  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.
