Stochastic Replicator Dynamics Research Articles

Stochastic replicator dynamics: A theoretical analysis and an experimental assessment

We study the dynamics of strategic choice in an environment where payoffs are perturbed by strategy-specific noise and strategy revision is governed by the simple proportional imitation protocol. Applying the stochastic replicator dynamic to a two-by-two symmetric game, we derive the steady-state frequencies of the strategies and show that the dominated strategy can persist in equilibrium if the variance of the payoff shocks exceeds a critical value. Additionally, the rate of decay of the dominated strategy decreases with increasing variance. The behaviours of our experimental subjects accord with these tendencies. Our findings add to the growing experimental literature that demonstrates that evolutionary models of human behaviour with simple adaptive rules have predictive ability, even in complex, volatile environments.

Weak selection and stochastic evolutionary stability in a stochastic replicator dynamics

It is a very challenging problem whether natural selection is able to effectively resist the continuous disturbance of environmental noise such that the direction or outcome of evolution determined by the deterministic selection pressure will not be changed. By analyzing the impact of weak selection on the evolutionary stability of a stochastic replicator dynamics with n possible pure strategies, we found that the weak selection is able to enhance the evolutionary stability, that is, under weak selection, the stochastic evolutionary stability of the system is determined by the mean payoff matrix. This finding strongly implies that the weak selection should be regarded as an important mechanism to ensure evolutionary stability in stochastic environments.

Darwinian evolution as Brownian motion on the simplex: A geometric perspective on stochastic replicator dynamics

We prove that stochastic replicator dynamics can be interpreted as intrinsic Brownian motion on the simplex equipped with the Aitchison geometry. As an immediate consequence, we derive three approximation results in the spirit of Wong–Zakai approximation, Donsker’s invariance principle and a JKO-scheme. Using the Fokker–Planck equation and Wasserstein-contraction estimates, we also study the long time behavior of the stochastic replicator equation, as an example of a nongradient drift diffusion on the Aitchison simplex.

Stochastic replicator dynamics and evolutionary stability.

To develop the concept of evolutionary stability in a stochastic environment, we investigate the continuous-time dynamics of a two-phenotype linear evolutionary game with generally correlated random payoffs in pairwise interactions. By using the Gram-Schmidt orthogonalization procedure and Itô's formula, we deduce a stochastic differential equationfor the phenotype frequencies that extends the replicator equation, called the stochastic replicator equation. We give conditions for stochastic stability of a fixation state or a constant interior equilibrium point with respect to the stochastic dynamics of the two phenotypes. We show that, if a fixation state is stochastically stable, then the pure strategy corresponding to this fixation state must be stochastically evolutionarily stable with respect to mixed strategies. However, this is not the case for a mixed strategy that corresponds to a stochastically stable constant interior equilibrium point with respect to the two phenotypes.

Two‐strategy evolutionary games with stochastic adaptive control

SummaryWe investigate the stochastic stability of replicator dynamics for evolutionary games with multiplicative noises and time delay in this article. We study evolutionary games in infinite well‐mixed populations, among which each individual has two strategies, to cooperate or defect, each time when it is matched up with other individuals to play the two‐strategy game together. This article is concerned with the extension of the well‐known two‐strategy evolutionary games to the stochastic case with adaptive control. We analyze the stochastic replicator dynamics with time delays and provide conservative bounds on the strength of noise and the time delay for the almost sure exponential stability of stochastic systems. Finally, numerical simulations based on triple classical game models (snowdrift game, hunt stag game, and prisoner's dilemma game) are given to illustrate the preceding theoretical results.

