James W. Friedman

A Non-cooperative Equilibrium for Supergames

Presents a non-cooperative equilibrium concept, applicable to supergames, which fits John Nashs non-cooperative equilibrium and also has some features resembling the Nash cooperative solution. Description of an ordinary game; Definition and discussion of a non-cooperative equilibrium for supergames; Description of supergame and supergame strategies; Information on the Cournot strategy.

Reaction Functions and the Theory of Duopoly

Focuses on reaction functions and the theory of duopoly. Introduction and development of the reaction function; Crucial element of duopoly; Equilibrium points in a dynamic duopoly.

On Entry Preventing Behavior and Limit Price Models of Entry

Formally, the market is regarded as a supergame in which one established firm and one potential entrant are players. Both players know all relevant demand and cost functions, and throughout the paper, noncooperative behavior is assumed. The model has two distinct stages: pre- and post entry. In the pre-entry stage, the monopolist chooses his price and capital stock so as to maximize his discounted profits, noting that his investment decision may affect the entry plans of the entrant. Existence of equilibrium is proved, entry preventing behavior is characterized and conditions are shown under which it will be employed.

A Noncooperative View of Oligopoly

THERE ARE TWO BASIC APPROACHES to the theory of oligopoly which may be called the cooperative and the noncooperative. Typically a cooperative approach will utilize a bargaining model under which the several firms in the oligopoly are supposed to bargain among themselves in order to agree on some joint decision (say, a set of prices to be charged-one per firm) which yields to the industry one of the outcomes which is Pareto optimal for them. A noncooperative approach will, by contrast, involve each firm in isolated decision making. This is not to imply that each firm ignores the effects of its rivals decisions on its own profit or of its own decisions on its rivals behavior (and hence its own profits). Noncooperative formulations will generally assume the firms not to make decisions jointly. An equilibrium of a noncooperative nature typically features a set of strategies (say a strategy is a rule for choosing ones price), one for each firm, having the following property: For each firm, its equilibrium strategy gives at least as much profit as any other strategy it might choose, given the strategy choices of its rivals. One possible method of finding a cooperative solution is to define a bargaining process, or a set of axioms which the solution must obey, which turns the process of determining a solution into a noncooperative game. The Nash cooperative solution [6] is a case in point. Here, each firm may be regarded as (a) agreeing to abide by the Nash axioms to find the solution point which results from the threat point and (b) choosing a threat strategy. The several threat strategies determine a threat point, which determines a solution; while the choice of a threat strategy is dictated by the wish to arrive at the threat point whose associated solution yields maximum profit to the firm. In the literature some discussion may be found of the relative merits of cooperative and noncooperative theories. The former have appeal because it seems so obviously sensible and in the interest of all oligopolists to jointly exploit their market to the full. On the other hand, making joint agreements is difficult. The more firms there are, the more interests to reconcile; hence, the harder it is to come to agreement. In addition, where each firm has knowledge not shared by others (say knowledge of ones own profit function), it becomes even harder to bargain because no one knows the full alternatives open to the group and it may be in a firms own interest to misrepresent

Reaction Functions as Nash Equilibria

A reaction function for the ith firm, xit = i(xt.1), is a decision rule which selects a price for the firm in period t as a function of the observed price vector of period t -1. A Nash [11] non-cooperative equilibrium for this model, in which the equilibrium strategies were reaction functions, would be characterized by n reaction functions (xt), ..., +P*(xt), one for each firm, which have the following property: For the ith firm, no sequence of prices, xit, xi,t+ , ... will yield a higher value for equation (1) than will Xi?= t4(x1) ( = t, t+ 1, ...), given that all other firms will choose their prices by Xjt = ql*(x,_1) in all periods. This condition holds.for i = 1, ..., n.1 The purpose of this paper is to establish conditions under which the equilibrium outlined above approximately exists. Thus this paper is in the reaction function tradition of oligopoly which has its earliest roots in Cournot [2] and its main early development at the hands of Bowley [1], Stackelberg [12] and Fellner [5].2 These writers worked with single period models of the firm, while discussing how the firms ought to behave given that they are really concerned with profits over a long time horizon. Taking explicit account that the firms objective is to maximize a discounted stream of profits (as in equation (1) above) is first done, so far as I am aware, in my article on duopoly [6]. The present paper continues the research begun there and carried on in [7], [9]. While each of these papers makes steps in the direction of Nash type reaction function equilibria, existence was not proved in them for any class of models, nor was any satisfactory approximate equilibrium found to exist; therefore, the present paper is an advance in this line of research. Section 1 gives the basic assumptions and pertinent results from earlier work, and Section 2 defines and shows existence of the approximate equilibrium for a large class of models.

