We consider evolutionary games on a population whose underlying topology of interactions is determined by a binomial random graph $G(n,p)$. Our focus is on 2-player symmetric games with 2 strategies played between the incident members of such a population. Players update their strategies synchronously. At each round, each player selects the strategy that is the best response to the current set of strategies its neighbours play. We show that such a system reduces to generalised majority and minority dynamics. We show rapid convergence to unanimity for $p$ in a range that depends on a certain characteristic of the payoff matrix. In the presence of a bias among the pure Nash equilibria of the game, we determine a sharp threshold on $p$ above which the largest connected component reaches unanimity with high probability. For $p$ below this critical value, where this does not happen, we identify those substructures inside the largest component that remain discordant throughout the evolution of the system.
翻译:我们考虑的进化游戏是针对一个其互动基本结构由二流随机图 $G(n,p) 美元决定的人口。 我们的焦点是2个玩家对称游戏,在此类人群的事故成员之间玩了2个策略。 玩家同步更新策略。 每一轮, 每个玩家都选择最适合其邻居玩耍的当前一套策略的战略。 我们显示, 这样的系统会降低为普遍多数和少数民族的动态。 我们显示, 在取决于报酬矩阵的某个特点的范围之内, 我们迅速一致使用美元。 在游戏纯净的Nash equilibria 中存在偏差的情况下, 我们确定一个尖锐的门槛值是$p$, 上方的最大连接部分会以很高的概率达到一致。 对于低于这个关键值的 $p$, 如果这种情况没有发生, 我们就会找出在系统演进过程中仍然不和谐的最大部分内的各个子结构。