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. 

Lecture 10 -- Running Ascending Price Auctions that Make Sincere Bidding an Ex-Post Dominant Strategy

In the 10th lecture in our privacy and mechanism design class, we consider the problem of running an ascending price auction. An ascending price auction is just a generalization of what you normally see as an "auction" on TV -- rather than submitting your valuation in some kind of one-shot protocol, the prices of the goods gradually rise, and you take turns with other bidders making bids on the goods as a function of the current prices.

Why would you want to run such an auction when the VCG mechanism already can provide welfare optimal outcomes for every social choice function, while making truthful reporting a dominant strategy? People quote a couple of reasons:

1. It might be hard to actually report your full valuation: in principle, you need to figure out exactly your value for every bundle you might receive, and its difficult to pin down a number. In an ascending price auction, all you need to do is be able to point to your favorite good (or bundle of goods) that you would buy if the current prices were the final prices, which is often an easier task. 
2. An ascending price auction can end without you having to reveal your full type. For example, in a single item second price auction, the highest bidder never has to reveal (even to the auctioneer) his value for the good -- only that it is higher than that of the second highest bidder. Hence, people might prefer such auctions for "privacy" reasons. 

In an ascending price auction, "truthful" reporting doesn't make sense, since nobody ever asks you to report your type. But we can ask for "sincere bidding", in which bidders truthfully bid on the item at each round that is their favorite, given the current prices. But there is a problem: we typically can't implement sincere bidding as a dominant strategy, because of the problem of threats. Consider the following simple example:

Suppose we have two unit demand bidders 1 and 2, and two goods for sale a and b. We have v_{1,a} = 1, v_{1,b} = epsilon and v_{2,a} = 1/2, v_{2, b} = 1/2 - \epsilon. Suppose moreover that bidder 2 takes the following strategy: "Bid on good a. If bidder 1 bids on good a, then outbid him on whatever he bids on until the price is > 1.'' Against this strategy, bidder 1 cannot obtain non-negative utility if he bids on his favorite good (a), and so his best response is to place an insincere bid on good 2. Moreover, bidder 2 has a clear motivation to take this threatening position -- he obtains substantially higher payoff than if players followed sincere bidding, since he gets his most preferred good without any competition. As a result of instances like these, typically ascending price auctions can implement sincere bidding at best as an (ex-post) Nash equilibirum. 

In this lecture, we talk about how to implement an ascending auction such that the prices are differentially private in the bidding strategies of the players (and the allocation in the end is jointly differentially private). This fixes two of the problems above:

1. The privacy guaranteed by the ascending price auction is no longer hand-wavy and qualitative, but rather precise and quantitative. 
2. We get sincere bidding as an asymptotic ex-post dominant strategy for all players.

To get this result, we need only a mild large-market assumption: that the "supply" of each good is modestly large compared to the number of different types of goods -- but crucially we need to assume nothing about how bidder preferences are generated. 

The intuition, which we will appeal to again later, is that by running the auction privately, we have eliminated the possibility that players can distort incentives by threatening each other.