Privacy as a Desideratum – Making the Exponential Mechanism Truthful with Payments

In the last lecture, we saw (one instance of) how differential privacy can be used as a tool in mechanism design. We used it to design a prior-free near-revenue optimal digital goods auction. As this course continues, we will see several other instances in which privacy is used purely as a tool to solve other mechanism design problems, but today I want to shift gears and think of differential privacy itself as a goal of the mechanism designer. Why might a mechanism designer care about providing privacy for the participants in a mechanism? The simplest answer is that participants might not participate if their privacy is not protected. But why? – Shouldn’t we just model all concerns of the agents in their utility function? The issue is that this might be difficult to do. Ideally we can come up with a utility function that accurately models an agent’s gains in some restricted setting: for example, in an auction, we imagine the agent has some utility vi for receiving an item, which might accurately capture the agent’s true utility while interacting with the auctioneer. However, an agent might unbeknownst to the auctioneer be participating in some other interaction tomorrow. For example, the agent might be involved in a negotiation for some other good, from some other buyer. If the agent’s value for the good he is purchasing today is correlated with his value for the good he might be purchasing tomorrow, then he has a very real concern about how information leaked about his interaction today might affect his interaction tomorrow. We might worry that this unmodeled utility might substantially affect the agents incentives in the otherwise dominant strategy truthful mechanism that we are running today We can view privacy as a way of making mechanisms robust to unmodeled future utility that might be affected by agents’ participation in a mechanism today. Suppose that we are designing a mechanism M : T n → O, and we know that agents have utility functions ui (o) ≡ u(ti, o) over outcome space O. However, suppose that (unbeknownst to us), these agents will also experience some outcome oi ∈ O′ i tomorrow, that will be chosen in a way that might depend on the outcome that our mechanism chooses today. We make no assumptions about how this event is chosen – we imagine only that there is some (randomized) function fi(o) : O → O′ i that determines the distribution over outcomes o′ ∈ O′ i. In total, agents obtain utility ui(o, o ′) = ui (o) + u 2 i (o ′). We observe that if the mechanism we design M is dominant strategy truthful with respect to the known utility functions ui , then it remains -dominant strategy truthful even with respect to the unknown utility function ui. This is important, since we must always expect that our models are at most good approximations of people’s true utilities.