Introduction

It is intended here to offer a possible characterization of the concept of complete ignorance. Like other formulations, the problem is taken to be that of choice of an action from a given set when the consequences of any action are functions of an unknown state of nature. However, the properties regarded as defining an optimal choice are designed to reflect completely the idea that there is no a priori information available which gives any state of nature a distinguished position. Most importantly, the optimality criterion differs from those in the now more standard subjective probability framework by not presupposing a fixed list of states of nature. As we note shortly, the arguments and conclusions are much closer to Shackle's than to those of Ramsey, de Finetti, and Savage.

The axiom systems of these last authors imply the existence of subjective probabilities as weights to be assigned to the different possible states of nature. These authors thus provide a foundation for the centuries-old use of probability as a guide to action. The concept of complete ignorance can be expressed in this subjective probability framework only by the assignment of equal probabilities to all the states of nature, which is the principle of indifference or insufficient reason implicit in the earliest combinatorial probability calculations of Pascal and Fermat and explicit in Jacob Bernoulli, Bayes, and Laplace.

This paper deals with the application of certain computational methods to evaluate constrained extrema, maxima, or minima. To introduce the subject, we will first discuss nonlinear games. Under certain conditions, the finding of the minimax of a certain expression is closely related to, in fact identical with, the finding of a constrained minimum or maximum. Let us consider then a game (in a generalized sense) where player 1 has the choice of a certain set of numbers x1, …, xm that are constrained to be nonnegative for present purposes and player 2 selects numbers y1, …, yn also constrained to be nonnegative but otherwise unrestricted. The payoff of the game, the payment made by player 2 to player 1, will be a function of the decisions made by the two players, the x's and the y's. This pay off will be designated by φ(x1, …, xm; y1, …, yx, …, yn). To play the game in an ideal way is to find the minimax solution; we know this solution exists under certain conditions. That is, we arrive at a choice of strategies by the two players where player 1 is maximizing his payoff given the strategy of player 2, and player 2 is minimizing the payoff, given the strategy of player 1.

The role of price adjustment equations in economic theory

In this essay, it is argued that there exists a logical gap in the usual formulations of the theory of the perfectly competitive economy, namely, that there is no place for a rational decision with respect to prices as there is with respect to quantities. A suggestion is made for filling this gap. The proposal implies that perfect competition can really prevail only at equilibrium. It is hoped that the line of development proposed will lead to a better understanding of the behavior of the economy in disequilibrium conditions.

In the traditional development of economic theory, the usual starting point is the construction for each individual (firm or household) of a pattern of reactions to events outside it (examples of elements of a reaction pattern: supply and demand curves, propensity to consume, liquidity preference, interindustrial movements of capital and labor in response to differential profit and wage rates). This point of view is explicit in the neoclassicists (Cournot, Jevons, Menger, and successors) and strongly implicit in the classicists (from Smith through Cairnes) in their discussion of the motivations of capitalists, workers, and landlords which lead to establishment of the equilibrium price levels for commodities, labor, and the use of land. The basic logic of Marx's system brings it, I believe, into the same category, although some writers have referred to his theories as being “class” economics rather than “individual” economics.

This book draws together a long series of papers by the two senior authors, alone and in collaboration with each other and with other friends and colleagues, to whose thinking and stimulation we are grateful. Both of us have had a strong primary concern with the workings of the economic system as a mechanism for achieving the optimal allocation of resources. The theme is of course an old one in economic thought; its importance was especially reinforced to us through our teachers, Harold Hotelling and Oskar Lange, and our colleague, Jacob Marschak. The very concept of optimization with resource constraints links the theory with the classical mathematical theory of constrained optimization and the more modern versions of so-called mathematical programming, where emphasis has been placed on inequality constraints.

The important property of the market as a resource allocation mechanism is its decentralization. It has always been assumed in the mainstream of economic theory, though frequently only implicitly, that the transmission of detailed information about tastes or technology is costly and that there is a virtue to systems in which decisions are made at the point where the information already existed. This desire for decentralization has, however, to be reconciled with the need for balance in the economy as a whole, most especially, in the need to respect limitations on the overall availability of resources. The mathematical characterizations of constrained optima, at least in the Lagrangian formulation and its generalizations, suggests the possibility of decentralization.

Introduction

A great deal of work has been done recently on what one may call the static aspects of competitive equilibrium, its existence, uniqueness, and optimality. This work is characterized, in the main, by being based on models whose assumptions are formulated in terms of certain properties of the individual economic units, although in the last analysis it is the nature of the aggregate excess demand functions that determines the properties of equilibria.

