Ruihao Zhu and Kang G. Shin Differentially Private and Strategy-Proof Spectrum Auction with Approximate Revenue Maximization Ruihao Zhu and Kang G. Shin Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor Hello, ladies and gentlemen, I am Ruihao Zhu, a senior student in University of Michigan. Today, I will introduce my work, differentially private… to all of you.

Outline Background Design Goal Primers: differential privacy, exponential mech., truthfulness , revenue maximization Near Optimal Mechanism PASS Evaluation Results The talk will be divided into the following parts.

Spectrum Need Forecast ‐ Table of Results This table comes from a whitepaper (“Mobile Broadband: The Benefits of Additional Spectrum”) published by FCC in Oct. 2010. On one hand, mobile broadband traffic is growing dramatically; on the other hand, spectrum is limited resource. FCC predicts that the spectrum deficit is likely to approach 300 megahertz this year. FCC whitepaper, Oct. 2010

Secondary Spectrum Market Traditionally, static, long-term licenses Radio spectrum is not fully utilized Unlicensed bands are getting crowded =>Dynamic spectrum redistribution/auction needed! Traditional spectrum allocation is carried out by the government in a static and long term way. Therefore, radio spectrum is not fully utilized, and unlicensed bands get crowded. Therefore, redistribution of idle radio spectrum is highly important. Open markets, such as Spectrum Bridge, have already appeared to improve spectrum utilization by providing services for buying, selling, and leasing idle spectrum.

Unique Challenge in Spectrum Auctions Spatial Reusability Bidders far away can use the same channel Channel 1 Channel 2 Different from traditional goods, spectrum has a unique characteristic. That is the spatial reusability, which allows bidders far away enough from each other to use the same channel.

Traditional Spectrum Auctions Auctioneer Channels Bidders However, spectrum valuations are the private information of the bidders. It may disclose the bidders' profits for serving their subscribers or their economic situations, which are highly desirable information for rivals and stock market speculators. Once the valuations are revealed to a corrupt auctioneer, she may exploit such knowledge to her advantage. Auctioneer’s Revenue Truthfulness

Privacy in Spectrum Auctions Channels are for short-term usage. Sequential auctions make inference of bidding information possible even with secure channel. spectrum valuations are the private information of the bidders. It may disclose the bidders' profits for serving their subscribers or their economic situations, which are highly desirable information for rivals and stock market speculators. Once the valuations are revealed to a corrupt auctioneer, she may exploit such knowledge to her advantage.

Privacy in Spectrum Auctions spectrum valuations are the private information of the bidders. It may disclose the bidders' profits for serving their subscribers or their economic situations, which are highly desirable information for rivals and stock market speculators. Once the valuations are revealed to a corrupt auctioneer, she may exploit such knowledge to her advantage. How to infer?

Privacy in Spectrum Auctions, cont’d Single channel First time: 𝑏 𝑎𝑡&𝑡 =2, 𝑏 𝑣𝑒𝑟𝑖𝑧𝑜𝑛 =1.8 Second time: 𝑏 𝑎𝑡&𝑡 =2.02, 𝑏 𝑣𝑒𝑟𝑖𝑧𝑜𝑛 =2.2 spectrum valuations are the private information of the bidders. It may disclose the bidders' profits for serving their subscribers or their economic situations, which are highly desirable information for rivals and stock market speculators. Once the valuations are revealed to a corrupt auctioneer, she may exploit such knowledge to her advantage. 0.01% revenue for channel cost

Outline Background Design Goal Primers: differential privacy, exponential mech., truthfulness Near Optimal Mechanism PASS Evaluation Results

Goal Design a truthful auction mechanism that maximizes auctioneer’s revenue while keeping participants’ bidding prices confidential Bearing the concern in mind, our goal here is design…

Outline Background Design Goal Primers: differential privacy, exponential mech., truthfulness, revenue maximization Near Optimal Mechanism PASS Evaluation Results

Differential Privacy Differential privacy aims to provide means to maximize the accuracy of queries from statistical databases while minimizing the chances of identifying its records. Intuitively, it means that a single change in the input dataset does not affect the outcome much.

Differential Privacy, cont’d Def’n. A mechanism M is (𝜖,𝛿)-differential private if for any two data profiles D1 and D2 differing on a single element, and all S ⊆ Range(M), Pr[M(D1) ∈ S] ≤ exp(𝜖)×Pr[M(𝐷2) ∈ S]+𝛿

Differential Privacy cont’d Randomness (no deterministic DP): Input perturbation Exponential mechanism

Exponential Mechanism Bids:𝑏=( 𝑏 1 , 𝑏 2 ,…, 𝑏 𝑛 ). ∆= range of bids. Revenue: REV(𝐱)= 𝑖=1 𝑛 𝑏 𝑖 𝑥 𝑖 . Choose outcome x with probability Pr[x] ∝exp(𝜖𝑅𝐸𝑉(𝐱)/2∆). Logarithmic loss in revenue (2𝜖∆)-differentially private

Truthful (in Expectation) A bidder always maximize expected utility by bidding true valuation, i.e., 𝐸[ 𝑢 𝑖 (𝑠 𝑖 , 𝑠 −𝑖 )] ≥𝐸[ 𝑢 𝑖 (𝑠′ 𝑖 , 𝑠 −𝑖 )].

Truthful Mechanism A mechanism is truthful in expectation if and only if, for any agent 𝑖 , and any fixed choice of bids by the other agents 𝑏 −𝑖 , 1. 𝑖 ′s winning probability is monotone in 𝑏 𝑖 ; 2. 𝑝 𝑖 𝑏 = 𝑏 𝑖 𝑦 𝑖 𝑏 − 0 𝑏 𝑖 𝑦 𝑖 𝑧 𝑑𝑧 , where 𝑦 𝑖 (𝑧) is the probability that 𝑖 wins when his bid is 𝑧.

