Belief-invariant and quantum 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 the familiar Nash equilibria. The notion of belief-invariant equilibrium appeared in game theory in the 90s. However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 80s 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, together with quantum correlated equilibria, using tools from quantum information but the language of (algorithmic) game theory. We present a general framework of belief-invariant communication equilibria, which contains correlated equilibria and quantum correlated equilibria as special cases. Our framework also contains the theory of Bell inequalities and their violations due to non-locality, which is a question of intense interest in the foundations of quantum mechanics, and it was indeed the original motivation for the aforementioned studies. Moreover, in our framework we can also model quantum games where players have conflicting interests, a recent developing topic in physics. We then use our framework to show new results related to the social welfare of equilibria. 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.