Lecture 12 -- Privacy Yields an Anti-Folk Theorem in Repeated Games

Last week, Kobbi Nissim gave us an excellent guest lecture on differential privacy and machine learning. The semester has gone by fast -- this week is our last lecture in the privacy and mechanism design class. (But stop by next week to hear the students present their research projects!)

Today we'll talk about infinitely repeated games. In an infinitely repeated game, n players repeatedly, in an infinite number of stages, play actions and obtain payoffs based on some commonly known stage game. Since the game is infinitely repeated, in order to make sense of players total payoff, we employ a discount factor delta that specifies how much less valuable a dollar is tomorrow compared to a dollar today. (delta is some number in [0, 1) ). In games of perfect monitoring, players perfectly observe what actions each of their opponents have played in past rounds, but in large n player games, it is much more natural to think about games of imperfect monitoring, in which agents see only some noisy signal of what their opponents have played.

For example, one natural signal players might observe in an anonymous game is a noisy histogram estimating what fraction of the population has played each type of action. (This is the kind of signal you might get if you see a random subsample of what people play -- for example, you have an estimate of how many people drove on each road on the way to work today by looking at traffic reports). Alternately, there may be some low dimensional signal (like the market price of some good) that everyone observes that is computed as a randomized function of everyone's actions today (e.g. how much of the good each person produced).

A common theme in repeated games of all sorts are folk theorems. Informally, these theorems state that in repeated games, we should expect a huge multiplicity of equilibria, well beyond the equilibria we would see in the corresponding one-shot stage game. This is because players observe each other's past behavior, and so can threaten each other to behave in prescribed ways or else face punishment. Whether or not a folk theorem is a positive result or a negative result depends on whether you want to design behavior, or predict behavior. If you are a mechanism designer, a folk theorem might be good news -- you can try and encourage equilibrium behavior that has higher welfare than any equilibrium of the stage game. However, if you want to predict behavior, it is bad news -- there are now generically a huge multiplicity of very different equilibria, and some of them have much worse welfare than any equilibrium of the stage game.

In this lecture (following a paper joint with Mallesh Pai and Jon Ullman) we argue that:

1. In large games, many natural signaling structures produce signal distributions that are differentially private in the actions of the players, where the privacy parameters tends to 0 as the size of the game gets large, and
2. In any such game, for any discount factor delta, as the size of the game gets large, the set of equilibria of the repeated game collapse to the set of equilibria of the stage game. In other words, there are no "folk theorem equilibria" -- only the equilibria that already existed in the one shot game. 

This could be interpreted in a couple of ways. On the one hand, this means that in large games, it might be harder to sustain cooperation (which is a negative result). On the other hand, since it shrinks the set of equilibria, it means that adding noise to the signaling structure in a large game generically improves the price of anarchy over equilibria of the repeated game, which is a positive result.