  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 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 sometimes be
achieved by quantum mechanical correlations, which do not need an informed
mediator to be implemented, and go beyond the classical shared randomness
approach.
