title: 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.
authors.affiliation: IPMC, Institute of Physics of Complex Matter, IPMC/FSB/EPFL, Lausanne, Switzerland.
authors.forename: D
authors.initials: D
authors.lastname: Ariosa
authors.forename: H
authors.initials: H
authors.lastname: Fort