Transition matrix model for evolutionary game dynamics

Abstract

We study an evolutionary game model based on a transition matrix approach, in which the total change in the proportion of a population playing a given strategy is summed directly over contributions from all other strategies. This general approach combines aspects of the traditional replicator model, such as preserving unpopulated strategies, with mutation-type dynamics, which allow for nonzero switching to unpopulated strategies, in terms of a single transition function. Under certain conditions, this model yields an endemic population playing non-Nash-equilibrium strategies. In addition, a Hopf bifurcation with a limit cycle may occur in the generalized rock-scissors-paper game, unlike the replicator equation. Nonetheless, many of the Folk Theorem results are shown to hold for this model.

Consensus and Information Cascades in Game-Theoretic Imitation Dynamics with Static and Dynamic Network Topologies

“…Completing this program of study by proving or finding counter-examples to our conjectures on convergence is essential. Proving that unbiased cyclic games almost always converge to consensus in this model is of substantial interest as it would represent a fundamental result on a large class of games that normally induce oscillations in evolutionary game theory [55].…”

“…Among the models for interactions in these systems, evolutionary game theory offers a conveniently adjustable and straightforward model for well-characterized strategic interactions, which include aspects such as sub-optimal stable equilibria and multiple equilibria [51]. Work in theoretical biology has begun to use evolutionary games on graphs in similar ways to understand network topologies for which evolutionary stability can be expected [52,53] and develop variations on the replicator dynamic (see e.g., [54,55]). Hussein [56] investigated a similar problem for generic network social behaviors, while Pantoja and Quijano [57] investigate a distributed optimization problem on a network with the replicator.…”

Cooperation can emerge in prisoner’s dilemma from a multi-species predator prey replicator dynamic

“…In addition, variations like discrete 3 time replicator dynamics [5] and replicator dynamics with mu-4 tations [6,7] have been more recently studied within the physics 5 community for understanding dynamics of populations in social 6 and complex systems. Variations of the replicator dynamics, in- 7 cluding imitation [2,4] dynamics have also been studied exten-8 sively (see [1,4,[8][9][10] ). In addition to this, evolutionary games have 9 come under criticism for their lack of adherence to experimental 10 data, particularly for modeling social systems [11] .…”

“…Variations of the replicator dynamics, in- 7 cluding imitation [2,4] dynamics have also been studied exten-8 sively (see [1,4,[8][9][10] ). In addition to this, evolutionary games have 9 come under criticism for their lack of adherence to experimental 10 data, particularly for modeling social systems [11] . Recent work by 11 Press and Dyson [12] has called into question (again) the origin of 12 cooperation in systems where non-cooperative strategies dominate.…”

Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games

“…Since this early work, several authors have investigated various cyclic games and games characterized by circulant matrices. Among many other works: Hofbauer and Schlag [15] consider imitation in cyclic games; Diekmann and Gils specifically study the cyclic replicator dynamics and focus on the properties of low-dimensional cyclic games [6]; Ermentrout et al consider a transition matrix evolutionary dynamic in which a limit cycle emerges in the rock-paper-scissors game [7]; and Griffin and Belmonte [12] study a triple public goods game and show that is is diffeomorphic to generalized rock-paper-scissors. Each of these works focuses explicitly on classes of circulant games, while recent work by Granić and Kerns [11] characterizes the Nash equilibria of arbitrary circulant games, but does not focus on the evolutionary game context.…”