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 firm's 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.