Evolution of social behaviors in noisy environments

TL;DR

This work extends evolutionary game theory to account for random changes in the social environment, so that mutual cooperation may bring different rewards today than it brings tomorrow, for example.

Abstract

Evolutionary game theory offers a general framework to study how behaviors evolve by social learning in a population. This body of theory can accommodate a range of social dilemmas, or games, as well as real-world complexities such as spatial structure or behaviors conditioned on reputations. Nonetheless, this approach typically assumes a deterministic payoff structure for social interactions. Here, we extend evolutionary game theory to account for random changes in the social environment, so that mutual cooperation may bring different rewards today than it brings tomorrow, for example. Even when such environmental noise is unbiased, we find it can have a qualitative impact on the behaviors that evolve in a population. Noisy payoffs can permit the stable co-existence of cooperators and defectors in the prisoner's dilemma, for example, as well as bistability in snowdrift games and stable limit cycles in rock-paper-scissors games -- dynamical phenomena that cannot occur in the absence of noise. We conclude by discussing the relevance of our framework to scenarios where the nature of social interactions is subject to external perturbations.

Figures (4)

• Figure 1: Evolution of behavior in a noisy social environment. We model a noisy social environment as a payoff matrix with a deterministic component and a random component. a, In the noise-free setting, the payoff structure is the same in all generations. b, In a noisy environment, a random perturbation $\xi$ is sampled independently from the standard Gaussian distribution each generation. c, In each generation, every individual interacts pairwise with all others and derives an average payoff, i.e. $\Pi_C$ for cooperators and $\Pi_D$ for defectors, which depends on the current payoff matrix. The figure illustrates an example of $\Pi_C$ and $\Pi_D$ in a population consisting of $2$ cooperators and $3$ defectors. Each individual's payoff $\Pi$ determines their fitness according to $f=\text{exp}(s\Pi)$. d, Following all social interactions, an individual is selected to serve as the role model, chosen randomly proportional to fitness (dashed circle), and their strategy is imitated by another individual chosen uniformly (birth-death updating).
• Figure 2: Behavioral outcomes in noisy social environments. The figure illustrates evolutionary dynamics for three types of games (prisoner's dilemma, coordination, and snowdrift games) in either a noise-free environment (panels adg) or a noisy environment with different fluctuation intensities (panels bcefh). Each panel shows several evolutionary trajectories starting from various initial states; and the $y$-axis also indicates the direction of evolution as well as unstable (open circles) and stable (solid circles) equilibria. Environmental fluctuations increase the diversity of long-term outcomes. In the prisoner's dilemma, for instance, the unique dominance of defection in the noise-free environment (a) can be augmented with a new equilibrium containing a mixture of cooperation and defection (b), or, if the fluctuation intensity is larger, coexistence may become the only possible outcome (c). For other games, noisy environments can produce new qualitative outcomes such as defection/coexistence (e) or bistable coexistence (f,h) that cannot occur in the absence of noise. Parameters: $s=0.1$, $N=10000$.
• Figure 3: Classification of all behavioral outcomes for $2\times2$ games in noisy and noise-free social environments. Two key variables, $x^*=(d-b)/(a-b-c+d)$ and $K=1/[k^2(a-b-c+d)]$, which depend on the deterministic payoff matrix and the intensity of noise, effectively categorize all possible dynamical outcomes. Insets display how the change in cooperator frequency, $\dot{x}$, depends on the current frequency of cooperators, $x$, with solid (open) circles denoting stable (unstable) equilibria. a, In the absence of noise ($k=0$) there are four distinct dynamical outcomes that depend only on the sign of $K$ and the value of $x^*$. For example, games in the regime $x^*<0$ and $K<0$ (bottom-left region, green) have a unique outcome dominated by defection. b, In the presence of noise ($k>0$) there are seven possible dynamical outcomes, three of which cannot occur without environmental noise (i.e., $k=0$): an interior stable equilibrium and a stable equilibrium on the boundary $x=0$ (red region), an interior stable equilibrium and a stable equilibrium on the boundary $x=1$ (peach region), and two interior stable equilibria (teal region). We have derived an analytic classification of which dynamical pattern will arise, determined by lines and parabolas in the parameters $K$ and $x^*$, as indicated in the figure.
• Figure 4: Evolution of behavior in Rock-Paper-Scissors games. In the classical setting without noise, the dynamical pattern of behaviors in a rock-paper-scissors (RPS) game hinges on the relative magnitudes of payoffs $\alpha$ (paper vs rock) and $\beta$ (rock vs scissors). a, For $\alpha>\beta$, the interior equilibrium with a mixture of types ($x=y=z=1/3$) is an unstable focus, with all trajectories spiralling out towards the boundary. d, In the case $\beta=\alpha$, the interior equilibrium becomes a center, and all trajectories manifest as isolated closed orbits, which are unstable. g, For $\alpha<\beta$, the interior equilibrium becomes a stable focus inducing a spiral sink towards the mixture of all three types. Introducing environmental noise produces a variety of new dynamical phenomena. For $\alpha>\beta$, noise of intermediate intensity can induce a stable limit cycle (b) which is globally attractive: all trajectories ultimately converge to this orbit. When the intensity of noise is yet stronger, there is another stable limit cycle (c), although it is not globally attractive and it co-occurs with three stable equilibria each featuring a mixture of two strategies. In scenarios when $\alpha=\beta$ or $\alpha<\beta$ environmental noise can render the interior equilibrium asymptotically stable (ef) and increase its stability (hi). As the intensity of noise increases, trajectories converge more rapidly towards the interior equilibrium, accompanied by the emergence of new equilibria with stable mixtures of two strategies. Parameters: $s=0.1$.