Introduction
The literature on game theory (gt) is usually concerned with providing formulations of the same that are enough to consider specific applications (almost always in economics or political science), or to prove the existence of equilibria.
In contradistinction, this chapter is devoted to provide a systematic logical reconstruction of GT whose aim is to determine its basic theory-element through the formulation of its fundamental law. After defining the concepts of strategy, it proceeds to show how any behavioral strategy determines a probability space over the histories of the game, as a prolegomenon to the definition of the concept of game and the treatment of the problem of its empirical applicability.The basic theory-element of “neoclassical economics” is obtained, as a specialization, out of some axiomatic formulation of the concept of a dynamic game. My main aim in this chapter is to offer a rather general, rigorous and abstract formulation of this concept. This is tantamount to an extrinsic characterization of the models of game theory. The second is to discuss the meaning of the fundamental law of game theory, seen as a positive discipline. The third is to discuss its empirical applicability and to show that the conditions usually known as “axioms of revealed preference” (Samuelson 1938, 1947) are nothing but special cases of a certain constraint (in the sense of svt).
The class of dynamic games is sufficiently general for almost all economic applications. Their temporal horizon can be finite or infinite, their temporal sequence discrete or continuous, but I will consider here only the structure of games with finite horizon and discrete time for two reasons. The first is that a first approach to the logical structure of the concept of a dynamic game does not need to get involved with the details of continuous time or infinite horizon; the second is that the generalization is straightforward through the theorem of Kolmogorov-Bochner (cf. Rao 1981: 9, and chapter HI).
After introducing the conceptual apparatus required to define such games, I will discuss the concepts of strategy. Behavioral strategies determine probability measures over events that have as members, as elementary events, feasible histories of the game. I shall explain in detail how those measures are defined over the sets of such events.
Another central concept of the theory is that of equilibrium. The usual literature of game theory proceeds to find sufficient conditions for the existence of equilibria, but a general formulation of the logical foundations of a theory should not try to provide some set of such conditions, but rather to formulate the fundamental law that all games should satisfy. I will formulate this law before discussing the constraints of the theory and the general form of its empirical claim.
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