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.
Iterative combinatorial auctions (ICAs) are IT-based economic mechanisms where bidders submit bundle bids in a sequence and an auctioneer computes allocations and ask prices in each auction round. The literature in this field provides equilibrium analysis for ICAs with nonlinear personalized prices under strong assumptions on bidders' strategies. Linear pricing has performed very well in the lab and in the field. In this paper, we compare three selected linear price ICA formats based on allocative efficiency and revenue distribution using different bidding strategies and bidder valuations. The goal of this research is to benchmark different ICA formats and design and analyze new auction rules for auctions with pseudodual linear prices. The multi-item and discrete nature of linear price iterative combinatorial auctions and the complex price calculation schemes defy much of the traditional game theoretical analysis in this field. Computational methods can be of great help in exploring potential auction designs and analyzing the virtues of various design options. In our simulations, we found that ICA designs with linear prices performed very well for different valuation models even in cases of high synergies among the valuations. There were, however, significant differences in efficiency and in the revenue distributions of the three ICA formats. Heuristic bidding strategies using only a few of the best bundles also led to high levels of efficiency. We have also identified a number of auction rules for ask price calculation and auction termination that have shown to perform very well in the simulations.