Abstract. We consider a private variant of the classical allocation problem: given k goods and n agents with private valuation functions over bundles of goods, how can we allocate goods to agents to maximize social welfare? An important special case is when agents desire at most one good, and specifies their (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 for a 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 receive. We first show that if there are several identical copies of each good, it is possible to efficiently and accurately solve the matching problem while guaranteeing joint differential privacy. We then consider the more general allocation problem where 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.
REF subsequently studied a two-sided allocation problem, in which buyers with private valuation functions over bundles of goods must be allocated goods to maximize social welfare (note that here buyers are allocated goods, but do not provide them, unlike the problem we study in this work).
Private Matchings and Allocations
SIAM Journal on Computing 45(6) 1953--1984 (2016)
Mathematics Computer Science