Information Concealing Games

Abstract

A system with an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional state vector and a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">controller</i> and an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">actor</i> is considered. The controller has complete information about the system state, and reveals a certain “minimum” amount of information to the actor. The actor takes certain actions based on the information the controller reveals, and the actions fetch certain utilities for each entity. Both the controller and actor seek to maximize their individual utilities by respectively selecting the information to reveal and the actions to adopt. This decision problem forms the basis of several technical and social systems, and can be formulated as a signaling game. It is shown that the Perfect Bayesian Equilibrium of this game has several counterintuitive properties and can be obtained as a saddle point of a different two person zero sum game. The computation time for saddle points using standard linear programs however turns out to be superexponential in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , which leads to computational intractability even for moderate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> . Algorithms for computing saddle point policies using a computation time that is exponential in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> are presented. Finally, simple linear time computable policies that approximate the saddle-point policies within guaranteeable approximation ratios are obtained.