The developmental sequence: the first and second phases

Despite an appearance of relative stability in the years immediately following World War II, the unevenness and instability of accumulation remain what they have always been: reflections of contradictions embedded within the apparatus of industrial and circulatory accumulation, contradictions that culminate in economic breakdown and whatever may follow from that. To be sure, in the course of irregular advances the underlying situation gradually alters; times change, and with them come changes in the form of the crisis. As capital accumulates through successive phases, the laws of motion work ever more decisively and powerfully in the promotion of economic exhaustion, of an inability to continue the vital work of developing further the productive forces. In the end, even in periods of activity, the swimmer treads water rather than move against the current. The struggle begins then in earnest for a socialist release from an increasingly intolerable predicament.

But, as always, in order to see a complicated present one must return to the past for understanding. In order to perceive correctly the manner of working of laws of motion in the present phase, one must consider what has previously been. In contemplating the events of the post – World War II period, one must return to the historical setting that throws into proper relief the real character of the phase in which current events are taking place.

Introduction

The rationality postulates (axioms) that we will use in our analysis of game situations fall into two main classes:

A. Postulates of rational behavior in a narrower sense, stating rationality criteria for strategies to be used by the players.

B. Postulates of rational expectations, stating rationality criteria for the expectations that rational players can entertain about each other's strategies.

Postulates of Class A in themselves would not be sufficient. We have defined game situations as situations in which each player's payoff depends not only on his own strategy but also on the other players' strategies. If a player could regard the other players' strategies as given, then the problem of rational behavior for him would be reduced to a straightforward maximization problem, viz., to the problem of choosing a strategy maximizing his own expected payoff. But the point is precisely that he cannot regard the other players' strategies as given independently of his own. If the other players act rationally, then their strategies will depend on the strategy that they expect him to follow, in the same way that his own strategy will depend on the strategies that he expects them to follow. Thus there is, or at least appears to be, a vicious circle here. The only way that game theory can break this vicious circle, it seems to me, is by establishing criteria for deciding what rational expectations intelligent players can consistently hold about each other's strategies.

The rationality assumption

Like other versions of game theory – and indeed like all theories based on some notion of perfectly rational behavior – regarding its logical mode, our theory is a normative (prescriptive) theory rather than a positive (descriptive) theory. At least formally and explicitly it deals with the question of how each player should act in order to promote his own interests most effectively in the game and not with the question of how he (or persons like him) will actually act in a game of this particular type. All the same, the main purpose of our theory is very definitely to help the positive empirical social sciences to predict and to explain real-life human behavior in various social situations.

To be sure, it has been a matter of continual amazement among philosophers, and often even among social scientists (at least outside the economics profession), how any theory of rational behavior can ever be successful in explaining or predicting real-life human behavior. Yet it is hardly open to doubt that in actual fact such theories have been remarkably successful in economics and more recently in several other social sciences, particularly political science, international relations, organization theory, and some areas of sociology [see Harsanyi, 1969, and the literature there quoted].

Needless to say, theories based on some notion of rational behavior (we will call them rational-behavior or rational-choice theories or, briefly, rationalistic theories), just as theories based on different principles, sometimes yield unrealistic predictions about human behavior.

Need for a game-theoretical approach yielding determinate solutions

The purpose of this book is to present a new approach to game theory. Based on a general theory of rational behavior in game situations, it yields a determinate solution (i.e., a solution corresponding to a unique payoff vector) for each particular game and clearly specifies the strategies by which rational players can most effectively advance their own interests against other rational players.

This new approach, it seems to me, significantly increases the scope and the analytical usefulness of game-theoretical models in the social sciences. It furnishes sharp and specific predictions, both qualitatively and quantitatively, about the outcome of any given game and in particular about the outcome of bargaining among rational players. It shows how this outcome depends on the rewards and penalties that each player can provide for each other player, on the costs that he would incur in providing these rewards or penalties, and on each player's willingness to take risks. Thus it supplies the analytical tools needed for what may be called a bargaining-equilibrium analysis of social behavior and of social institutions, i.e., for their explanation in terms of a bargaining equilibrium (corresponding to the “balance of power”) among the interested individuals and social groups.

This new approach to game theory also has significant philosophical implications, because it throws new light on the concept of rational behavior and on the relationship between rational behavior and moral behavior.

