Stability and instability of the unbeatable strategy in dynamic processes

Abstract

A strategy is unbeatable if it is immune to any entrant strategy of any size. This paper investigates static and dynamic properties of unbeatable strategies. We give equivalent conditions for a strategy to be unbeatable and compare it with related equilibrium concepts. An unbeatable strategy is globally stable under replicator dynamics. In contrast, an unbeatable strategy can fail to be globally stable under best response dynamics even if it is also a unique and strict Nash equilibrium.

Basins of attraction and equilibrium selection under different learning rules

“…However, such conditions also imply a unique stable equilibrium. 4 One branch of the learning literature does consider games in which stability depends on the learning dynamic (Kojima, 2006) as well as games with distinct basins of attraction for different learning rules (Hauert et al, 2004). Those models rely on nonlinear payoff structures.…”

Evolutionary finance: A model with endogenous asset payoffs

“…EF models invoke ideas related to behavioral economics and finance (Tversky and Kahneman [51], Shiller [47], Bachmann et al [8]), evolutionary game theory (Weibull [52], Samuelson [42], Gintis [24], Kojima [29]) and games of survival (Milnor and Shapley [38], Shubik and Thompson [48]). Another important source for EF is capital growth theory, or the theory of growth-optimal investments: Kelly [27], Breiman [14], Algoet and Cover [1], and others.…”

An evolutionary finance model with a risk-free asset

“…Modelling frameworks studied in evolutionary finance combine the ideas of behavioural economics and finance (Tversky and Kahneman [68], Shleifer [64], Shiller [63], Thaler [66], Bachmann et al [5]), evolutionary game theory (Weibull [73], Vega-Redondo [69], Samuelson [57], Hofbauer and Sigmund [35], Kojima [39], Gintis [31]), stochastic games (Shapley [61], Dynkin [19], Haurie et al [34], Kifer [38], Neyman and Sorin [55], Vieille [70][71][72]), stochastic evolutionary games (Foster and Young [26], Fudenberg and Harris [27], Cabrales [15], Germano [30]), games of survival (Milnor and Shapley [53], Shubik and Thompson [65], Borch [8], Karni and Schmeidler [36],) and capital growth theory (Shannon [60], Kelly [37], Latané [41], Breiman [13], Algoet and Cover [1], Hakansson and Ziemba [32], Cover [16], Dempster et al [18], MacLean et al [49], Kuhn and Luenberger [40], Ziemba and Vickson [75], MacLean and Ziemba [50], and others; for a textbook treatment of capital growth theory see [22], Ch. 17.…”