Evolutionary Games with Randomly Changing Payoff Matrices

Abstract

Evolutionary games are used in various fields stretching from economics to biology. In most of these games a constant payoff matrix is assumed, although some works also consider dynamic payoff matrices. In this article we assume a possibility of switching the system between two regimes with different sets of payoff matrices. Potentially such a model can qualitatively describe the development of bacterial or cancer cells with a mutator gene present. A finite population evolutionary game is studied. The model describes the simplest version of annealed disorder in the payoff matrix and is exactly solvable at the large population limit. We analyze the dynamics of the model, and derive the equations for both the maximum and the variance of the distribution using the Hamilton-Jacobi equation formalism. The Master Equation and Its Solution via Hamilton-Jacobi EquationLet us consider the model with constant matrices: A ¼ fa ij g, B ¼ fb ij g, i; j ¼ 1; 2; . . . ; m. The total population size is N; variable X represents the number of players in the first population category. As shown in Fig. 1, the whole system can exist in two versions: the upper chain with matrix A and the lower chain with matrix B, and there are transitions between the two chains.Here we have the probability conservation condition at any moment of time τ:where PðX; Þ is the probability that the system is in state A and there are X players with the first strategy, and QðX; Þ is the probability that the system is in state B with X players with the first strategy.

Polymorphism in rapidly changing cyclic environment

“…We stress that this dependence is not a special feature of the considered stable polymorphic state. It will appear whenever there is a fast environment and the space of p 1 posseses several attractors, e.g., for the unstable polymorphic case studied in section V C; see (43). Fig.…”

“…for both I and II acting d yields a higher payoff than c (cooperating), no matter what the opponent does; see (51,52) and note that (d, d) is the only Nash equilibrium of game (51). Both players can get R > P , if they both act c. But acting c is vulnerable, since the opponent can change to d, gain out of this, and leave the cooperator with the minimal pay-off S. This makes the dilemma, which raised deep questions about rationality and cooperation [27,39,40] and produced a vast literature [41][42][43][44][45]. We focus on the case when pay-offs in (51) are timeperiodic functions.…”

Mathematical Models of Evolution for Replicator Systems: Fitness Landscape Adaptation

“…However, for practical applications, it is reasonable to consider the impact of changing environment and, thus, dynamical fitness landscapes. There are several ways to include these variations in the fitness landscape, e.g., considering random evolutionary parameters [2,3] or some optimization process over these parameters [4,5] In this paper, we discuss the underlying concept of the recent studies [4,5], which propose a new method of fitness landscape adaptation and describe it in the form of the optimization problem. One of the first researchers, who suggested using the extreme principle for Darwinian evolution, was R. Fisher.…”

“…To rephrase the problem in terms of the described system, we consider the following optimization problem: to find such elements a ij of matrix A() satisfying (4) such that the mean integral fitness () m  of the population, defined by (3), is maximized at some fixed time moment  = T of the evolutionary time.…”

On the stochastic evolution of finite populations

“…This is the same "surprising effect" presented in Ashcroft et al (2014), however with deterministic time evolution, i.e, no stochasticity was assumed in the time evolution of the model. See also Melbinger and Vergassola (2015); Yakushkina et al (2015).…”