On the Economic Efficiency of the Combinatorial Clock Auction

Abstract

Since the 1990s spectrum auctions have been implemented world-wide. This has provided for a practical examination of an assortment of auction mechanisms and, amongst these, two simultaneous ascending price auctions have proved to be extremely successful. These are the simultaneous multiround ascending auction (SMRA) and the combinatorial clock auction (CCA). It has long been known that, for certain classes of valuation functions, the SMRA provides good theoretical guarantees on social welfare. However, no such guarantees were known for the CCA.In this paper, we show that CCA does provide strong guarantees on social welfare provided the price increment and stopping rule are well-chosen. This is very surprising in that the choice of price increment has been used primarily to adjust auction duration and the stopping rule has attracted little attention. The main result is a polylogarithmic approximation guarantee for social welfare when the maximum number of items demanded C by a bidder is fixed. Specifically, we show that either the revenue of the CCA is at least an Ω 1 C 2 log n log 2 m -fraction of the optimal welfare or the welfare of the CCA is at least an Ω 1 log n -fraction of the optimal welfare, where n is the number of bidders and m is the number of items. As a corollary, the welfare ratio -the worst case ratio between the social welfare of the optimum allocation and the social welfare of the CCA allocation -is at most O(C 2 · log n · log 2 m). We emphasize that this latter result requires no assumption on bidders valuation functions. Finally, we prove that such a dependence on C is necessary. In particular, we show that the welfare ratio of the CCA is at least Ω C · log m log log m .

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

“…So, even if price increments depend on the excess demand, one cannot achieve a better welfare ratio . This contrasts sharply with the behavior of the CCA whose welfare ratio becomes polylogarithmic if the price increment is allowed to depend upon the excess demand and the size of the demand sets are bounded [7].…”

“…Consequently, a bidder cannot be allocated a subset of her bid; in particular, a bidder cannot be allocated an undesirable subset. Unfortunately, the basic CCA mechanism cannot provide for non-trivial approximate welfare guarantees, even for auctions with additive valuation functions and a small number of bidders and items [7]. It is perhaps surprising, then, that a minor adjustment to the CCA mechanism leads to good welfare guarantees for any class of valuation function.…”

“…It is perhaps surprising, then, that a minor adjustment to the CCA mechanism leads to good welfare guarantees for any class of valuation function. Specifically, if bid increments are made proportional to excess demand the welfare of the CCA is within an O(k 2 • log n log 2 m) factor of the optimal welfare [7]. Here n is the number of items, m is the number of bidders and k is the maximum cardinality of a set desired by the bidders.…”

“…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.…”

The Price of Anarchy in Auctions

“…This auction contains both sequential and simultaneous bidding elements, which complicates the analysis of its price of anarchy. Bousquet, Cai, Hunkenschrder, and Vetta (2016) recently provided a theoretical analysis of the efficiency of the auction when all players act truthfully. Extending this approximate welfare guaranteee to equilibria appears highly non-trivial.…”