Periodic orbit can be evolutionarily stable: Case Study of discrete replicator dynamics

Abstract

In evolutionary game theory, it is customary to be partial to the dynamical models possessing fixed points so that they may be understood as the attainment of evolutionary stability, and hence, Nash equilibrium. Any show of periodic or chaotic solution is many a time perceived as a shortcoming of the corresponding game dynamic because (Nash) equilibrium play is supposed to be robust and persistent behaviour, and any other behaviour in nature is deemed transient. Consequently, there is a lack of attempt to connect the non-fixed point solutions with the game theoretic concepts. Here we provide a way to render game theoretic meaning to periodic solutions. To this end, we consider a replicator map that models Darwinian selection mechanism in unstructured infinite-sized population whose individuals reproduce asexually forming non-overlapping generations. This is one of the simplest evolutionary game dynamic that exhibits periodic solutions giving way to chaotic solutions (as parameters related to reproductive fitness change) and also obeys the folk theorems connecting fixed point solutions with Nash equilibrium. Interestingly, we find that a modified Darwinian fitness-termed heterogeneity payoff-in the corresponding population game must be put forward as (conventional) fitness times the probability that two arbitrarily chosen individuals of the population adopt two different strategies. The evolutionary dynamics proceeds as if the individuals optimize the heterogeneity payoff to reach an evolutionarily stable orbit, should it exist. We rigorously prove that a locally asymptotically stable period orbit must be heterogeneity stable orbit-a generalization of evolutionarily stable state.

Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games

“…In line with the opening sentence of this paper, a very useful discrete replicator equation [9,23,41,[45][46][47][48][49] is as follows:…”

“…However, one must take this non-unanimity positively and constructively since the replicator equations are merely minimal models that easily incorporate the Darwinian tenet of natural selection to investigate mathematical system mimicking exclusively the replication-selection aspect of the real evolutionary systems. In this spirit, what we have achieved in this paper is to provide a more sound (microscopic) footing to the type I replicator map that has immense potential to serve as a convenient testbed for investigating the connections [41,48] between the complex dynamical outcomes and game-theoretic equilibria.…”

“…The connection of the evolutionarily stable strategy/state with the fixed points of the replicator dynamics is less general; results differ depending on whether the equation is discrete or continuous. In this context, it should be borne in mind that it is rather unexplored [41] how the non-fixed point equilibrium sets (like periodic orbits and chaotic orbits) correspond to the game theoretic equilibria (like Nash equilibria or evolutionarily stable strategies).…”

Deciphering chaos in evolutionary games

“…A two-player-two-strategy (one dimensional) discretetime replicator map [37][38][39] may be written as follows:…”

Dynamic Evolution Game of Travelers’ Air-to-HSR Choice under the Scenario of HSR Speed-Up

“…In the dynamic models that simulate the learning and strategies of game players, replication selection mechanism is the most common one [31]. Based on this, the study has found out that the HSR-air competition has a significant impact on the ticket price and market share of both sides [32], which is further shown in the three major aviation data reports in China [33]; similarly, the data also show that the ticket price and the frequency are the two main factors that lead to the decrease of civil aviation market share [17]; in the process of strategy selection, when the individual travel utility has certain constraints, the system model has a unique solution.…”