Welfare and Rationality Guarantees for the Simultaneous Multiple-Round Ascending Auction

Abstract

The simultaneous multiple-round auction (SMRA) and the combinatorial clock auction (CCA) are the two primary mechanisms used to sell bandwidth. Under truthful bidding, the SMRA is known to output a Walrasian equilibrium that maximizes social welfare provided the bidder valuation functions satisfy the gross substitutes property [20]. Recently, it was shown that the combinatorial clock auction (CCA) provides good welfare guarantees for general classes of valuation functions [7]. This motivates the question of whether similar welfare guarantees hold for the SMRA in the case of general valuation functions.We show the answer is no. But we prove that good welfare guarantees still arise if the degree of complementarities in the bidder valuations are bounded. In particular, if bidder valuations functions are α-near-submodular then, under truthful bidding, the SMRA has a welfare ratio (the worst case ratio between the social welfare of the optimal allocation and the auction allocation) of at most (1 + α). The special case of submodular valuations, namely α = 1, was studied in [12] and produces individually rational solutions. However, for α > 1, this is a bicriteria guarantee, to obtain good welfare under truthful bidding requires relaxing individual rationality. In particular, it necessitates a factor α loss in the degree of individual rationality provided by the auction. We prove this bicriteria guarantee is asymptotically (almost) tight. Truthful bidding, though, is not reasonable assumption in the SMRA [10]. But, bicriteria guarantees continue to hold for natural bidding strategies that are locally optimal. Specifically, the welfare ratio is then at most (1 + α 2 ) and the individual rationality guarantee is again at most α, for α-near submodular valuation functions. These bicriteria guarantees are also (almost) tight. Finally, we examine what strategies are required to ensure individual rationality in the SMRA with general valuation functions. First, we provide a weak characterization, namely secure bidding, for individual rationality. We then show that if the bidders use a profit-maximizing secure bidding strategy the welfare ratio is at most 1 + α. Consequently, by bidding securely, it is possible to obtain the same welfare guarantees as truthful bidding without the loss of individual rationality. Unfortunately, we explain why secure bidding may be incompatible with the auxiliary bidding activity rules that are typically added to the SMRA to reduce gaming.