We analyse the interplay between license auctions and market structure in a model with several incumbents and several potential entrants. The focus is on the competitiveness induced by the number of auctioned licences. Moreover, we study how the auction format affects the incentives for explicit or tacit collusion among incumbents. A crucial role is played by the relation between the number of incumbents and the number of licenses. We show that auctioning more licences need not result in greater competitiveness when the number of incumbents is greater than the number of new licences. When the number of licenses exceeds the number of incumbents, we display plausible conditions under which all incumbents get a licence. Finally, we suggest a positive role for some auction formats in which the number of licences is endogenously determined at the auction. We illustrate some results with examples drawn from the German and UK licence auctions for 3G mobile telephony.