice set was M(A) H jAj, where M(A) is the set of probability measures on A, X is a compact interval, and H is the subset of X S containing all the simple acts (that is, acts that have finite range). In many applications of agency theory, however, it is assumed that the agent’s preferences are additively separable in income and actions (e.g., Holmstrom, 1979). In this paper, I develop an axiomatic model that dispenses with the use of probabilities as a primitive concept. In this model, the agent’s behavior is characterized by additive representation on A F. Moreover, in this paper, I invoke the algebraic approach and a representation result of Karni and Safra (1998) to obtain a model that is more general in the sense of not assuming a topological structure on the choice set.
2. Axioms and representations 2.1. Axioms Assume, henceforth, that the state-space is infinite and that the preference relation + on A F is a weak order (that is, + is complete and transitive). The strict preference relation, d, and the indifference relation, f , are the asymmetric and symmetric parts of +, respectively. Moreover, suppose that, given any action, the preference relations on the set of acts satisfy all the axioms of Savage (1954). Formally, for every given aaA, define a conditional preference relation +a on F by f +ag if and only if (a, f )+(a,g), for all f, gaF, then (A.0). For every given aaA, +a on F satisfies the postulates (P.1) – (P.7) of Savage (1954). The alternatives that the agent must choose from have two components, namely, action and act. The next axiom requires that the preference relation + displays independence of equal component. Formally: (A.1) (Independence). For all (a, f ),(aV, f V)aA F, (a, f )+(a, f V) implies (aV, f )+(a V, f V) and (a, f )+(a V, f ) implies (a, f V)+(aV, f V).
E. Karni / Mathematical Social Sciences 48 (2004) 113–119
115
Independence implies that +a = +aV, for all a,aVaA. To emphasize this fact and to simplify the notations, I denote the induced relation +a on F by +F. Similarly, define an induced binary relation +A on A by a+Aa V if there exist faF such that (a, f )+(a V, f ). Again, independence implies that this relation is well-defined. Actions and acts are essential if the asymmetric parts of the induced relations +A and +F, A and F, respectively, are nonempty. A sequence {an}aA is an increasing standard sequence if there exist f, f Va F such that f VdF f and, for all n z 1, (an, f V) f (an + 1, f ). A decreasing standard sequence is defined analogously. A standard sequence is bounded if there exist a,a VaA such that, for all n, a+Aan+Aa V. The standard (bounded) sequence in F is defined similarly. Definition. The relation + is Archimedean if every bounded standard sequence is finite. The relation + satisfies solvability if, whenever a,a V,aWaA and f, f Va F for which (aV, f V)+(a, f )+(aW, f V), then there is a¯aA such that (a¯, f V) f (a, f ) and a similar condition holds with A and F interchanged. (A.2) (Solvability). The relation + satisfies solvability. Because there are at most two essential components, independence is not sufficient for the sought-after additive representation. The next well-known hexagon condition is also required. (A.3) (Hexagon condition). For all a,a V,aWaA and f, f V, f Wa F, if (a,f V) f (a V, f ) and (aW, f ) f (aV, f V) f (a, f W), then (a V, f W) f (aW, f V). 2.2. Additively separable representation An array of real-valued functions (Vj)jn= 1 is said to be a jointly cardinal representation n of Pn + on a product set D ¼ jj¼1Ds if, for all d,dVaD, d+d Vif and only if Pna binary relation j¼1 Vj ðdj Þz j¼1 Vj ðdj VÞ , and the class of all functions that constitute an additive representation of + consists of those arrays of functions, (Wj)jn= 1, that satisfy Wj = bVj + aj, b>0, j = 1,. . .,n. Theorem 1 . Let + be a binary relation on A F and suppose that both actions and acts are essential. Then the following conditions are equivalent: (i) + is an Archimedean weak order satisfying (A.0) (A.1) (A.2) (A.3). (ii) There exist a nonatomic probability measure p on S, a real-valued function, u, on R, a real-valued function, v, on A, and a strictly monotonic increasing, function U: R ! R, such that, for all (a,f ) and (aV,f V) in A F, Z Z ða; f Þ+ðaV; f VÞZU uðf ðsÞÞdpðsÞ þ vðaÞzU uðf VðsÞÞdpðsÞ þ vðaVÞ: S
S
ð1Þ Moreover, p is unique, U and v are jointly cardinal representation of +, and u is unique up to positive affine transformation.
