Principles of Game Theory

Abstract

The prehistory of game theory is relatively short, devoted to an algorithm for the resolution of extensive-form games (Zermelo) and an equilibrium notion for zero-sum normal-form games (Borel, von Neumann). The theory appeared in an already elaborated form in the pioneering work " Theory of Games and Economic Behavior " (1944), issued from the collaboration between the mathematician von Neumann and the economist Morgenstern. It saw a first period of development during the 1950s, when its main equi-librium concept -Nash equilibrium -was introduced (Nash). It then fell into neglect for a time, before enjoying a second burst of life in the 1970s with the explicit integration of time (Selten) and uncertainty (Harsanyi) into the equilibrium notions. It enjoyed a further boost in the 1990s, with the explicit internalization of players' beliefs (Aumann) and the advent of evolutionary game theory. The central purpose of game theory is to study the strategic relations between supposedly rational players. It thus explores the social structures within which the consequences of a player's action depend, in a conscious way for the player, on the actions of the other players. To do so, it takes as its basis the rational model of individual decision, although current work is increasingly focused on the limited rationality of the players. It studies direct multilateral relations between players, non-mediatized by prior insti-tutions. Game theory is generally divided into two branches, although there are bridges that connect the two. Noncooperative game theory studies the equilibrium states that can result from the autonomous behavior of players unable to define irrevocable contracts. Cooperative game theory studies the results of games governed by both individual and collective criteria of rationality, which may be imposed by an agent at some superior level. Game theory's natural field of application is economic theory: the eco-nomic system is seen as a huge game between producers and consumers, who transact through the intermediation of the market. It can be more specifically applied to situations outside the realm of perfectly competitive markets, i.e. situations in which the agents acquire some power over the fixing of prices (imperfect competition, auction mechanisms, wage negoti-ations). It can be applied equally well to relations between the state and agents, or to relations between two states. Nonetheless, it is situated at a level of generality above that of economic theory, for it considers non-ii specialized -though heterogeneous -agents performing actions of one na-ture or another within an institution-free context. It can therefore be con-sidered a general matrix for the social sciences and be applied to social relations as studied in political science, military strategy, sociology, or even relations between animals in biology. In this paper, we shall restrict ourselves to the study of noncooperative game theory. This is the prototype of formalized social science theory which, though making enormous use of diverse mathematical tools, transmits very simple, even simplistic, literary messages. From a syntactic point of view, it provides explanations of some social phenomena on the basis of a small number of concepts and mechanisms, thus endowing these phenomena with a duly signposted domain of validity. From a semantic point of view, it is the subject of laboratory experimentation, although this is aimed more at testing the consequences of existing theories than at inducing original regularities from the results. From a pragmatic point of view, it provides a unifying language for parties faced with decision-making, helping them better to express their shared problems, but supplying few instructions able to help them solve these problems. In what follows, we shall only consider two-player games, for the sake of simplicity (thus avoiding the question of coalitions between players). The games will be examined not only from the point of view of the modelizer overlooking them but also from the points of view of the players them-selves. Each player is characterized by three " choice determinants " : his opportunities (sets of possible actions), his beliefs (representations of the environment) and his preferences (value judgments on the effects of ac-tions). Interaction between the players is liable to lead to an " equilibrium state " defined -as it is in mechanics -as a situation that remains stable in the absence of perturbations from the environment. We shall start with the simplest forms of game and make the model more complex by the gradual introduction of the concepts of time and uncertainty. In each section, we shall present behavior assumptions illustrated by simple examples and then deal with the equilibrium concepts introduced with their properties. 0.3 Static games without uncertainty