Definitions and assumptions

We will first consider only vocal cooperative games. We make the following assumptions. The two players can achieve any payoff vector u = (u1, u2) within the payoff space P of the game, if they can agree which particular payoff vector u to adopt, i.e., if they can agree how to divide the payoffs between them. The players are free to use jointly randomized mixed strategies, which make the payoff space P a convex set. Moreover,P is assumed to be bounded and closed, i.e., compact. We also exclude the degenerate case in which the payoff space is a segment of a straight line with ui = const. for either player i: Any such case will always have to be treated as a strictly noncooperative game, since player i would have no incentive whatever to cooperate with the other player.

Among cooperative games we will distinguish two cases, depending on the nature of the conflict situation that would emerge if the two players could not agree on their final payoffs u1 = ū1 and u2 = ū2. In a simple bargaining game the rules of the game themselves fully specify the conflict-payoff vector or conflict point c = (c1, c2) to which the players would be confined in such a conflict situation. This means that the players have essentially only one conflict strategy, viz., simple noncooperation.

Introduction

In an n-person simple bargaining game the n players have to choose a payoff vector u = (u1,…, un) from a compact and convex set P of possible payoff vectors, called the payoff space of the game. The choice of u must be by unanimous agreement of all n players. If they cannot reach unanimous agreement, then they obtain the conflict payoffs c1, …, cn. The payoff vector c = (c1, …, cn) is called the conflict point of the game. We will assume that c∈P.

That region P* of the payoff space P which lies in the orthant defined by the n inequalities ui ≧ ci for i = 1, …, n, is called the agreement space. Like P itself, P* is always a compact and convex set.

We will exclude the degenerate case where the payoff (s) of some player(s) is (or are) constant over the entire agreement space P*. For in this case this player (or these players) would have on interest in cooperating with the other player(s), and so the game would not be a truly cooperative game.

The set of all points u in the payoff space P undominated, even weakly, by any other point u* in P is called the upper boundary H of P. In other words, H is the set of strongly efficient points in P. In general the payoff space P is a set of n dimensions.

Introduction

The main concern of this book is with game situations (situations of strategic interdependence), in which the outcome depends on mutual interaction between two or more rational individuals, each of them pursuing his own interests (and his own values) against the other individuals, who are likewise pursuing their own interests (and their own values). In earlier chapters we discussed situations of individual independence (certainty, risk, and uncertainty), in which the outcome depends on the actions of only one individual (and possibly on chance). We also discussed moral situations, in which the outcome does depend on interaction between two or more individuals but in which this outcome and these individuals' actions are evaluated, not in terms of their own individual interests but rather in terms of the interests of society as a whole – as seen by an impartial but sympathetic observer. However, all of this was merely a preliminary to our analysis of game situations.

Following von Neumann and Morgenstern [1944] it is customary to analyze what we call game situations by using parlor games – already existing ones or ones specially constructed for this very purpose – as analytical models. (More specifically, what are used as models are games of strategy, where the outcome depends at least in part on a rational choice of strategy by the participants rather than on mere physical skill or on mere chance.) Hence the term “game situations” and the name “game theory”, for the theory analyzing such situations, arise.

How to define rational behavior (practical rationality) is a philosophical problem of fundamental importance – both in its own right and by virtue of its close connection with the problem of theoretical rationality. The concept of rational behavior is equally fundamental to a number of more specialized disciplines: to normative disciplines such as decision theory (utility theory), game theory, and ethics; and to some positive Social sciences, such as economics and certain, more analytically oriented, versions of political science and of sociology.

This book presents what I believe to be the first systematic attempt to develop a conceptually clear, and quantitatively definite, general theory of rational behavior. No doubt, technically more advanced and philosophically more sophisticated versions of such a theory will soon follow. In fact, the first version of this book was completed in 1963, but game theory has been advancing at a very rapid rate since then, and my own thinking has also been changing. Thus, I have revised this manuscript several times to bring it more in line with new developments, but this process must stop if this material is ever to be published. I hope the reader will bear with me if he finds that this book does not cover some recent results, even some of my own. In such a rapidly growing subject as game theory, only journal articles – indeed, only research reports – can be really up to date.