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Proof. (i) ! (ii). Since both actions and acts are essential, Karni and Safra (1998) imply that the binary relation + on A F is an Archimedean weak order satisfying (A.1), (A.2), and (A.3) if and only if there exist real-valued functions v on A and V on F such that, for all (a, f ) and (aV, f V) in A F, ða; f Þ+ðaV; f VÞZV ð f Þ þ vðaÞzV ð f VÞ þ vðaVÞ:
ð2Þ
Moreover, V and v are jointly cardinal representation of +. By (A.0) and Savage’s (1954) theorem, there exist a unique, nonatomic, probability measure, p, on S and a real-valued function, u, on R, unique up to positive linear transformation, such that, for every f,gaF, Z
uðf ðsÞÞdpðsÞ z
f +F g Z S
Z uðgðsÞÞdpðsÞ;
ð3Þ
S
However, by Eq. (2), f +Fg if and only if V( f ) z V ( g). Hence, Z V ðf Þ ¼ U uðf ðsÞÞdpðsÞ ;
ð4Þ
S
where U is strictly monotonic increasing function. (ii) ! (i). The proof that Eq. (2) implies that + is an Archimedean weak order satisfying (A.1), (A.2), and (A.3) is straightforward. The proof that Eq. (4) implies (A.0) is an implication of Savage’s (1954) theorem. 5 2.3. Subjective expected utility representation In many applications, it is assumed that U is a positive affine function. This specification requires additional restrictions on the preference relation. One such restriction is given in the following condition whose statement requires the following additional notation: For all ya R, denote by y the constant function faF, such that f(s) = y for all saS. For every given EoS and x, yaR, denote by xEy the act faF satisfying f(s) = x for all saE and f (s) = y otherwise. (A.4) (Uniform utility differences). Let a,aVaA and y,y VaR satisfy (a, f )d(aV, f ) for all faF, (a,y)d(a,y V) for all aaA, and (a,y V) f (aV,y). Then, for all x, x VaR, (a, x V) f (aV, x) if and only if (a, x VEy) f (a, xEy V) for all EoS such that (a,wE z) f (a, zEw) for all z,waR and aaA. Axiom (A.4) looks complicated, but it is easy to understand when dissected. To begin with the last part of (A.4), namely, the requirement that(a,wEz) f (a,zEw) for all z,waR and aaA, implies that E is an event whose (subjective) probability is one-half. In other words, E and its complement in S are ethically neutral events in the sense of Ramsey (1931). (Notice that the nonatomicity of the subjective probability measure in Savage
E. Karni / Mathematical Social Sciences 48 (2004) 113–119
117
(1954) model assures that such events exist.) According to this interpretation of E, the condition that, for all x,x VaR, (a, x V) f (aV, x) if and only if (a, x VEy) f (a, x Ey V) implies that x and xVhave the same von Neumann – Morgenstern (that is, u) utility difference as y and y V. The equivalences (a, yV) f (a V,y) and (a, x V)f (a V,x) imply that x and x Vhave the same utility differences as y and yV in terms of the transformation U of the subjective expected utility. The addition of Axiom (A.4) implies that the transformation U must be positive linear. Formally, Theorem 2. Let + be a binary relation on A F and suppose that both actions and acts are essential. Then the following conditions are equivalent: (i) + is an Archimedean weak order satisfying (A.0) (A.1) (A.2) (A.3) (A.4). (ii) There exist a unique, nonatomic, probability measure p on S, a real-valued function, u, on R, and a real-valued function, v, on A such that, for all (a,f ) and (aV,f V) in A F, ða; f Þ+ðaV; f VÞZ
Z
uð f ðsÞÞdpðsÞ þ vðaÞz
S
Z
uð f VðsÞÞdpðsÞ þ vðaVÞ:
ð5Þ
S
Moreover, u and v are jointly cardinal representations of +. Proof. From Theorem 1, if E satisfies (a,wEz) f (a, zEw) for all z,waR and aaA then p(E) = 1/2. Moreover, (a,x V) f (aV, x) if and only if U(u(x V)) U(u(x)) = v(aV) v(a) and (a, x VE y) f (a,xEy V) if and only if u(x V) u(x) = u( y V) u( y) (where used has been made of the fact that p(E ) = 1 p (E )). Now, (a,y V) f (a V,y) implies that u( y V) u( y) = v(aV) v(a). Hence, by (A.4), for all x,x VaR, Uðuðx VÞÞ UðuðxÞÞ ¼ uðx VÞ uðxÞ: Thus, U is a positive affine transformation and the representation follows.
