How Much Coordination is Possible via Correlation?

Players in a game are said to coordinate their behavior if they jointly choose strategies. (The idea goes back to von Neumann [19, 1928] and von Neumann-Morgenstern [21, 1944].) Correlation in behavior arises if players choose their strategies independently, but the choices are conditioned on observations of correlated signals (Aumann [1, 1974]). This note establishes an impossibility result: Under certain conditions, some coordinated behavior in a game cannot be derived from correlated behavior. The result rests on an impossibility theorem from the hidden-variable program in quantum mechanics (Hardy [9, 1992]-[10, 1993], following Bell [3, 1964]). We explain the connection.