Additive representations over actions and acts

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.

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, s′∈S, 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,s′)∈X. 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 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×$\text{R}$)^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^∣A∣, 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.