Belief-invariant equilibria in games of incomplete information?

Drawing on ideas from game theory and quantum physics, we investigate nonlocal correlations from the point of view of equilibria in games of incomplete information. These equilibria can be classified in decreasing power as general communication equilibria, belief-invariant equilibria and correlated equilibria, all of which contain Nash equilibria. The notion of belief-invariant equilibrium has appeared in game theory before (in the 1990s). However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 1980s in the foundations of quantum mechanics. In the present work, we explain and unify these two origins of the idea and study the above classes of equilibria.We present a general framework of belief-invariant communication equilibria, which contains correlated equilibria as special cases. We then use our framework to show new results related to the social welfare of games. Namely, we exhibit a game where belief-invariance is socially better than any correlated equilibrium, and a game where all non-belief-invariant communication equilibria have a suboptimal social welfare. We also show that optimal social welfare can in certain cases be achieved by quantum mechanical correlations, which do not need an informed mediator to be implemented, and go beyond the classical "sunspot" or shared randomness approach.