Private Interdependent Valuations

We consider the single-item interdependent value setting, where there is a single item sold by a monopolist, n buyers, and each buyer has a private signal si describing a piece of information about the item. Additionally, each bidder i has a valuation function vi(s1; : : : ; sn) mapping the (private) signals of all buyers into a positive real number representing their value for the item. This setting captures scenarios where the item's information is asymmetric or dispersed among agents, such as in competitions for oil drilling rights, or in auctions for art pieces. Due to the increased complexity of this model compared to the standard private values model, it is generally assumed that each bidder's valuation function vi is public knowledge to the seller or all other buyers. But in many situations, the seller may not know the bidders' valuation functions|how a bidder aggregates signals into a valuation is often their private information. In this paper, we design mechanisms that guarantee approximately-optimal social welfare while satisfying ex-post incentive compatibility and individually rationality for the case where the valuation functions are private to the bidders, and thus may be strategically misreported to the seller. When the valuations are public, it is possible for optimal social welfare to be attained by a deterministic mechanism when the valuations satisfy a single-crossing condition. In contrast, when the valuations are the bidders' private information, we show that no finite bound on the social welfare can be achieved by any deterministic mechanism even under single-crossing. Moreover, no randomized mechanism can guarantee better than n-approximation. We thus consider valuation functions that are submodular over signals (SOS), introduced in the context of combinatorial auctions in a recent breakthrough paper by Eden et al. [EC'19]. Our main result is an O(log2 n)-approximation randomized mechanism for buyers with private signals and valuations under the SOS condition. We also give a tight (k)-approximation mechanism for the case each agent's valuation depends on at most k other signals even for unknown k.