5
3. Discussion The study of the designs of incentive contracts that mitigate the welfare loss associated with moral-hazard problems has been the focal point of economic analysis in the last three decades. The modeling of principal – agent relationships in this context admits alternative formulations, including the state-space formulation and the parameterized distribution formulation (see Hart and Holmstrom, 1987; Chambers and Quiggin, 2000). Either formulation requires the assignment of objective functions to the parties involved, typically assuming that both the principal and the agent are expected utility maximizers. Despite the widespread interest in the problem little attention was devoted to the study of the axiomatic foundations of agency theory. This lack of interest may be due to the belief that the axiomatic underpinnings of the behavior of the contracting
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parties already exists in the theory of Savage (1954). In the case of the parameterized distribution formulation, this belief is utterly misguided (see Karni, 2003). In the case of the state-space formulation, this paper demonstrates clearly, that the traditional subjective expected utility theory does not capture some essential aspects of the agent’s behavior. Even if the agent is a subjective expected utility maximizer, the axiomatic structure of expected utility theory must be amended to accommodate the direct effect of the actions on the agent’s preferences. In a typical principal –agent problem, both parties are supposed to have preference relations over actions and contracts. The preference relations axiomatized in this paper are over action –acts pairs. To translate the model and results of this paper into the terms used in the literature dealing with contract theory, let a contract, w, be a function from X to R, representing the monetary payoff of the agent contingent on the realized consequence. Given a contract, w, the agent’s action-dependent contingent monetary payoff is given by (wBt)(a, ). In other words, each action– contract pair, (a,w), induces an act in F given by (wBt)(a, ): S ! R. The preference relation + on A F induces a preference relation +ˆ on ˆ A W as follows: (a,w) +ˆ (aVwV) if and only if (a,(wBt)(a, ))+(aV,(wBt)(aV, )). Let +P and + denote the preference relations of the principal and the agent, respectively, on A W. Then the principal’s problem may be stated as follows: Choose (a*,w*) such that: ða*; ðw*BtÞða*; ÞÞ+P ða; ðwBtÞða; ÞÞ; for all ða; wÞa A W subject to the incentive compatibility constraints: ˆ ða*; ðw*BtÞða*; ÞÞ+ða; ðw*BtÞða; ÞÞ; for all aa A and the participation constraint ˆ a; wÞ ¯ ða*; ðw*BtÞða*; ÞÞ+ð¯ where (¯a ,w¯) represents the action – contract pair that is equivalent to the agent’s best alternative course of action should he reject the contract. Assume next that the agent’s preferences satisfy (A.0) (A.1) (A.2) (A.3) (A.4) and that the principal’s preferences on (X R)S satisfy Savage’s (1954) postulates. Let P(x;a) = pP(t 1(a, x)) for every xaX, where pP is the subjective probability measure of the principal on S. Let U: X R ! R denote the principal’s von Neumann –Morgenstern utility function. Similarly, let p(x;a) = p(t 1(a,x)) for every xaX, where p is the subjective probability measure of the agent on S. Let u: R ! R denote the agent’s von Neumann– Morgenstern utility function. Then the principal’s problem may be restated as follows: Choose (a*,w*) so as to maximize Z U ðx; wðxÞÞdPðx ; aÞ X
subject to the incentive compatibility constraints: Z Z uðw*ðxÞÞdpðx ; a*Þ þ vða*Þz uðw*ðxÞÞdpðx ; aÞ þ vðaÞ; for all aaA; X
X
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and the participation constraint Z uðw*ðxÞÞdpðx ; aÞ þ vða*Þz uo X
where u0 is the reservation utility level of the agent.
Acknowledgements Financial support by the NSF under grant SES-0314249 is gratefully acknowledged.
References Chambers, R.J., Quiggin, J., 2000. Uncertainty, Production, Choice, and Agency Cambridge Univ. Press, Cambridge. Hart, O., Holmstrom, B., 1987. The theory of contracts. In: Bewley, T. (Ed.), Advances in Economic Theory. Cambridge Univ. Press, Cambridge, pp. 71 – 155. Holmstrom, B., 1979. Moral hazard and observability. The Bell Journal of Economics 10, 74 – 91. Karni, E., 2003. Subjective expected utility theory with costly actions. Games and Economic Behavior (in press). Karni, E., Safra, Z., 1998. The hexagon condition and additive representation for two dimensions: an algebraic approach. Journal of Mathematical Psychology 42, 393 – 399. Ramsey, F.P., 1931. Truth and probability. In: Paul, K. (Ed.), The Foundations of Mathematics and Other Logical Essays. Trench, Truber and Co, London, pp. 156 – 198. Savage, L.J., 1954. The Foundations of Statistics. Wiley, New York.