A Prior-Free Revenue Maximizing Auction for Secondary Spectrum Access

Ajay Gopinathan and Zongpeng LiIEEE INFOCOM 2011, Shanghai, China

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The Secondary Spectrum Market

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

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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

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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? “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

Achieving all four properties simultaneously is non-trivial

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Example - Interference-Free Assignment

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Interference

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{ CH1, CH2 }Channels

CH1

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

Achieving all four properties simultaneously is non-trivial

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Best known truthful auction in economics 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

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

We can no longer rely on the VCG mechanism

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Solution? 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 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

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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

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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

bidsWhat if prior knowledge is unavailable?

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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

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The random sampling auction1. Randomly assign bidders to one of two sets,

A and B Flip a coin for each agent. Heads => A, Tails =>

B2. Compute optimal revenue for A, $A3. Compute optimal revenue for B, $B 4. Attempt to “extract” $A from bidders in B5. Attempt to “extract” $B from bidders in A

[Fiat et al., ACM STOC 2002][Goldberg et al., Games and Economic Behavior, 2006]

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Random sampling auction - Analysis 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 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

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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

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Conclusion 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?