  The Combinatorial Multi-Round Ascending Auction (CMRA) is a new auction
format that has already been used in several recent European spectrum auctions.
We characterize ex-post equilibria that feature auction-specific forms of
truthful bidding, demand expansion, and demand reduction for settings where
bidders have either decreasing or non-decreasing marginal values. In
particular, we show that the truthtelling equilibrium is fragile to small
asymmetries in the bidders' caps. On the other hand, if bidders are
sufficiently symmetric, the CMRA is vulnerable to risk-free collusion. We
propose an alternative activity rule that prevents such collusive strategies
but keeps the other equilibria intact. We discuss to what extent our theory is
consistent with outcomes in Danish spectrum auctions and how our predictions
can be tested using bidding data.
