The value of information structure in zero-sum games with lack of information on one side

Two players are engaged in a zero-sum game with lack of information on one side, in which player 1 (the informed player) receives some stochastic signal about the state of nature. I consider the value of the game as a function of player 1's information structure, and study the properties of this function. It turns out that these properties reflect the fact that in zero sum situation the value of information for each player is positive.