Effects of Stochastic Games on Evolutionary Dynamics in Structured Populations

TL;DR

This work analyzes how stochastic transitions among social games shape the evolution of cooperation on heterogeneous networks. By combining coalescent methods on graphs with an effective-payoff formulation, it derives conditions under which cooperation can dominate or coexist with defection across donation, public goods, and snowdrift dilemmas. Key findings show that exogenous or endogenous transitions can either promote or inhibit cooperation depending on the game type and network topology, with explicit thresholds (e.g., $(b/c)^*$) computed for complete, ceiling-fan, conjoined-star, and empirical networks. The results illuminate how environmental variability and structural diversity interplay to foster or hinder altruistic behavior, offering insights for designing interaction rules in artificial and natural systems. The study also discusses limitations under strong selection and mutation, and outlines avenues for extending the framework to local transitions and broader game families.

Abstract

Continuously changing environments have a paramount role in the evolution of cooperative behavior. Previous works have shown that the transitions among different games, as the feedback between behaviors and environments, can promote cooperative behavior significantly. Quantitative analysis, however, is limited to homogeneous populations, while realistic populations in nature are often more complex and highly heterogeneous. We hereby provide an analytical treatment of when the evolution of cooperation can be supported in stochastic games, applying to arbitrary spatial heterogeneity and payoff structure. We highlight that the rule and frequency of game changes can have surprisingly diverse effects on evolutionary outcomes, depending on the governing social dilemmas. While stochastic games favor the evolution of cooperation in donation games, this is not the case for public goods games and snowdrift games. Hence, our framework and model results offer a more subtle insight into the long-standing enigma.

Figures (13)

• Figure 1: Evolutionary dynamics in stochastic games on population structures. The population structure can be described by a graph, in which each node represents an individual and each edge represents the interaction between two individuals (panel (a)). Each individual can choose one of the two strategies, Cooperation (blue node) and Defection (red node), in each round. The payoff of each individual is obtained by playing games (games are possibly different for different neighbors in the same round, illustrated by different edge colors, orange (cyan) edges representing game $1$ ($2$)) with its neighbors, influenced by both its opponent's strategy and the current game (the payoff matrix of three considered games, i.e., the donation game (DG), the public goods game (PGG), and the snowdrift game (SG), are shown in panel (b)). After the accumulation of payoffs (averaged by the number of neighbors of each individual), an individual is uniformly chosen for death (gray circle) and all its neighbors (black circles) compete to reproduce at the empty site (dashed lines) proportional to their fitness, which is determined by their corresponding payoffs. Then, all games update, driven by either exogenous (independent of the population state) or endogenous factors (dependent on the population state) (panel (c)). An example is given in panel (d), in which mutual cooperation leads to a more profitable donation game (left game), and otherwise, a less profitable one would be reached (right game). The payoff structure of the donation game is a specific parameter setting of the general payoff matrix in panel (c), i.e., $R=b-c$, $S=-c$, $T=b$, and $P=0$. The population eventually terminates in either an all-cooperator (fixation of cooperation) or all-defector (fixation of defection) state.
• Figure 2: Evolutionary game with exogenous game transitions on diverse population structures under death-birth updating. We present the critical benefit-to-cost ratio, $(b/c)^*_{\text{DS}}$, as a function of the probability of the donation game (game 1), $p_1$, on the star graph with one central node and $n$ peripheral nodes (panel (a)), complete graph with size $N$ (panel (b)), and wheel graph with size $N$ (panel (c)). These graphs represent structures in which the critical ratio is infinite, negative, and positive, respectively. Panels show different trajectories of $(b/c)^*$ as $p_1$ changes. The vertical line is the discontinuous point $(p_1)^*$, which leads to $(b/c)^* = \infty$, and the horizontal line indicates the critical benefit-to-cost ratio in a single game, i.e, $p_1 = 0$ (corresponding to the single snowdrift game) and $p_1 = 1$ (corresponding to the single donation game). In general, $(p_1)^*$ is within the range $[0,1]$ when the critical ratio in the donation game is negative. In contrast, the positive critical ratio leads to $(p_1)^*$ out of $[0,1]$, the discontinuous change of $(b/c)^*$ (vertical line in panel (c)), therefore, cannot be observed. The insets of each subgraph are the corresponding network structures.
• Figure 3: Evolutionary dynamics with endogenous game transitions. We consider the death-birth (DB) (panels (a) and (b)) and pairwise-comparison (PC) updating (panels (c) and (d)) in the donation game (DG) (panels (a) and (c)) and snowdrift game (SG) (panels (b) and (d)). The fixation probability of cooperation, $\rho_{\textrm{C}}$, is presented as a function of the benefit-to-cost ratio $(b_1/c)$ on Barabási-Albert (BA) graph barabasi1999emergence. Cooperation is favored if $N\rho_\textrm{C}$ exceeds $1$ (horizontal lines), i.e., $\rho_{\textrm{C}}>1/N$. Filled (open) squares are the results of evolution with game transitions (resp. single game) obtained from Monte Carlo simulations. Vertical lines represent corresponding analytical predictions of the critical benefit-to-cost ratios $(b_1/c)^*$, above which the evolution of cooperation is facilitated. All games are initialized as game $2$. Simulations on fixation probabilities are repeated for $10^7$ independent runs. The parameters under DB are: the benefit difference between two games $\Delta b = 1$ in DG and $\Delta b = 0.6$ for SG. The parameters under PC are: the benefit in game 2 $b_2 = 2$ in DG and $b_2 = 1$ in SG. Other parameters are: BA with linking number $m=3$, the cost for cooperation $c = 1$, and selection strength $\delta = 0.01$.
• Figure 4: Effects of game transitions on ceiling fan graphs and conjoined star graphs. The critical benefit-to-cost ratio $(b_1/c)^*$ for cooperation as a function of the number of branches $n$ on ceiling fan graphs (panels (a) and (b)) and the subgraph with size $n$ on conjoined star graphs (panels (c) and (d)). The dashed (resp. solid) curves correspond to the analytical results for finite $n$ varying from $5$ to $50$ under deterministic game transitions (resp. single game). The dash-dotted horizontal lines display theoretical results for infinite $n$. The three cooperative games are the donation game (DG), public goods game (PGG), and snowdrift game (SG), as indicated in the legend.
• Figure 5: Effects of game transitions on empirical social networks under death-birth updating. Four empirical datasets are taken into account: (a) grooming behaviors observed among a cohort of 24 primates griffin2012community; (b) social associations of a group comprising 30 beetles formica2016consistency; (c) the friendship network of its $34$ members within a karate club zachary1977information; (d) the social connections among a sample of 32 women from the Davis community davis1941deep. The theoretical predictions under death-birth updating are provided for each social network in the context of the donation game $(b/c)^*_{\text{DG}}$ (resp. snowdrift game $(b/c)^*_{\text{SG}}$) and its corresponding value under the deterministic game transition $(b_1/c)^*_{\text{DG}}$ (resp. $(b_1/c)^*_{\text{SG}}$).