Evolutionary Matrix-Game Dynamics Under Imitation in Heterogeneous Populations

Decision-making individuals often imitate their highest-earning fellows rather than optimize their own utilities, due to bounded rationality and incomplete information. Perpetual fluctuations between decisions have been reported as the dominant asymptotic outcome of imitative behaviors, yet little attempt has been made to characterize them, particularly in heterogeneous populations. We study a finite well-mixed heterogeneous population of individuals choosing between the two strategies, cooperation, and defection, and earning based on their payoff matrices that can be unique to each individual. At each time step, an arbitrary individual becomes active to update her decision by imitating the highest earner in the population. We show that almost surely the dynamics reach either an equilibrium state or a minimal positively invariant set, a \emph{fluctuation set}, in the long run. In addition to finding all equilibria, for the first time, we characterize the fluctuation sets, provide necessary and sufficient conditions for their existence, and approximate their basins of attraction. We also find that exclusive populations of individuals playing coordination or prisoner's dilemma games always equilibrate, implying that cycles and non-convergence in imitative populations are due to individuals playing anticoordination games. Moreover, we show that except for the two extreme equilibria where all individuals play the same strategy, almost all other equilibria are unstable as long as the population is heterogeneous. Our results theoretically explain earlier reported simulation results and shed new light on the boundedly rational nature of imitation behaviors

Controlling Networks of Imitative Agents

We consider how asynchronous networks of agents who imitate their highest-earning neighbors can be efficiently driven towards a desired strategy by offering payoff incentives, either uniformly or targeted to individuals. In particular, if for each available strategy, agents playing that strategy receive maximum payoff when their neighbors play that same strategy, we show that providing incentives to agents in a network that is at equilibrium will result in convergence to a unique equilibrium. When a uniform incentive can be offered to all agents, one can compute the optimal incentive using a binary search algorithm. When incentives can be targeted to individuals, we propose an algorithm to select which agents should be chosen based on iteratively maximizing a ratio of the number of agents who adopt the desired strategy to the payoff incentive required to get those agents to do so. Simulations demonstrate that this algorithm computes near-optimal targeted payoff incentives for a range of networks and payoff distributions in coordination games

Controlling Networks of Imitative Agents

Imitation is widely observed in populations of decision-making agents. Using our recent convergence results for asynchronous imitation dynamics on networks, we consider how such networks can be efficiently driven to a desired equilibrium state by offering payoff incentives for using a certain strategy, either uniformly or targeted to individuals. In particular, if for each available strategy, agents playing that strategy receive maximum payoff when their neighbors play that same strategy, we show that providing incentives to agents in a network that is at equilibrium will result in convergence to a &lt;italic&gt;unique&lt;/italic&gt; new equilibrium. For the case when a uniform incentive can be offered to all agents, this result allows the computation of the optimal incentive using a binary search algorithm. When incentives can be targeted to individual agents, we propose an algorithm to select which agents should be chosen based on iteratively maximizing a ratio of the number of agents who adopt the desired strategy to the payoff incentive required to get those agents to do so. Simulations demonstrate that the proposed algorithm computes near-optimal targeted payoff incentives for a range of networks and payoff distributions in coordination games.</p