Information Geometry of Influence Diagrams and Noncooperative Games
Nils Bertschinger, David Wolpert, Eckehard Olbrich, Juergen Jost.
In some multi-player games, having more information will hurt a player. For example, in games with first-mover advantage, the second mover is hurt by having more information. Similarly, in Braess' paradox, *all* players are hurt when they are all told about a road that is newly opened for commuting.
What determines whether a game has such negative "value of information'' for some of its players? More generally, what relates the parameters specifying a game to the value of information for the players in that game? More generally still, what relates those parameters to the trade-off between value of information, value of tax rates, etc.?
To answer these questions, we quantify information using concepts from Shannon's information theory and combine them with concepts from economics. Specifically, we generalize the economics concept of marginal utility in decision scenarios to apply to infinitesimal changes of the parameters specifying a noncooperative game. This allows us to derive general conditions for negative value of information. We show that these conditions hold generically, in all games. Specifically, in any game in which a player values some aspect of the game's specification beyond just 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. We then derive analogous results that hold simultaneously for all players, i.e., derive general conditions for information to have negative value 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.