Inducing Approximately Optimal Flow Using Truthful Mediators

We revisit a classic coordination problem from the perspective of mechanism design: how can we coordinate a social welfare maximizing flow in a network congestion game with selfish players? The classical approach, which computes tolls as a function of known demands, fails when the demands are unknown to the mechanism designer, and naively eliciting them does not necessarily yield a truthful mechanism. Instead, we introduce a weak mediator that can provide suggested routes to players and set tolls as a function of reported demands. However, players can choose to ignore or misreport their type to this mediator. Using techniques from differential privacy, we show how to design a weak mediator such that it is an asymptotic ex-post Nash equilibrium for all players to truthfully report their types to the mediator and faithfully follow its suggestion, and that when they do, they end up playing a nearly optimal flow. Notably, our solution works in settings of incomplete information even in the absence of a prior distribution on player types. Along the way, we develop new techniques for privately solving convex programs which may be of independent interest.Comment: Version with latencies not normalize

Private Matchings and Allocations

We consider a private variant of the classical allocation problem: given k goods and n agents with individual, private valuation functions over bundles of goods, how can we partition the goods amongst the agents to maximize social welfare? An important special case is when each agent desires at most one good, and specifies her (private) value for each good: in this case, the problem is exactly the maximum-weight matching problem in a bipartite graph. Private matching and allocation problems have not been considered in the differential privacy literature, and for good reason: they are plainly impossible to solve under differential privacy. Informally, the allocation must match agents to their preferred goods in order to maximize social welfare, but this preference is exactly what agents wish to hide. Therefore, we consider the problem under the relaxed constraint of joint differential privacy: for any agent i, no coalition of agents excluding i should be able to learn about the valuation function of agent i. In this setting, the full allocation is no longer published---instead, each agent is told what good to get. We first show that with a small number of identical copies of each good, it is possible to efficiently and accurately solve the maximum weight matching problem while guaranteeing joint differential privacy. We then consider the more general allocation problem, when bidder valuations satisfy the gross substitutes condition. Finally, we prove that the allocation problem cannot be solved to non-trivial accuracy under joint differential privacy without requiring multiple copies of each type of good.Comment: Journal version published in SIAM Journal on Computation; an extended abstract appeared in STOC 201

Approximately Stable, School Optimal, and Student-Truthful Many-to-One Matchings (via Differential Privacy)

We present a mechanism for computing asymptotically stable school optimal matchings, while guaranteeing that it is an asymptotic dominant strategy for every student to report their true preferences to the mechanism. Our main tool in this endeavor is differential privacy: we give an algorithm that coordinates a stable matching using differentially private signals, which lead to our truthfulness guarantee. This is the first setting in which it is known how to achieve nontrivial truthfulness guarantees for students when computing school optimal matchings, assuming worst- case preferences (for schools and students) in large markets