John H. Nachbar

“Evolutionary” selection dynamics in games: convergence and limit properties

This paper discusses convergence properties and limiting behavior in a class of dynamical systems of which the replicator dynamics of (biological) evolutionary game theory are a special case. It is known that such dynamics need not be well-behaved for arbitrary games. However, it is easy to show that dominance solvable games are convergent for any dynamics in the class and, what is somewhat more difficult to establish, weak dominance solvable games are as well, provided they are “small” in a sense to be made precise in the text. The paper goes on to compare dynamical solutions with standard solution concepts from noncooperative game theory.

Prediction, optimization, and learning in repeated games.

This paper shows that, in many infinitely repeated games, if players optimize with respect to beliefs that satisfy a diversity condition termed neutrality, then each player will choose a strategy that his opponent was certain would not be played. This is an obstacle to formulation of a learning theory in which Nash equilibrium behavior is a necessary long-run consequence of optimization by cautious players.

Non-computable Strategies and Discounted Repeated Games

SummaryA number of authors have used formal models of computation to capture the idea of “bounded rationality” in repeated games. Most of this literature has used computability by a finite automaton as the standard. A conceptual difficulty with this standard is that the decision problem is not “closed.” That is, for every strategy implementable by an automaton, there is some best response implementable by an automaton, but there may not exist any algorithm forfinding such a best response that can be implemented by an automaton. However, such algorithms can always be implemented by a Turing machine, the most powerful formal model of computation. In this paper, we investigate whether the decision problem can be closed by adopting Turing machines as the standard of computability. The answer we offer is negative. Indeed, for a large class of discounted repeated games (including the repeated Prisoners Dilemma) there exist strategies implementable by a Turing machine for whichno best response is implementable by a Turing machine.

Beliefs in Repeated Games

Consider a two-player discounted infinitely repeated game. A players belief is a probability distribution over the opponents repeated game strategies. This paper shows that, for a large class of repeated games, there are no beliefs that satisfy three conditions, learnability, consistency, and a diversity condition, CS. This impossibility theorem generalizes results in Nachbar (1997).

Evolution in the finitely repeated prisoner's dilemma

Abstract This paper examines some aspects of ‘evolutionary’ dynamic behavior in the finitely repeated prisoners dilemma. The ‘fitness’ of cooperation found in the best known simulation of this type, that by Robert Axelrod, stems from strategy set restrictions: The game used for the simulation has a continuum of pure cooperation equilibria and no pure defection equilibrium. Some new simulations are presented here for the finitely repeated game. Although cooperation is ultimately exploited and extinguished, dynamic paths can ‘pseudo converge’ in ways that allow partial cooperation to flourish for extended periods of time.

On the finiteness of the number of critical equilibria, with an application to random selections

Abstract This paper establishes the following result for exchange economies with l commodities and n agents: for a generic set of utility functions of the first consumer, the number of critical equilibria is at most ln at every endowment. This result has application to, for example, the theory of general equilibrium Cournot competition.

Bayesian learning in repeated games of incomplete information

Abstract. In Nachbar [20] and, more definitively, Nachbar [22], I argued that, for a large class of discounted infinitely repeated games of complete information (i.e. stage game payoff functions are common knowledge), it is impossible to construct a Bayesian learning theory in which player beliefs are simultaneously weakly cautious, symmetric, and consistent. The present paper establishes a similar impossibility theorem for repeated games of incomplete information, that is, for repeated games in which stage game payoff functions are private information.

Sunk Costs, Accommodation, and the Welfare Effects of Entry

Although economists usually support the unrestricted entry of firms into an industry, entry may lower social welfare if there are setup costs or if entrants have a cost disadvantage. The authors consider the welfare effects of entry within a standard Cournot model where some of an incumbent firms costs are sunk. They find that the range of parameter values over which entry can harm welfare declines monotonically in the fraction of costs that are sunk. Furthermore, the presence of even a small fraction of sunk costs often reverses an assessment that entry harms welfare. Copyright 1998 by Blackwell Publishing Ltd

General equilibrium comparative statics: discrete shocks in production economies

Abstract Nachbar [Econometrica 79 (5) (2002) 2065] established minimal conditions under which, following an infinitesimal shock to endowments in an exchange economy, changes in equilibrium prices are negatively related to changes in aggregate consumption. The present paper extends Nachbar (2002) to cover discrete shocks to technologies, ownership shares, and endowments in production economies. As in Nachbar (2002), the analyst’s choice of price normalization plays a key role. The required normalization is nonstandard but has a sensible interpretation.

A comment on 'evolution in economic games'

Abstract ‘Evolution in Economic Games’ by Hansen and Samuelson (1988, this journal) motivates an evolutionary dynamical system for games and applies these dynamics to a number of games of economic importance. This comment points out that Hansen-Samuelson dynamics are mathematically identical to the replicator dynamics of biological evolutionary game theory. Some simple examples are provided to illustrate that these dynamics need not be ‘well behaved’.