Lecture 5 -- Implementing Nash equilibria ex-post in incomplete information games (Part 1)

Welcome back (after a brief hiatus) to the privacy and mechanism design course. Last week we had a guest lecture by Rachel Cummings about her work on the Empirical Implications of Privacy Aware Choice.  We will have notes and a blog post about that soon, but for now, welcome to Lecture 5!

In this lecture, we consider the following problem. Suppose we have an n player game. We might want to implement a Nash equilibrium of the complete information game (which often have nice properties), but naturally find ourselves in the common situation when the n players do not know each others types. Is there a modification of the game that can implement the complete-information game equilibrium as a simple ex-post Nash equilibrium, which does not require that players know anything about each others types, and which requires no coordination?

The modification of the game we consider is the introduction of a very weak "mediator", which essentially only has the power to listen and to give advice, but not to verify or to enforce. Players will have the option of reporting their type to the mediator, in which case it will suggest an action for them to play. However, they can:

1. Completely ignore the mediator and choose an action to play on their own
2. Lie to the mediator about their types, and/or
3. Use the mediator's suggestion however they like, not necessarily following it (i.e. the mediator can not enforce its suggestion)

What we would like is that players should all follow "good behavior", which involves

1. Truthfully reporting their type to the mediator, and then
2. Faithfully following the mediator's suggestion.

We prove that if the mediator offers its suggestions by computing a Nash equilibrium of the complete information game defined by players reported types, and satisfies (a variant of) differential privacy, then the mediator makes "good behavior" an asymptotic ex-post Nash equilibrium of the incomplete information game, and that in this ex-post equilibrium, players end up playing a Nash equilibrium of the original game of complete information (defined by the realized player types).

This leaves us with the minor task of actually figuring out how to compute a Nash equilibrium privately. Here, we realize that thus far in the class, we have been skating by knowing nothing about private algorithm design except for the exponential mechanism. So we take a detour to learn some more basics. We get as far as learning how to count, which turns out to be enough. This will set us up for next lecture, when we can actually implement this agenda and design a mediator that satisfies the hypotheses we need.

(The idea of designing a mediator by privately computing an equilibrium comes from joint work with Michael Kearns, Mallesh Pai, and Jon Ullman, which shows how to privately compute correlated equilibria in arbitrary large games. We will cover this later. The implication in this lecture about mediators that privately compute Nash equilibria comes from joint work with Ryan Rogers, which shows how to privately compute Nash equilibria in large congestion games. )