## A Response Strategy for the Battle of the Sexes Game with Intrinsic Correlations

##### Summary

In this Thesis we apply a recently proposed model by Correia et al. to the Battle of the Sexes game. This model considers games with intrinsic correlations between the final possible outcomes, and lets players respond to these correlations with a response strategy. We studied all possible Nash equilibria in these response strategies, which we found to be 9 in total, including equilibria corresponding to the previously-known pure and mixed-strategy Nash equilibria. We finally computed which response-strategy profile leads to the optimal payoff for the players. We found that for a large part of the space of correlation probabilities the equilibrium corresponding to the pure-strategy Nash equilibrium is still optimal, since the average payoff of the other Nash equilibria is often less than one would obtain from consistently choosing one's least-preferred pure-strategy option.
We furthermore study this model when applied to a simple 1-dimensional network, i.e. a ring, by incorporating the exact solution of the 1-dimensional Ising model. We find that the external field that players experience is always the same logarithmic function of the
correlation probabilities, but the way players react to this field differs for the various Nash equilibria. The interaction strength between the players, which is mostly of the ferromagnetic type, is a different function for dominant diagonal correlations and dominant off-diagonal
correlations. However, the change between these two seems to be continuous, but we lack an analytical solution for part of the probability space of the correlations. We finally find a magnetization which has the same dependence on the correlation probabilities for all the
Nash equalibria.