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The combinatorial clock auction (CCA) has frequently been used in recent spectrum auctions. It combines a dynamic clock phase and a one-off supplementary round. The winning allocation and the corresponding prices are determined by the Vickrey-Clarke-Groves rules. These rules should encourage truthful bidding, whereas the clock phase is intended to reveal information. We inquire into the role of the clock when bidders have lexicographic preferences for raising rivals' costs. We show that in an efficient equilibrium, the clock cannot fully reveal bidders' types. In the spirit of the ratchet effect, in the supplementary round competitors extract surplus from strong bidders whose type is revealed. We also show that if there is substantial room for information revelation, that is, if the uncertainty about the final allocation is large, all equilibria of the CCA are inefficient. Qualitative features of our equilibria are in line with evidence concerning bidding behavior in some recent CCAs.

In this chapter we analyze the Combinatorial Clock Auction (CCA) with budgetconstrained bidders. By means of illustrative examples, we highlight positive and critical aspects of bidding under a budget in the clock and in the supplementary phase. Since the supplementary phase of a CCA without constraints from the clock phase is just a VCG auction, we also relate the CCA to the VCG. In the VCG auction, bidding under a budget constraint can be strategically complicated. However, on the positive side, the information revealed in the clock can facilitate bidding since bidders can compute upper (and lower) bounds on the final VCG price. This information might allow them to bid above budget without facing the risk of having to pay more than budget, which might lead to an efficient allocation. In one example we show that the clock might actually last longer under a budget constraint. More critical is that the information provided in the clock can actually be used to bid above budget in order to raise rivals' costs. There are asymmetric equilibria that have a Hawk-Dove flavor: the most aggressive bidder gets the highest surplus. Coordination issues can arise and in the case of mis-coordination bidders have to pay more than their budget.

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The Combinatorial Multi-Round Auction (CMRA) is a new auction format which has already been used in several recent European spectrum auctions. We characterize equilibria in the CMRA that feature auction-specific forms of truthful bidding, demand expansion, and demand reduction for settings in which bidders have either decreasing or non-decreasing marginal values. In particular, we establish sufficient conditions for riskless collusion. Overall, our results suggest that the CMRA might be an attractive auction design in the presence of highly complementary goods on sale. We discuss to what extent our theory is consistent with outcomes data in Danish spectrum auctions and how our predictions can be tested using bidding data.