Stochastic Evolutionary Game Analysis Between Special Committees and CEO: Incentive and Supervision

Focusing on the problem of CEO incentive and supervision, this paper constructs a three-party stochastic evolutionary game model of the CEO, compensation committee and audit committee, and analyzes the boundary conditions of the stability based on stochastic replicator dynamics equations. In addition, it carries out the numerical approximations with Taylor expansion and $$It{\hat{o}}$$ -type stochastic differential equation, and further analyzes the influence of key variables on the direction and convergence rate of game players' strategy choice. The results show that: (1) The probability that the CEO takes self-interested behavior falls faster and the probability that audit committee adopts supervisory strategy rises faster when the self-interested behavior of the CEO gets larger. (2) The audit committee has a fast convergence rate, and arrives earlier in the neighborhood of the stochastically stable state where the incentive coefficient adopted by the compensation committee is larger. (3) The increase in supervision intensity increases the convergence rate of the CEO, but decreases the convergence rate of the compensation committee. (4) According to a comparative analysis of noise intensity, the influence of random interference on the decision-making of the audit committee is more volatile.

An Evolutionary Game to Study Banks–Firms Relationship: Monitoring Intensity and Private Benefit

The paper analyzes a dynamic evolutionary game between banks and firms whose interaction has always been characterized by conflictual relationships. Banks would like that the funding is spent to achieve the objectives of the projects submitted, whereas firms would allocate these loans to obtain private benefits. Following replicator dynamics, we show that banks and firms have predator-prey interactions of the Lotka–Volterra type. Misbehaving firms who seek private benefits are “predators” and banks are their “prey”. We analyze the dynamics emerging from the model and we prove that the stability of equilibria depending on the fundamental parameters which describe the banks–firms interaction. In addition, we compare equilibria in terms of Pareto efficiency computing welfare through the average profits with some numerical applications. Finally, we propose a stochastic replicator dynamics approach in order to assume a perturbation in the population growth rate and we suppose as endogenous the monitoring intensity.

Metastability in Stochastic Replicator Dynamics

We consider a novel model of stochastic replicator dynamics for potential games that converts to a Langevin equation on a sphere after a change of variables. This is distinct from the models of stochastic replicator dynamics studied earlier. In particular, it is ill-posed due to non-uniqueness of solutions, but is amenable to a natural selection principle that picks a unique solution. The model allows us to make specific statements regarding metastable states such as small noise asymptotics for mean exit times from their domain of attraction, and quasi-stationary measures. We illustrate the general results by specializing them to replicator dynamics on graphs and demonstrate that the numerical experiments support theoretical predictions.

Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

Imitation dynamics with payoff shocks

We investigate the impact of payoff shocks on the evolution of large populations of myopic players that employ simple strategy revision protocols such as the “imitation of success”. In the noiseless case, this process is governed by the standard (deterministic) replicator dynamics; in the presence of noise however, the induced stochastic dynamics are different from previous versions of the stochastic replicator dynamics (such as the aggregate-shocks model of Fudenberg and Harris in J Econ Theory 57(2):420–441, 1992). In this context, we show that strict equilibria are always stochastically asymptotically stable, irrespective of the magnitude of the shocks; on the other hand, in the high-noise regime, non-equilibrium states may also become stochastically asymptotically stable and dominated strategies may survive in perpetuity (they become extinct if the noise is low). Such behavior is eliminated if players are less myopic and revise their strategies based on their cumulative payoffs. In this case, we obtain a second order stochastic dynamical system where non-equilibrium states are no longer attracting and dominated strategies become extinct (a.s.), no matter the noise level.

Stochastic replicator dynamics subject to Markovian switching

Population dynamics are often subject to random independent changes in the environment. For the two strategy stochastic replicator dynamic, we assume that stochastic changes in the environment replace the payoffs and variance. This is modeled by a continuous time Markov chain in a finite atom space. We establish conditions for this dynamic to have an analogous characterization of the long-run behavior to that of the deterministic dynamic. To create intuition, we first consider the case when the Markov chain has two states. A very natural extension to the general finite state space of the Markov chain will be given.

On the Existence and Stability of Pareto Optimal Endogenous Matching with Fairness

In the current paper, we study the asymmetric normal-form game between two heterogeneous groups of populations by employing the stochastic replicator dynamics driven by Levy process. A new game equilibrium, i.e., the game equilibrium of a stochastic differential cooperative game on time, is derived by introducing optimal-stopping technique into evolutionary game theory, which combines with the Pareto optimal standard leads us to the existence of Pareto optimal endogenous matching. Moreover, stability of the Pareto optimal endogenous matching is confirmed by essentially using the well-known Girsanov Theorem.