With regard to dynamics, especially the stability of equilibrium, much remains to be done. The concept of stability, used already by the nineteenth century economists in its modern sense, did not receive systematic treatment in the context of economic dynamics until Samuelson's paper of 1941. Samuelson, however, did not fully explore the implications of the assumptions underlying the perfectly competitive model. He (as well as Lange, Metzler, and Morishima) focused attention on the relationship between “true dynamic stability” and the concept of “stability” as defined by Hicks in Value and Capital, rather than on whether under a given set of assumptions stability (in either sense) would prevail or not. Even though the Hicksian concept does not, in general, coincide with that of “true dynamic stability,” it is of considerable interest to us for two reasons: first, as shown by the writers just cited, there are situations where the two concepts are equivalent; second, because the equilibrium whose “stability” Hicks studied is indeed competitive equilibrium.

The conditions under which a Walrasian system of multiple markets will be stable have been investigated by a number of authors under the implicit assumption of static expectations. It often has been assumed that expectations based upon an extrapolation of current rates of change (rather than upon the assumption that the future would be like the present) would prevent the system from converging onto its equilibrium position at all. Since interesting results have been scarce and difficult to achieve even in the case of static expectations, it is not surprising that little has been done with the relationship between extrapolative expectations and dynamic stability. In this paper, we shall introduce, under rather restrictive assumptions, a type of extrapolative expectations and we shall test its effects on the stability of a dynamic system.

Excess demands in a multiple market system are usually taken to be functions of the current prices of all goods. Ideally, it would be desirable also to include expected prices for all future time periods and for all individuals and all assets as arguments of the excess demand functions. A theory of such formal generality, however, would be necessarily devoid of much content. Abstractions and simplifying assumptions are necessary. There are many possible expectations functions by which people might relate current and expected prices, and there is a variety of ways to represent plausibly the type of extrapolative expectations which we wish to describe. Our choice was made largely on the grounds of mathematical simplicity.

The increasing span of government control over economic life in the last fifty years has directed the attention of economic theorists to the relative merits of centralization and decentralization in economic decision-making.

If the aim of the economic system is something like the maximization of national income, it may be asked whether it is better to make the economic decisions in a central agency where information relating to the entire system can be used or in the many independent units which characterize a capitalist economy such as ours.

In the course of this great debate, the role of the price system in coordinating many individual decisions has been given stronger and stronger recognition, although the idea itself already appears in Adam Smith's famous “invisible hand.”

There is also growing concern in the field of industrial management with the administration of large business corporations. Again there arises the issue of centralization vs. decentralization. To what extent is it necessary for the efficiency of a corporation that its decisions be made at a high level where a wide degree of information is, or can be made, available? How much, on the other hand, is gained by leaving a great deal of latitude to individual departments which are closer to the situations with which they deal, even though there may be some loss due to imperfect coordination?

A representative collection of management viewpoints on this issue is found in the proceedings of a conference held in The Netherlands some years ago.

Traditionally, economic analysis treats the economic system as one of the givens. The term “design” in the title is meant to stress that the structure of the economic system is to be regarded as an unknown. An unknown in what problem? Typically, that of finding a system that would be, in a sense to be specified, superior to the existing one. The idea of searching for a better system is at least as ancient as Plato's Republic, but it is only recently that tools have become available for a systematic, analytical approach to such search procedures. This new approach refuses to accept the institutional status quo of a particular time and place as the only legitimate object of interest and yet recognizes constraints that disqualify naive Utopias.

A wealth of ideas, originating in disciplines as diverse as computer theory, public administration, games, and control sciences, has, in my view, opened up an exciting new frontier of economic analysis. It is the purpose of this paper to survey some of the accomplishments and to consider outstanding unsolved problems and desirable directions for future efforts.

It is not by accident that the terms “analytical” and “institutional” were only a few words apart in the preceding statement of scientific goals of our inquiry. In the past, especially in the nineteenth century, cleavage developed between analysts who tended to focus on the competitive and monopolistic market models and institutionalists who, either as historians or as reformers, felt the need for a broader framework, but found the existing analytical tools inadequate for their purposes.

The frequent and loud complaints of a shortage of engineers and scientists heard over the past eight years or so might be taken as indicating a failure of the price mechanism and indeed have frequently been joined with (rather vaguely stated) proposals for interference with market determination of numbers and allocation. It is our contention that these views stem from a misunderstanding of economic theory as well as from an exaggeration of the empirical evidence. On the contrary, a proper view of the workings of the market mechanism, recognizing, in particular, the dynamics of market adjustment to changed conditions, would show that the phenomenon of observed shortage in some degree is exactly what would be predicted by classical theory in the face of rapidly rising demands.

In this paper we present a model which explains the dynamics of the market adjustment process and apply the conclusions drawn from this analysis to the scientist-engineer “shortage.”

Equality of supply and demand is a central tenet of ordinary economic theory, but only as the end result of a process, not as a state holding at every instant of time. On the contrary, inequalities between supply and demand are usually regarded as an integral part of the process by which the price on a market reaches its equilibrium position. Price is assumed to rise when demand exceeds supply and to fall in the contrary case.

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.