Effects of mobility in a population of prisoner’s dilemma players

We address the problem of how the survival of cooperation in a social system depends on the motion of the individuals. Specifically, we study a model in which prisoner’s dilemma players are allowed to move in a two-dimensional plane. Our results show that cooperation can survive in such a system provided that both the temptation to defect and the velocity at which agents move are not too high. Moreover, we show that when these conditions are fulfilled, the only asymptotic state of the system is that in which all players are cooperators. Our results might have implications for the design of cooperative strategies in motion coordination and other applications including wireless networks.

Figure 1

Average level of cooperation $⟨c⟩$, as a function of time (Monte Carlo steps) for $v=0.01$ and two different values of $b$: $b=1.1$ and $b=1.3$, as indicated. Other model parameters have been fixed to $\rho =1.30$ and $N={10}^3$ agents.

Figure 2

Fraction of realizations in which the system ends up in an all-C configuration $F_c$, as a function of the density of players $\rho$ for a fixed value of $b=1.1$ and $v=0.01$. The system is made up of $N={10}^3$ agents. The results are averages taken over 100 different realizations.

Figure 3

(Color online) The color code shows the fraction of realizations in which the whole system is made up of cooperators $F_c$, as a function of the velocity at which the agents move $\left(v\right)$ and the temptation to defect $\left(b\right)$. The $Y$ axis is in logarithmic scale for clarity. The rest of parameters are $N={10}^3$ agents and $\rho =1.30$. Each point is an average over 100 different realizations

Figure 4

Fraction of realizations ending up in an all-C configuration as a function of the velocity $v$ of the agents for $b=1.1$. The inset shows the smallest value of the temptation to defect $b_c$, for which the probability of achieving a fully cooperator asymptotic state is zero, as a function of $v$. In both cases, $N={10}^3$ agents, $\rho =1.30$, and results correspond to averages over 100 realizations.