The coevolutionary dynamics in finite populations currently is investigated in a wide range of disciplines, as chemical catalysis, biological evolution, social and economic systems. The dynamics of those systems can be formulated within the unifying framework of evolutionary game theory. However it is not a priori clear which mathematical description is appropriate when populations are not infinitely large. Whereas the replicator equation approach describes the infinite population size limit by deterministic differential equations, in finite populations the dynamics is inherently stochastic which can lead to new effects. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection in finite populations based on microscopic processes. In asymmetric conflicts between two populations with a cyclic dominance, a finite-size dependent drift reversal was demonstrated, depending on the underlying microscopic process of the evolutionary update. Cyclic dynamics appears widely in biological coevolution, be it within a homogeneous population, or be it between disjunct populations as female and male. Here explicit analytic address is given and the average drift is calculated for the frequency-dependent Moran process and for different pairwise comparison processes. It is explicitely shown that the drift reversal cannot occur if the process relies on payoff differences between pairs of individuals. Further, also a linear comparison with the average payoff does not lead to a drift towards the internal fixed point. Hence the nonlinear comparison function of the frequency-dependent Moran process, together with its usage of nonlocal information via the average payoff, is the essential part of the mechanism.
Biology offers a rich laboratory of various types of oscillatory, chaotic and stochastic dynamics. Recently cyclic evolutionary dynamics has been observed in E.coli in vitro [1] and in vivo [2], and attracted interest as a possible mechanism to stabilize biodiversity. This contributes to a long-standing debate how the emergence of new mutants is maintained in biological evolution: Cyclic dynamics has been one of the first proposals for such mechanisms [3]. Cyclic coevolution is not only observed in asexual reproduction. A prominent observation of cyclic domination is observed in side-blotched lizards [4,5]. Three territorial mating behaviour strategies of the male lizards occur, and coincide genetically with orange, blue and yellow blotches. While the cyclic dynamics of rock-paper-scissors type can be demonstrated without taking males and females into account explicitely, hereby an improved quantitative understanding is possible [6].
Social and economic systems are a likewise interesting class of systems in which cyclic dynamics is observed. Social individua deciding in economic situations [7,8,9,10,11,12,13] can fall into oscillatory cycles, e.g. when loners, not participating in the game, are added as a third strategy to a Prisoner’s Dilemma [14,15]. Evolutionary game theory is a unifying approach for such systems [16,17,18].
In this paper, a paradigmatic cyclic game is analyzed, which is of likewise importance, for biological mating behaviour, as well as in human social decision dynamics: Dawkins’ “Battle of the Sexes” (BOTS), a 2×2 bimatrix game, is the simplest possible type of a cyclic game played by individuals between two homogeneous, mixed populations [19]. In the mating behaviour of females and males, a cyclical dominance of ‘slow’ and ‘fast’ strategies can lead to a “Battle of the Sexes”, or oscillations, in the infinite population replicator dynamics [19,20,21]. The inherent stochasticity in a finite population [22,23,24] refines this picture, depending on population size and underlying process [25,26,27,28,29,30].
The aim of the paper is to investigate in detail the drift reversal firstly reported in [25], and to corrobate the simulations with an analytic result, allowing for a refined insight into the drift reversal. The main results are (i) that for a generic class of pairwise comparison processes a drift reversal does not occur, and (ii) that for the Moran process, the combination of both a nonlinear reproductive fitness and the comparison with a global function (average fitness) are necessary for a drift reversal.
The paper is organized as follows. In Section 1, the BOTS payoff matrices are introduced and the infinite population description of evolutionary game theory by the replicator equation is recalled. In Section 2, evolutionary birth-death processes are defined in a unifying framework for comparison. In Section 3 the influence of the stochasticity on the time evolution of the population densities is motivated by computer simulations of the process.
In the main Section 4, the average drift (more formally introduced in Sect. 4.1) of the BOTS dynamics is calculated explicitely in finite populations, and the population size corrections are obtained to first and second order analytically for four microscopic interaction processes, for neutral evolution as well as for two linear processes, the Local Update and the linearized Moran process. The paper concludes by discussing and summarizing the results.
1 Battle of the Sexes: Replicator dynamics This is the simplest cyclic game between two populations. Following Dawkins [19], the elementary payoffs read
for agents interacting with a population of (x, y, 1-x, 1-y) = (i/N, j/N, (N -i)/N, (N -j)/N ) agents in the strategies (π | A , π Ã , π | B , π B ). As the opponent’s payoff matrix is not the simple transpose of the proponent’s payoff matrix, such games are called bimatrix games or asymmetric conflicts. While in the replicator equations [20] picture the population cannot go extinct due do lack of discreteness, for the stochastic description in this paper the population size will be fixed to N female and N male individuals.
For the relative frequencies, or abundance densities, evolutionary game theory, with the implicit assumption of an infinite population, considers the replicator equation
For the standard parameter choice (equivalent to “Matching Pennies”) of the BOTS the replicator equation has a constant of motion
The replicator equations exhibits neutrally stable oscillations around the x = y = 1/2 fixed point; the adjusted replicator equation (see [20]) has an attractive stable fixed point. In so-called asymmetric conflicts (bimatrix games)
where members of the two (sub)populations can receive different payoffs, both populations may gain different average payoffs, and the denominators in the adjusted replicator equations (being the proper N → ∞ limit of the Moran process, see [27])
are different, as π | = -π ~ . Thus the deno
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