Journal Article Reaction Functions as Nash Equilibria Get access James W. Friedman James W. Friedman University of Rochester Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 43, Issue 1, February 1976, Pages 83–90, https://doi.org/10.2307/2296602 Published: 01 February 1976 Article history Received: 01 January 1974 Accepted: 01 April 1975 Published: 01 February 1976

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 n. n 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). n 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 p associated with it in the simplex), can map into a distinct (j in the simplex. Thus the Brouwer theorem cannot be used. While n 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 n 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

Journal Article A Non-cooperative Equilibrium for Supergames Get access James W. Friedman James W. Friedman University of Rochester Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 38, Issue 1, January 1971, Pages 1–12, https://doi.org/10.2307/2296617 Published: 01 January 1971

[This paper is concerned with "supergames" in which the action taken in a given time period by a player will affect the payoff to any other player in the subsequent period. A supergame consists of a set of players and a countable sequence of "ordinary" games. To illustrate "time-dependence," consider an economic market in discrete time. Say each firm must choose a price in each time period. This market has time-dependence if the amount demanded of a firm today is a function of the prices chosen today and of the prices chosen in the preceding period. Conditions are given for the existence of non-cooperative equilibria of two types: (i) steady state, in which the individual moves of the players converge over time to some s extasciicircum0 and (ii) balance temptation equilibria of the sort developed by Friedman [6] for games lacking time dependence.]

This paper contrasts consumer choice under uncertain lifetimes with the behavior that would arise if each individual's lifetime were announced at birth. In a model that includes life insurance and excludes investments in human capital, the expected utility under uncertain lifetimes exceeds that under known lifetimes when the latter expectation is based on preannouncement survival probabilities. This conclusion emerges, first, because the model without human capital contains no planning benefits from knowledge of the horizon and, second, because the prior announcement of lifetimes forces risk-averse consumers to undertake an extra gamble that they could otherwise avoid by using life insurance.

We report here a few results of oligopoly experiments which we carried out several years ago. These are laboratory experiments with human subjects along the lines of Lawrence Fouraker and Sidney Siegel (1963), Friedman (1967) and Hoggatt (1969). We sought to observe the predictive power of various theoretical equilibria. Then, to the extent that theoretical equilibria do not predict observed behavior, we seek to understand that behavior as best we can in the hope that the existing body of theoretical knowledge can be improved upon. We used a simple textbook model of an oligopoly, then arranged for undergraduate student subjects to be the in the market. Our markets have 2, 3, 4 or 6 firms, each represented by one subject. One single consists of a period of 1 X2 to 2 hours during which one group of subjects forms a market. For example, in a 3-firm market there would be 3 subjects each in a different room in the laboratory. Each would make the price and output decisions for his firm, and the pay to each for his participation in the game would consist only of the profits of his firm. At the beginning of the game, each would make the price and output decisions for his firm, and the pay to each for his participation in the game would consist only of the profits of his firm. At the beginning of the game, each subject would make his initial price and output decision. These initial decisions would remain in effect until changed by a subject; and he would be free to change either his price or output (or both) at any time. A particular game or market with its fixed set of firms (subjects) would remain in continuous operation for approximately an hour and a half. The time periods are approximately 40 seconds each; and every fourth period, the subject receives a report on his sales, production, inventory, profit, etc. These time periods are sufficiently short that they approximate continuous time for the subject, especially given that there is not time in the whole game, except the beginning, when the subject must actively choose a price and an output level. Anytime afterward he can change his price and/or output level by entering new ones. They take effect at the start of the time period following. Our experimental design was chosen with several objectives in mind: casual evidence, formal evidence (Friedman 1967), and intuition all suggest that sub* Professor of business administration, University of California, Berkeley, Professor of economics, University of Rochester, and Research Assistant, Center for Research in Management Science, University of California, Berkeley, respectively. We are grateful for research support to the National Science Foundation for GS-2423 and GS-2463, to the University of Rochester and to the University of California. The Center for Research in Management Science, Berkeley, where this research was carried out, operates a computer controlled laboratory for the study of human behavior. A manual describing the experimental control programs and the observations on which this paper is based is available on request.