Time averages, recurrence and transience in the stochastic replicator dynamics

We investigate the long-run behavior of a stochastic replicator process, which describes game dynamics for a symmetric two-player game under aggregate shocks. We establish an averaging principle that relates time averages of the process and Nash equilibria of a suitably modified game. Furthermore, a sufficient condition for transience is given in terms of mixed equilibria and definiteness of the payoff matrix. We also present necessary and sufficient conditions for stochastic stability of pure equilibria.

Robust permanence and impermanence for stochastic replicator dynamics

Garay and Hofbauer (SIAM J. Math. Anal. 34 (2003)) proposed sufficient conditions for robust permanence and impermanence of the deterministic replicator dynamics. We reconsider these conditions in the context of the stochastic replicator dynamics, which is obtained from its deterministic analogue by introducing Brownian perturbations of payoffs. When the deterministic replicator dynamics is permanent and the noise level small, the stochastic dynamics admits a unique ergodic distribution whose mass is concentrated near the maximal interior attractor of the unperturbed system; thus, permanence is robust against small unbounded stochastic perturbations. When the deterministic dynamics is impermanent and the noise level small or large, the stochastic dynamics converges to the boundary of the state space at an exponential rate.

Evolutionary dynamics of finite populations in games with polymorphic fitness equilibria

The hawk–dove (HD) game, as defined by Maynard Smith [1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge], allows for a polymorphic fitness equilibrium (PFE) to exist between its two pure strategies; this polymorphism is the attractor of the standard replicator dynamics [Taylor, P.D., Jonker, L., 1978. Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156; Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge] operating on an infinite population of pure-strategists. Here, we consider stochastic replicator dynamics, operating on a finite population of pure-strategists playing games similar to HD; in particular, we examine the transient behavior of the system, before it enters an absorbing state due to sampling error. Though stochastic replication prevents the population from fixing onto the PFE, selection always favors the under-represented strategy. Thus, we may naively expect that the mean population state (of the pre-absorption transient) will correspond to the PFE. The empirical results of Fogel et al. [1997. On the instability of evolutionary stable states. BioSystems 44, 135–152] show that the mean population state, in fact, deviates from the PFE with statistical significance. We provide theoretical results that explain their observations. We show that such deviation away from the PFE occurs when the selection pressures that surround the fitness-equilibrium point are asymmetric. Further, we analyze a Markov model to prove that a finite population will generate a distribution over population states that equilibrates selection-pressure asymmetry; the mean of this distribution is generally not the fitness-equilibrium state.

The long-run behavior of the stochastic replicator dynamics

Fudenberg and Harris’ stochastic version of the classical replicator dynamics is considered. The behavior of this diffusion process in the presence of an evolutionarily stable strategy is investigated. Moreover, extinction of dominated strategies and stochastic stability of strict Nash equilibria are studied. The general results are illustrated in connection with a discrete war of attrition. A persistence result for the maximum effort strategy is obtained and an explicit expression for the evolutionarily stable strategy is derived.

On ?badly behaved? dynamics

Adaptive (path dependent) processes of growth modeled by urn schemes are important for several fields of applications: biology, physics, chemistry, economics. In this paper we present a general introduction to urn schemes, together with some new results. We review the studies that have been done in the technological dynamics by means of such schemes. Also several other domains of economic dynamics are analysed by the same machinery and its new modifications allowing to tackle non-homogeneity of the phase space. We demonstrate the phenomena of multiple equilibria, different vonvergence rates for different limit patterns, locally positive and locally negative feedbacks, limit behavior associated with non-homogeneity of economic environment where producers (firms) are operating. It is also shown that the above urn processes represent a natural and convenient stochastic replicator dynamics which can be used in evolutionary games.