Nash Equilibrium Of Game Theory Research Articles

Fluctuations around Nash equilibria in game theory

We investigate the fluctuations induced by irrationality in simple games with a large number of competing players. We show that Nash equilibria in such games are “weakly” stable: irrationality propagates and amplifies through players' interactions so that huge fluctuations can result from a small amount of irrationality. In the presence of multiple Nash equilibria, our statistical approach allows to establish which is the globally stale equilibrium. However, characteristic times to reach this state can be very large.

Closure and upper semicontinuity results in mathematical programming, Nash and economic equilibria1

A result on Kubatowski's convergence of the intersection of two converging sequences of closed convex subsets of an Euclidean space enables to obtain closedness results for the solution sets of problems with constraints. Such results are applied to minimization problems in mathematical programming, to Nash equilibria in game theory and to competitive equilibria in mathematical economics. Upper semicontinuity results are also derived under usual hypotheses guaranteeing existence of solutions.

Competition, kin selection, and evolutionary stable strategies

In his investigations on animal fighting behavior, John Maynard Smith (Ref. 1) coined the term “evolutionarily stable strategy” (ESS) to denote a behavioral strategy that is stable against invasion by a small number of individuals who employ a “mutant,” or deviant strategy. The notion of an ESS is quite similar—but not identical to—the concept of a Nash equilibrium in game theory. Several authors had previously attempted to apply game theoretic formalisms to evolutionary problems (e.g., Lewontin, Ref. 2; Slobodkin and Rappoport, Ref. 3; Rocklin and Oster, Ref. 4). With the exception of Maynard Smith’s analyses, however, few empirically verifiable predictions were generated. Moreover, with the exception of Stewart (Ref. 5), the models were mostly restricted to static games. In this study we shall present a number of models that treat ESSs from a dynamic viewpoint. In particular, we shall attempt to generalize conventional competition theory by permitting the competing parties to adjust their strategies. Rather than seeking dynamically stable equilibria, as in Volterra-Lotka theory, we shall look for strategically stable solutions, or ESSs (cf. Maynard Smith, Ref. 6). Ultimately, one must extend competitive models to the level of the genetic loci influencing the strategies. Unfortunately, the efforts in this direction usually lead to models that are mathematically intractible (cf. Rocklin and Oster, Ref. 4); therefore, there is some justification for taking a phenomeno-logical approach such as game theory.