Interdependent Values without Single-Crossing
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We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation of each agent is a known function of all n private signals. This captures settings such as valuations for artwork, oil drilling rights, broadcast rights, and many more. In the interdependent value setting, all previous work has assumed a so-called single-crossing condition. Single-crossing means that the impact of agent i's private signal, s_i, on her own valuation is greater than the impact of s_i on the valuation of any other agent. It is known that without the single-crossing condition an efficient outcome cannot be obtained. We study welfare maximization for interdependent valuations through the lens of approximation. We show that, in general, without the single-crossing condition, one cannot hope to approximate the optimal social welfare any better than the approximation given by assigning the item to a random bidder. Consequently, we introduce a relaxed version of single-crossing, c-single-crossing, parameterized by c≥ 1, which means that the impact of s_i on the valuation of agent i is at least 1/c times the impact of s_i on the valuation of any other agent (c=1 is single-crossing). Using this parameterized notion, we obtain a host of positive results. We propose a prior-free deterministic mechanism that gives an (n-1)c-approximation guarantee to welfare. We then show that a random version of the proposed mechanism gives a prior-free universally truthful 2c-approximation to the optimal welfare for any concave c-single crossing setting (and a 2√(n)c^3/2-approximation in the absence of concavity). We extend this mechanism to a universally truthful mechanism that gives O(c^2)-approximation to the optimal revenue.
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