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A Prior-Free Revenue Maximizing Auction for Secondary Spectrum Access Ajay Gopinathan and Zongpeng Li IEEE INFOCOM 2011, Shanghai, China

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The Secondary Spectrum Market 1 We require an auction protocol for secondary spectrum access that is Revenue-Maximizing Strategyproof (truthful) Interference-free Efficiently Computable We require an auction protocol for secondary spectrum access that is Revenue-Maximizing Strategyproof (truthful) Interference-free Efficiently Computable

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The myth of spectrum scarcity Growing number of wirelessly equipped devices Demand for usable spectrum is increasing Limited available spectrum How scarce is spectrum? Utilization varies over time and space 15%-85% variation in spectrum utilization [FCC, ET Docket No 03-222, 2003] Existing allocated spectrum is badly utilized! Solution: Secondary spectrum access Allow secondary users to utilize idle spectrum 2

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Dynamic Spectrum Allocation Secondary Spectrum Market Primary users (AT&T, Verizon etc) Secondary users (smaller ISPs) Secondary users lease spectrum from the primary user Idle spectrum divided into channels Secondary users pay for obtaining a channel 3

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Dynamic Spectrum Allocation - Challenges Allocation How do we allocate spectrum? Avoid interference Exploit spatial reusability Payment How much should secondary users be charged? 4 “Who gets the spectrum, and at what price?” Auctions!

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Auction Desiderata Maximize Revenue Primary user has incentive to lease spectrum Strategyproof (truthful) Secondary users have no incentive to lie about valuation Interference-free allocation Limited number of channels to be assigned Channel assignment = Graph colouring (NP-Hard!) Computationally efficient Protocol runs in polynomial time 5 Achieving all four properties simultaneously is non-trivial

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Example - Interference-Free Assignment 6 1 1 Interference 2 2 3 3 4 4 { CH1, CH2 } Channels CH1 CH2

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Auction Desiderata Maximize Revenue Primary user has incentive to lease spectrum Strategyproof (truthful) Secondary users have no incentive to lie about valuation Interference-free allocation Limited number of channels to be assigned Channel assignment = Graph colouring (NP-Hard!) Computationally efficient Protocol runs in polynomial time 7 Achieving all four properties simultaneously is non-trivial

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Best known truthful auction in economics 8 Vickrey-Clarke-Groves (VCG) mechanism Family of auction type mechanisms Best known, widely used mechanism in economics Versatile and provably strategyproof Main drawback Requires access to the optimal allocation Loses strategyproof property otherwise

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Auction Desiderata Maximize Revenue Primary user has incentive to lease spectrum Strategyproof (truthful) Secondary users have no incentive to lie about valuation Interference-free allocation Limited number of channels to be assigned Channel assignment = Graph colouring (NP-Hard!) Computationally efficient Protocol runs in polynomial time 9 Must resort to approximation algorithms and suboptimal allocation

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Auction Desiderata Maximize Revenue Primary user has incentive to lease spectrum Strategyproof (truthful) Secondary users have no incentive to lie about valuation Interference-free allocation Limited number of channels to be assigned Channel assignment = Graph colouring (NP-Hard!) Computationally efficient Protocol runs in polynomial time 10 We can no longer rely on the VCG mechanism

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Solution? 11 Forget about VCG - design auction from scratch How do we get a truthful auction? Examine characterization of truthfulness in an auction

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Mathematical description of auctions 12 Auctions can specified as function of bids Allocation function Probability of winning as a function of the bid Payment rule Bidders have private valuation “How much is a channel worth to me?” Bidders want to maximize

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Characterizing truthfulness If an agent wins the auction, charge her the minimum bid that guarantees winning Charge winning agents a bid independent price 13

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Auction Desiderata Maximize Revenue Primary user has incentive to lease spectrum Strategyproof (truthful) Secondary users have no incentive to lie about valuation Interference-free allocation Limited number of channels to be assigned Channel assignment = Graph colouring (NP-Hard!) Computationally efficient Protocol runs in polynomial time 14

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What about revenue? Vickrey-type auctions have bad revenue properties E.g. 2 bids of $x > 0 and $0 has no revenue Solution: reserve price $R Add imaginary bidder with bid $R Run Vickrey auction on set of bids Vickrey auction with reserve prices are optimal How to compute the optimal $R? Need prior knowledge of probability distribution of bids What if prior knowledge is unavailable? 15

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The prior-free setting Assume no knowledge of agent valuations Worse-case setting Online optimization problem First studied by Fiat et al. [Fiat et al., ACM STOC 2002] Random Sampling Auction Context of selling digital goods – unlimited supply of items Key idea: acquire knowledge by sampling bids 16

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The random sampling auction 1. Randomly assign bidders to one of two sets, A and B Flip a coin for each agent. Heads => A, Tails => B 2. Compute optimal revenue for A, $A 3. Compute optimal revenue for B, $B 4. Attempt to “extract” $A from bidders in B 5. Attempt to “extract” $B from bidders in A 17 [Fiat et al., ACM STOC 2002] [Goldberg et al., Games and Economic Behavior, 2006] [Fiat et al., ACM STOC 2002] [Goldberg et al., Games and Economic Behavior, 2006]

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Random sampling auction - Analysis 18 Equivalent to Vickrey auction with 2 bidders Each set is a “bidder” Guarantees minimum of ($A, $B) Offer price is bid independent – truthful! 4-approximate revenue guarantee – constant! Assumes unlimited supply of item being auctioned

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An idea for reduction 19 Step 1: Compute a feasible, interference-free channel assignment Step 2 : All bidders that can be feasibly assigned spectrum participate in the Random Sampling Auction “Unlimited supply” of channels Challenges What is the best type of assignment in Step 1? Maximize potential revenue in Step 2 How do we make Step 1 truthful? Still need to use suboptimal assignment Can we make the Random Sampling Auction better?

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Our Contributions A two-phase auction protocol for maximizing revenue Phase 1: Truthful and interference-free channel allocation Highest potential revenue Works with any MAX-K-CIS approximation algorithm Tailored payment scheme to ensure truthfulness Phase 2: Iterative Random Partitioning Auction Based on the random sampling auction Only bidders allocated in phase 1 participate (unlimited supply of channels) Achieves a 3-approximate revenue guarantee 20

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Iterative Partitioning Auction Improving random sampling auction – “Rinse and repeat!” Choose the set that loses the auction, repeat sampling auction Participation in future round is bid independent – still truthful! Analysis is difficult Revenue in each round is a random variable Number of rounds is a random variable Solution: Don’t sample, partition set instead Revenue is still random variable Number of rounds is fixed at log n This achieves asymptotically a 3-approximate revenue guarantee 21

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Conclusion 22 We design 2-phase auction protocol for secondary spectrum access Phase 1: Compute interference-free assignment Phase 2: Maximize revenue from bidders assigned in Phase 1 Our two main tools Myerson’s characterization of truthful mechanisms Randomization Questions?

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