Chapter 11 Oligopoly theory

Publisher Summary The oligopoly theory usually refers to the partial equilibrium study of markets in which the demand side is competitive, while the supply side is neither monopolized nor competitive. It is exclusively concerned with single period models. The two models that are mainly discussed are Cournots and a model based on Chamberlin. In Cournots model, it is assumed that the products of the firms are perfect substitutes and each firm decides the output levels with price being determined in the market. The Chamberlinian model allows for imperfect substitutability among the outputs of the firms, and each firm is assumed to name prices for their products. The chapter also presents some of the basic structure for the models of multi-period oligopoly. The time structure of none of these models is explicit; however, none of them makes sense when interpreted as strictly single-period models. Thus, for the best understanding of them, it is necessary to augment them by making the time relations explicit and by making an appropriate reformulation of the firms objective functions. The chapter also examines various models in which the time structure is explicit and the firms seek to maximize discounted profit streams. The chapter also discusses models with entry and exit of firms.

A Non-cooperative Equilibrium for Supergames: A Correction

The purpose of this note is to correct an error in my paper [1]. Under the assumptions of the paper, Proposition 3 is not, in general, true. The point at which the proof goes awry is in the use of the mapping Ω. Ω is treated in the paper as if it were a function (i.e. a point to point mapping); whereas it is in fact a correspondence (a point to set mapping). Ω maps points of the unit simplex into itself, using a subset of the Pareto optimal set to obtain the simplex. A given point in the payoff space may be the image of more than one point in the strategy space. Each strategy, s, which maps into a given point in the Pareto optimal set (i.e. which has a given I associated with it in the simplex), can map into a distinct δ in the simplex. Thus the Brouwer theorem cannot be used. While Ω is surely upper semi-continuous, it need not have convex image sets, ruling out use of the Kakutani fixed point theorem. Furthermore, it need not be lower semi-continuous, ruling out another route to the use of the Brouwer theorem. That route is to use the results of E. Michael [2], which establish that if Ω is lower semi-continuous, then it has a selection which is a continuous function from the unit simplex into itself. The upshot is that the axioms of [1] must be somehow strengthened in one of the ways suggested by the preceding paragraph in order to make the proposition true. One way would be to strengthen A3 so as to make the payoff functions quasi-concave on S. Then the images of Ω would be convex and the Kakutani theorem could be applied. An alternative, which I prefer, is to strengthen A5. Let T = {s | s ∊ S and τ(s) ∊ H*}. T is the subset of strategies in S which map onto the Pareto optimal set, H*. Thus τ−1 is a mapping from H onto S, in general, a correspondence; and, when restricted to H* it is a correspondence from H* onto T. The stronger A5 is: A5′ H* is concave and the mapping τ−1 from H* on to T is lower semi-continuous. A5′ would, of course, be satisfied if the mapping from T to H* were one-to-one.In that case, Ω would be a continuous function and the Brouwer theorem could be directly applied. As A5′ stands, Ω must have a continuous selection to which the latter theorem could be applied. I have seen no way to prove Proposition 3 from the assumptions in [1], nor have I found a more satisfactory method of strengthening them. The only alternatives not mentioned above of which I am aware are essentially minor variations of the two methods.