Efficiency with Linear Prices? A Game-Theoretical and Computational Analysis of the Combinatorial Clock Auction

Combinatorial auctions have been suggested as a means to raise efficiency in multi-item negotiations with complementarities among goods because they can be applied in procurement, energy markets, transportation, and the sale of spectrum auctions. The combinatorial clock (CC) auction has become very popular in these markets for its simplicity and for its highly usable price discovery, derived by the use of linear prices. Unfortunately, no equilibrium bidding strategies are known. Given the importance of the CC auction in the field, it is highly desirable to understand whether there are efficient versions of the CC auction providing a strong game theoretical solution concept. So far, equilibrium strategies have only been found for combinatorial auctions with nonlinear and personalized prices for very restricted sets of bidder valuations. We introduce an extension of the CC auction, the CC+ auction, and show that it actually leads to efficient outcomes in an ex post equilibrium for general valuations with only linear ask prices. We also provide a theoretical analysis on the worst case efficiency of the CC auction, which highlights situations in which the CC leads to highly inefficient outcomes. As in other theoretical models of combinatorial auctions, bidders in the field might not be able to follow the equilibrium strategies suggested by the game-theoretical predictions. Therefore, we complement the theoretical findings with results from computational and laboratory experiments using realistic value models. The experimental results illustrate that the CC+ auction can have a significant impact on efficiency compared to the CC auction.