Efficiency with linear prices: a theoretical and experimental analysis of the combinatorial clock auction.

Combinatorial auctions have been suggested as a mean to raise efficiency in multi-item negotiations with complementarities as they can be found in procurement, in energy markets, in transportation, and for the sale of spectrum auctions. Anonymous linear ask prices are desirable and sometimes even essential for many of these applications. The Combinatorial Clock (CC) auction [4] has become very popular in these markets for its simplicity and as it "produces highly usable price discovery, because of the item prices (linear pricing)" [1]. Unfortunately, the CC auction fails to lead always to efficient outcomes, and there is no theory on equilibrium bidding strategies in such auctions. Given the importance of the CC auction in the field, it is desirable to better understand this auction format. In our paper, we show that the worst-case efficiency in the CC auction can actually be as low as 0%, if bidders follow the straightforward strategy. We identify demand-masking valuations, i.e., the characteristics of valuation functions, which can cause such low efficiencies. As an alternative we evaluate a powerset strategy, which also leads to 0% efficiency in the worst case. Knowing which situations cause inefficiencies the key question is, if there is any way to make the CC auction fully efficient and incentive compatible. We show what is necessary to achieve these properties and that this is actually possible, which is not obvious given the negative results on linear competitive equilibrium prices in the literature. We introduce an extension of the CC auction, the CC+ auction, with a simple, modified price update rule and a VCG payment rule. We show that powerset bidding in the CC+ auction leads to efficient outcomes and an ex-post Nash equilibrium for general valuations, while maintaining linear ask prices. Efficient iterative combinatorial auctions have already been proposed in the literature [2, 3], but all these auctions use nonlinear and personalized prices. Additionally they need a strong assumption on the coalitional value function to achieve an ex-post Nash equilibrium in straightforward bidding. The assumptions on the CC+ auction are also strong to achieve the desired properties. For example the powerset strategy is restrictive and might only be possible in very small scenarios. In computational experiments we relax the strong assumptions and show that both the CC and CC+ auctions achieve high levels of efficiency, even if bidders are restricted to submit only a small number of bids in each round. The results indicate that the CC+ auction is robust against various deviations from powerset bidding strategies which also explains high levels of efficiency of the CC auction that have been observed in the lab. Comparing the number of bids and auction rounds with efficient non-linear price auction formats, we observe considerably lower numbers in the CC and CC+ auctions on average.