IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. , NO. , 1 Dynamic Spectrum Sharing Auction with Time-Evolving Channel Qualities Mehrdad Khaledi, Student Member, IEEE, Alhussein A. Abouzeid, Senior Member, IEEE, Abstract—Spectrum auction is considered a suitable approach to efficiently allocate spectrum among unlicensed users. In a typical spectrum auction, Secondary Users (SUs) bid to buy spectrum bands from a Primary Owner (PO) who acts as the auctioneer. Existing spectrum auctions assume that SUs have static and known values for the channels. However, in many real world settings, the SUs do not know the exact value of channel access at first, but they learn it and adapt it over time. In this paper, we study spectrum auctions in a dynamic setting where SUs can change their valuations based on their experiences with the channel quality. We propose ADAPTIVE, a dynAmic inDex Auction for sPectrum sharing with TIme-evolving ValuEs that maximizes the social welfare of the SUs. ADAPTIVE is based on multi-armed bandit models where for each user an allocation index is independently calculated in polynomial time. Then we generalize ADAPTIVE to Multi-ADAPTIVE that auctions multiple channels at each time. We provide a sufficient condition under which Multi-ADAPTIVE achieves the maximum social welfare. Both ADAPTIVE and Multi-ADAPTIVE have some desired economic properties that are formally proven in the analysis. Also, we provide a numerical performance comparison between our proposed mechanisms and the well known static auctions, namely the Vickrey second price auction and the VCG mechanism. Keywords—Cognitive Radio Networks, Spectrum Sharing, Game Theory, Auction, Multi-armed Bandit. I. I NTRODUCTION W ITH the ever-increasing demand for wireless commu- nications, the wireless spectrum is becoming over- crowded. Currently, the Federal Communications Commission (FCC) allocates the spectrum for a long period of time through auctions among giant wireless operators. However, this static allocation results in inefficient use of the wireless spectrum. According to the measurements by the FCC’s Spectrum Policy Task Force, most of the allocated spectrum is under-utilized [1]. Dynamic spectrum sharing has been proposed recently to improve the spectrum utilization [2]. Dynamic spectrum sharing enables new methods of cooperation and competition where a Primary Owner (PO) can re-allocate its idle spectrum bands to unlicensed or Secondary Users (SUs). Therefore, This material is based upon work supported by the National Science Foundation under grant numbers 1422153 and 1456887, and a Tekes FiDiPro Fellow award. M. Khaledi is with the Department of Electrical and Computer Engi- neering, Rensselaer Polytechnic Institute, Troy, NY, 12180 USA (e-mail: khalem@rpi.edu). A. A. Abouzeid is with the Department of Electrical and Computer Engi- neering, Rensselaer Polytechnic Institute, Troy NY 12180 USA, and also has a visiting appointment with the Department of Communications Engineering, University of Oulu, Oulu 90014, Finland (e-mail: abouzeid@ecse.rpi.edu). designing mechanisms that provide incentives for both PO and SUs is imperative. We focus on auction-based mechanisms as they are very well-suited to the spectrum sharing problem, compared to the other possible market mechanisms. For instance, in pricing mechanisms the seller is assumed to have prior knowledge about the value of items to the potential buyers. However, in an auction the seller gets this information through bidding and the prior knowledge is not necessary. Also, auction mechanisms are more practicable compared to other market mechanisms (e.g. bargaining games [3]), since they incur less communica- tion overhead. In a simple spectrum auction, SUs bid to buy spectrum bands from a PO that sells its idle bands for a profit. An underlying assumption in existing spectrum auctions is that SUs know the exact value of channel access, and they bid ac- cordingly (see section II for a review of prior work). However, in real world scenarios, the value of obtaining channel access is not exactly known to the SUs a priori, but they learn it over time. In fact, SUs revise their estimates of values for channel access based upon what they experience. In this paper, we study spectrum auctions with dynamically evolving values. The setting allows SUs to learn their valua- tions based on their recent experiences of the channel quality. In this context, an SU’s experience is estimated as a function of the channel quality or Signal to Noise Ratio (SNR) of the channel. We propose ADAPTIVE, a dynAmic inDex Auction for sPectrum sharing with TIme-evolving ValuEs. To the best of our knowledge, ADAPTIVE is the first spectrum auction that considers dynamically evolving values. ADAPTIVE is technically a repeated auction of a channel in which SUs learn their values over time. The proposed auction results in efficient allocation that maximizes the expected discounted social welfare. The challenge presented by dynamic valuations is that the allocation needs to consider evolution of values that requires taking into account the consequences of current allocation by looking at future values. With non-deterministic evolution of values, the channel allocation is a stochastic dynamic pro- gramming problem. However, dynamic programming, which utilizes standard techniques such as backwards induction, is computationally intensive. The novelty of this work is that, by integrating multi-armed bandit models [4] in our auction models, we develop index-type allocation policies in our pro- posed auction where dynamic allocation indices are computed via forward induction in polynomial time. Every auction is determined by a pair of functions (or rules); the allocation function and the payment function. We cast the allocation part of ADAPTIVE into an infinite horizon