Revenue Maximization Bids:𝑏=( 𝑏 1 , 𝑏 2 ,…, 𝑏 𝑛 ). Bid PDFs and CDFs: 𝑓= 𝑓 1 , 𝑓 2 ,…, 𝑓 𝑛 ,𝐹=( 𝐹 1 , 𝐹 2 ,…, 𝐹 𝑛 ) Virtual bid:  𝑏 𝑖 )= 𝑏 𝑖 −(1− 𝐹 𝑖 ( 𝑏 𝑖 ))/ 𝑓 𝑖 ( 𝑏 𝑖 Virtual Revenue: REV(𝐱)= 𝑖=1 𝑛  𝑏 𝑖 𝑥 𝑖 . Choose outcome x to maximize REV.

Outline Background Problem Definition Primers: differential privacy, exponential mech., truthfulness, revenue maximization Near Optimal Mechanism PASS Evaluation Results

Near Optimal Mechanism exponential mechanism + revenue maximization technique: Calculate virtual bid Determine feasible allocations Select x with probability Pr[x] ∝exp(𝜖𝑅𝐸𝑉(𝐱)/2∆). NP hard!

Outline Background Design Goal Primers: differential privacy, exponential mech., truthfulness , revenue maximization Near Optimal Mechanism PASS Evaluation Results

Illustrative Example 𝑐=1 channel 𝑛=5 bidders with (𝑏 1 )=20, (𝑏 2 )=50,( 𝑏 3 )=80, (𝑏 4 )=70, (𝑏 5 )=30 location 4 2 5 1. Partition the geographic region into hexagons with unit side-length and color the region with 7 colors (as in the figure). 2. Grouping: Bidders in the same color region are in the same group and bidders in the same hexagon are in the same subgroup. Interference range 1 3

Random Selection and Allocation PASS Graph Partition Virtual Channel 1. Partition the geographic region into hexagons with unit side-length and color the region with 7 colors (as in the figure). 2. Grouping: Bidders in the same color region are in the same group and bidders in the same hexagon are in the same subgroup. Random Selection and Allocation

PASS Partition entire area uniformly into small hexagons with Graph Partition 4 Partition entire area uniformly into small hexagons with side length equal half interference range. 5 2 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup. 1 3

PASS 𝑟 1 ={ 𝑣𝑐 6,7 , 𝑣𝑐 7,12 } 𝑟 2 ={ 𝑣𝑐 6,7 } 𝑟 3 ={ 𝑣𝑐 7,12 } Virtual Channel 4 𝑟 1 ={ 𝑣𝑐 6,7 , 𝑣𝑐 7,12 } 𝑟 2 ={ 𝑣𝑐 6,7 } 𝑟 3 ={ 𝑣𝑐 7,12 } 𝑟 4 = 𝑣𝑐 13,14 𝑟 5 ={ 𝑣𝑐 13,14 } 5 2 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup. 1 3

Random Selection and Allocation PASS Random Selection and Allocation 4 exp( ε ′  𝑖 𝑏 𝑖 |𝑟 𝑖 | ). 5 2 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup. 1 3

Random Selection and Allocation PASS Random Selection and Allocation 4 Taking ε′=0.1 Pr W={1} ∝ exp(2/ 2 ) Pr W={2} ∝ exp(5 ) Pr W={3} ∝ exp(8 ) Pr W={4} ∝ exp(7 ) Pr W={5} ∝ exp(3 ) 5 2 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup. 1 3

Random Selection and Allocation PASS Random Selection and Allocation 4 Suppose bidder 1 is selected. 5 2 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup. 1 3

Random Selection and Allocation PASS Random Selection and Allocation 4 All the bidders conflict with bidder 1 is removed. 5 2 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup. 1 3

Random Selection and Allocation PASS Random Selection and Allocation 4 Taking ε′=0.1 Pr W={1,4} ∝ exp(7 ) Pr W={1,5} ∝ exp(3 ) 5 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup.

Random Selection and Allocation PASS Random Selection and Allocation 4 Suppose bidder 5 is selected. 5 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup.

Random Selection and Allocation PASS Random Selection and Allocation 4 All the bidders conflict with bidder 5 is removed. 5 In bidder grouping, dear use a hexagon based graph partition technique to divide the bidders into 7 groups, namely g1 to g7. each hexagon here is called a subgroup.

Properties of PASS Lemma 4. The size of the virtual channels bundle assigned to each bidder is less than or equal to 12, which is optimal for hexagon partition. Theorem 6. With the probability of at least 1−𝑛 𝑂(1) PASS can generate a set of winners with a revenue of at least 𝑅𝐸𝑉 ∗ 12 −𝑂( ln 𝑛 ), where 𝑅𝐸𝑉 ∗ is the optimal revenue. Theorem 7. For any 𝛿< 1 2 , PASS preserves ( ε ′ 𝑒−1 𝑒−1  ln 𝑒 𝛿 ,𝛿) differential privacy.

Outline Background Design Goal Primers: differential privacy, exponential mech., truthfulness , revenue maximization Near Optimal Mechanism PASS Evaluation Results

Revenue Revenue of PASS (5 channels)

Revenue Revenue of PASS (10 channels)

Revenue Revenue of PASS (15 channels)

Measuring Empirical ε Privacy of PASS (5 channels)

Measuring Empirical ε Privacy of PASS (10 channels)

Measuring Empirical ε Privacy of PASS (15 channels)

Conclusion PASS: First differentially private and truthful spectrum auction mechanism with approximate revenue maximization. Theoretically proved the properties in revenue and privacy. Implemented PASS and extensively evaluated its performance.

Thank you! rhzhu@umich.edu