Private Equilibrium Computation Yields Truthful Mediators ( Part 1 ) 1 Introduction

When we were being read fairy tails growing up, we first learned about games of “Complete information”. So the story goes, in such games, all n players know one another’s types, and use this information to somehow select and play one of the (possibly many) Nash equilibria of the game. To recap, a game G (here we will use costs rather than utilities, but it doesn’t matter) is defined by a set of n players, a set of actions A, a set of types T , and a cost function c : T × A → [0, 1]. Given a choice of actions a = (a1, . . . , an) for each player, we will sometimes write ci(a) = c(ti, a) to denote agent i’s cost. A game is one of complete information if the types t1, . . . , tn are common knowledge. We can define a pure strategy Nash equilibrium of a game of complete information: