Optimal Mixed Spectrum Auction

This work studies the revenue-maximizing auction of a single block of spectrum that can be awarded either for exclusive licensed use by one operator or reserved for unlic ensed use. A number of operators bid for exclusive licensed use and a group of non-colluding agents bid to keep the spectrum unlicensed. The revenue of this auction is compared to that of a Vickrey-Clarke-Groves (VCG) auction and that of another auction recently proposed. I. I NTRODUCTION The commercial success of WiFi demonstrates the value of unlicensed spectrum that can be shared by devices from many vendors. In contrast, cellular spectrum is licensed to individual providers for their exclusive use. Comparing th e benefits of these two models, one faces a typical conflict between social welfare and provider profits that may argue in favor of unlicensed spectrum. In this paper, we do not try to compare the merits of licensed versus unlicensed spectrum. Instead, we limit our study to a n exploration of auctions for spectrum that can be used either for an exclusive licensed provider or for a collective of unlice ns d users. The value of the spectrum for a licensed provider resu lts from the subscriptions by users. The value for a collective of unlicensed users may be more difficult to quantify but is very real. The collective could consist of content or servic e providers and equipment vendors for whom the additional access to unlicensed spectrum corresponds to an increase in usage. Thus, even though this collective does not get any exclusivity for spectrum usage, the increase in market size may justify paying for the spectrum. It may very well be that this increase generates enough user welfare and even tax revenue to warrant the government giving away the spectrum for free for unlicensed use. However, political fairness considera tions may justify an auction to avoid picking the winner in this conflict; moreover, the revenue that the auction generates c an also be used to improve user welfare in some other ways. The general situation is that of a set of spectrum blocks to be auctioned for either licensed or unlicensed use. A number of bidders are interested in exclusive licensed use o f spectrum blocks and a collective of bidders want spectrum blocks for unlicensed use. The problem is to design auctions for this situation and to study their characteristics, such as the revenue they generate and the net utility for the bidders. He re, we explore simple versions of the problem where a single spectrum block is auctioned. II. REVENUE MAXIMIZING AUCTION The mixed spectrum auction described in the abstract is a particular case of the following auction. There is a single item to be offered andK groupsG1, . ,GK of non-colluding agents. The item is given to one of the groups and every agenti of that group derives a valuation Vi. The valuations are independent random variables and Vi has a probability density functionfi(v) that is positive on[ai, bi] and zero elsewhere. LetFi(v) = P (Vi ≤ v). We discuss the revenuemaximizing auction among all auctions that are incentive compatible and individually rational. Incentive compatible means that no bidder has any incentive to lie about his value estimate (honest responses must form a Nash equilibrium in the auction game). Individually rational means that bidder s are not forced to participate in the auction (the expected payof f for every user is nonnegative). Assume that ci(v) := v − 1− Fi(v) fi(v) , ai ≤ v ≤ bi (1) is non-decreasing for every agent. Notice that a sufficient condition is for the distribution to have a non-decreasing hazard ratefi(v)/(1− Fi(v)). This property holds for the uniform distribution, the exponential distribution, the G aussian distribution, and many other distributions. (Note: The opt imal auction can be derived when ci(v) is not non-decreasing, but we limit ourselves to this case for now. ) Theorem 1. The following auction maximizes the revenue. The item is given to the group k with a maximal value of Ck(V) := ∑