Extended estimator approach for 2 x 2 games and its mapping to the Ising Hamiltonian.

ABSTRACT

We consider a system of adaptive self-interested agents interacting by playing an iterated pairwise prisoner's dilemma (PD) game. Each player has two options either cooperate (C) or defect (D). Agents have no (long term) memory to reciprocate nor identifying tags to distinguish C from D. We show how their 16 possible elementary Markovian (one-step memory) strategies can be cast in a simple general formalism in terms of an estimator of expected utilities Delta*. This formalism is helpful to map a subset of these strategies into an Ising Hamiltonian in a straightforward way. This connection in turn serves to shed light on the evolution of the iterated games played by agents, which can represent a broad variety of individuals from firms of a market to species coexisting in an ecosystem. Additionally, this magnetic description may be useful to introduce noise in a natural and simple way. The equilibrium states reached by the system depend strongly on whether the dynamics are synchronous or asynchronous and also on the system connectivity.