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.