What restrictions do Bayesian games impose on the value of information?

In a Bayesian game players play an unknown game. Before the game starts some players may receive a signal regarding the specific game actually played. Typically, information structures that determine different signals, induce different equilibrium payoffs.In zero-sum games the equilibrium payoff measures the value of the particular information structure which induces it. We pose a question as to what restrictions do Bayesian games impose on the value of information. We provide answers in two kinds of information structures: symmetric, where both players are equally informed, and one-sided where only one player is informed.