Weight of fitness deviation governs strict physical chaos in replicator dynamics

Abstract

Replicator equation-a paradigm equation in evolutionary game dynamics-mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here, we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions-fixed points, periodic orbits, and chaotic trajectories-are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate on which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states, useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.

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

“…7where S = −0.5 and T = 2.0 stands for Prisoner's Dilemma game. The discrete replicator dynamics of this game doesn't have any physical 2-periodic orbit [27]. As periodic orbit must be HO(2), we remark that this game doesn't have any HO (2).…”

“…The replicator map maps frequency of a phenotype in kth generation to its frequency in (k + 1)th nonoverlapping generation. Specifically, the map is expressed in the following form [27,[32][33][34]:…”

“…In literature there are many different evolutionary dynamical models: some of them can be obtained by varying revision protocol [16,17], e.g., replicator dynamics [14], best response dynamics [18], Brownvon Neumann-Nash dynamics [19], Smith dynamics [20] and logit dynamics [21]; whereas some can be seen as variants of incentive dynamics [22], e.g., replicator dynamics [14], best response dynamics [18], logit dynamics [21] and projection dynamics [23]. All the aforementioned dynamics have the common property of converging towards NE [24][25][26][27]. It has also been shown in the literature that evolutionarily stable state can be related to locally asymptotically stable fixed point for both replicator map [27] and replicator flow [15].…”

“…In view of the above, in this paper, we take first step of proposing the hitherto ill-understood question: Is there any game theoretic argument possible that we can connect with the periodic outcomes of the dynamical models in population games and what could be the physical implication of the periodic outcome? Given the plethora of models of evolutionary dynamics, we decide to work with the replicator map adapted for the two-player-twostrategy games [27] because of the following concrete reasons: (i) this evolutionary game dynamic models Darwinian selection, (ii) relationship of its fixed points with NE and ESS is well established through standard folk theorems and related theorems, and (iii) it is able to show periodic solutions (that bifurcate into chaotic solution). Since our main aim is to find game theoretic connection for periodic orbits, we restrict our analyses only to the mixed strategy domain.…”

Evolutionary dynamics of the delayed replicator-mutator equation: Limit cycle and cooperation

“…Nevertheless, an important question may and should be raised: Two-player-two-strategy symmetric games can be classified into twelve ordinal classes [60] that can be collected into four game-types-the SH, the SD, the PD, and the HG-based on how cooperate strategy fares or, equivalently, what the Nash equilibria are. While the PD game has only one type of ordinal game, the SH, the SD, and the HG has respectively three, three, and five ordinally inequivalent payoff matrices and corresponding games.…”