In nature and society problems arise when different interests are difficult to reconcile, which are modeled in game theory. While most applications assume uncorrelated games, a more detailed modeling is necessary to consider the correlations that influence the decisions of the players. The current theory for correlated games, however, enforces the players to obey the instructions from a third party or correlation device to reach equilibrium, but this cannot be achieved for all initial correlations. We extend here the existing framework of correlated games and find that there are other interesting and previously unknown Nash equilibria that make use of correlations to obtain the best payoff. This is achieved by allowing the players the freedom to follow or not to follow the suggestions of the correlation device. By assigning independent probabilities to follow every possible suggestion, the players engage in a response game that turns out to have a rich structure of Nash equilibria that goes beyond the correlated equilibrium and mixed-strategy solutions. We determine the Nash equilibria for all possible correlated Snowdrift games, which we find to be describable by Ising Models in thermal equilibrium. We believe that our approach paves the way to a study of correlations in games that uncovers the existence of interesting underlying interaction mechanisms, without compromising the independence of the players.
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