Information geometry of influence diagrams and noncooperative games

What is the "value of information" in non-cooperative games with imperfect information? To answer this question, we propose to quantify information using concepts from Shannon's information theory. We then relate quantitative changes to the information structure of a game to changes in the expected utility of the players. Our approach is based on the Multi-Agent Influence Diagram representation of games. We develop a generalization of the concept of marginal utility in decision scenarios to apply to infinitesimal changes of the channel parameters in noncooperative games. Using that framework we derive general conditions for negative value of information, and show that generically, these conditions hold in all games unless one imposes a priori constraints on the allowed changes to information channels. In other words, in any game in which a player values some aspect of the game's specification beyond the information provided in that game, there will be an infinitesimal change to the parameter vector specifying the game that increases the information but hurts the player. Furthermore, we derive analogous results for N > 1 players, i.e., state general conditions for negative value of information simultaneously for all players. We demonstrate these results numerically on a decision problem as well as a leader-follower game and discuss their general implications.