Crucial role of strategy updating for coexistence of strategies in interaction networks

Network models are useful tools for studying the dynamics of social interactions in a structured population. After a round of interactions with the players in their local neighborhood, players update their strategy based on the comparison of their own payoff with the payoff of one of their neighbors. Here we show that the assumptions made on strategy updating are of crucial importance for the strategy dynamics. In the first step, we demonstrate that seemingly small deviations from the standard assumptions on updating have major implications for the evolutionary outcome of two cooperation games: cooperation can more easily persist in a Prisoner's Dilemma game, while it can go more easily extinct in a Snowdrift game. To explain these outcomes, we develop a general model for the updating of states in a network that allows us to derive conditions for the steady-state coexistence of states (or strategies). The analysis reveals that coexistence crucially depends on the number of agents consulted for updating. We conclude that updating rules are as important for evolution on a network as network structure and the nature of the interaction.

Figure 1

Equilibrium fraction of cooperators in the Prisoner's Dilemma game (PDG) and the Snowdrift game (SDG) as a function of the costs $c$ of cooperation for four values of $m$, the number of agents consulted for strategy updating. Left panels: PDG; right panels: SDG; upper panels: random-regular network with degree 10; lower panel: Barabási-Albert scale-free network with average degree 10. The benefit of cooperation was kept constant at $b=1$. In our simulations on scale-free networks, if a player's degree was smaller than $m$, she chose all of her neighbors for consultation.

Figure 2

(a) A single updating when two different individuals encounter each other. Strategy $A$ (blue) has a probability $u_{A\to B}$ to switch to $B$ (red), while $B$ switches to $A$ with probability $u_{B\to A}$. (b) Diagrams illustrating an updating event in a network scenario where each player chooses $m$ of her neighbors randomly for updating. Each arrow specifies a probabilistic switching because it is formed by different strategies. Dotted lines indicate neighbors which are not selected at the current time step.

Figure 3

Equilibrium frequency of strategy $A$ in a random-regular network of degree 10 for four values of $m$, the number of agents consulted for strategy updating. For (a) $u_{A\to B}=0.25$ and (b) $u_{A\to B}=0.75$, the analytical predictions based on Eq. (6) (solid lines) and the outcome of simulations (symbols) are shown for a spectrum of values of $u_{B\to A}$. Both panels clearly indicate that a larger value of $m$ favors the equilibrium coexistence of both strategies. In our simulations time is discretized in time steps, and in each step players choose to be an $A$ or a $B$ player with the probability determined when finishing the previous step. We start from a configuration in which each player adopts strategy $A$ with a probability chosen uniformly from the range [0, 1]. In each round, player $i$ updates her strategy and is correspondingly associated with a probability that she is an $A$ player in the next round. Each simulation result corresponds to a result of averaging over ${10}^3$ generations after a transient period of ${10}^4$ rounds in 100 independent realizations with the population size ${10}^4$.

Figure 4

Equilibrium coexistence of strategies $A$ and $B$ as a function of the updating probabilities $u_{A\to B}$ and $u_{B\to A}$ for four values of $m$, the number of agents consulted for strategy updating: (a) $m=1$; (b) $m=2$; (c) $m=4$; (d) $m=10$. Red: fixation of strategy $A$; blue: fixation of strategy $B$; yellow lines: boundaries of coexistence region based on Eq. (7); all other colors: frequency of $A$ $\left(0<\widehat{p}<1\right)$ resulting from Eq. (6).

Figure 5

Equilibrium frequency of strategy A in a Barabási-Albert scale-free network with average degree 10 for four values of $m$, the number of agents consulted for strategy updating. For (a) $u_{A\to B}=0.25$ and (b) $u_{A\to B}=0.75$, the outcome of simulations is shown for a spectrum of values of $u_{B\to A}$. As in case of a random-regular network (Fig. 3), a larger value of $m$ favors the equilibrium coexistence of both strategies.