Player-Optimization of Payoff in Correlated Coordination Games

Abstract

In this Thesis we describe two types of two-player games with binary choices as a realiza- tion of an Ising system with two spins. We do this for a particular symmetric game, namely "snowdrift/chicken", and for a particular asymmetric game, the "battle of the sexes". By as- sociating the energy of a spin configuration with a set of probabilities, through the Principle of Maximum Entropy, and we express the games as correlated. In the normal Ising model, these probabilities would describe the probabilities of achieving a final state, but here they correspond to the probabilities with which the players receive a certain set of information given by the correlating device, that they can act upon. While this correlation starts off as externally imposed, we develop a way to include the players choice into new correlated probabilities. We show that the payoffs that the players obtain by using their choice is al- ways the same or better when compared to what it would be if they would always follow the initial correlating device while it is in equilibrium, and that it allows for a better payoff than the best uncorrelated solution, the mixed strategy equilibrium, when the initial correlating device is out of equilibrium. Because these are the best payoffs that the players get after they choose, we renormalize the initial correlating device to one that the players always follow. We associate these new probabilities with the energies of a renormalized Ising model, that effectively represents the final statistics of the game, allowing us to treat the problem with standard tools from statistical physics.