K -anonymous Signaling Scheme
KK -anonymous Signaling Scheme Binyi ChenShanghai Jiaotong University Tao QinMicrosoft Research AsiaTie-Yan LiuMicrosoft Research AsiaSeptember, 2013
Abstract
We incorporate signaling scheme into Ad Auction setting, to achievebetter welfare and revenue while protect users’ privacy. We propose anew K -anonymous signaling scheme setting , prove the hardness of thecorresponding welfare/revenue maximization problem, and finally proposethe algorithms to approximate the optimal revenue or welfare. In real ad Exchange market, auctioneer and bidders will have long term busi-ness. On the one hand, leaking impression’s information will reduce bidders’competition and thus lose revenue. For example, advertisers usually favor acertain category of users, companies selling football kits only wish to advertiseto soccer fans. When the search engine informs the advertiser that the user is asoccer fan, only football companies are willing to bid, which leads to a thinnermarket and less revenue for the search engine. On the other hand, this willalso do harm to users’ privacy protection, which is a highly important issue inInternet business. Therefore, it is important to protect users’ privacy and hideinformation of coming impressions.In order to raise the revenue while protecting privacy, we can inform thatthe user belongs to a ball game category irrespective of his interest in soccer,basketball, or tennis. Most of the sports companies will bid for this user, lead-ing to higher revenue for the search engine. In the mathematical model, weconstrained the support (soccer, basketball, tennis, etc) of the categories (ballgame) to be larger than K and our objective is to build up the type (soccer)to category (ball game) map that enables the search engine to achieve the bestwelfare/revenue.In section 2, we propose the specific setting of the problem. In section 3, wefirstly prove that it is NP-hard to solve K-anonymous signaling welfare maxi-mization problem and it cannot be approximated in factor less than e/ ( e − a r X i v : . [ c s . G T ] N ov hen we give a 2 e/ ( e −
1) approximation algorithms. Finally, we propose sev-eral special settings that can be solved optimally in polynomial time. In section4, we prove the NP-Hardness of revenue maximization, and give a method totransfer welfare maximization approximation result into revenue approximation.In section 5, we note the possible future work.
There is one impression coming for auction, the impression belongs to one of m category, the prior probability that the impression belongs to the i th categoryis p i . The i th bidder of n bidders values the j th category as v ij . Auctioneerwill broadcast a signal S after knowing the impression’s category and raisea second price auction. The map between categories and signals is public tobidders. Denote φ ( j, S ) as the probability that auctioneer will broadcast signal S when category is j . In this setting, φ ( j, S ) ∈ { , } . Finally, auctioneerwants to construct the best map that can extract the most revenue or socialwelfare, while satisfies that each signal’s support(categories that will broadcastthis signal) sup φ,S = { j : φ ( j, S ) = 1 } is equal or larger than K .Without K-constraint , the model is a typical pure signaling scheme men-tioned in [3]. For a signal S , the probability of broadcasting S is (cid:80) j p j φ ( j, S ).Given the auctioneer broadcasting S , the probability that the category is j is P r [ j | S ] = p j φ ( j, S ) / ( (cid:80) j (cid:48) p j (cid:48) φ ( j (cid:48) , S ). The expected welfare achieved by broad-casting S is max i { E [ v i | S ] } = max i { (cid:80) j v i,j p j φ ( j, S ) (cid:80) j p j φ ( j, S ) } Therefore, the objective is to find a map φ that satisfies K-constraint to maxi-mize (cid:88) S P r [ S ] max i { E [ v i | S ] } = (cid:88) S max i { (cid:88) j ( v i,j p j ) φ ( j, S ) } We use V i,j to describe v i,j p j for simplicity later. The revenue maximizationproblem is similar by replacing maximum with the second maximum since sec-ond price auction is truthful in this setting.We observe that this problem is equivalent to the bundling scheme problem:there are m items for sale, bidders have different value to different item. Thetask is to partition items into several bundles, each bundle has size at least K .Then auctioneer sells bundles separately by second price auction. The goal ofauctioneer is to maximize social welfare/revenue. In this section we firstly prove the NP-Hardness of welfare maximization prob-lem and prove it cannot be approximated in factor less than e/ ( e − e/ ( e −
1) approximation algorithms. Finally, we propose several specialsettings that can be solved optimally in polynomial time.2 .1 Hardness Results
When both number of signals used and bundle size have no constraint, welfaremaximization problem is trivial by giving category j impression to bidder whovalues it most. However, when signal number is constrained (Cardinality Con-strained Signaling Problem) , or each signal’s support size has to be no less than K (K-anonymous Signaling Problem) , the problem becomes NP-Hard.We prove the NP-Hardness of K-anonymous welfare-maximization problemby a reduction from [1]. Proposition 1. [1] There is no polynomial-time c -approximation algorithm forwelfare-maximization with known valuation, when signal number is constrained,for any constant c < ee − , unless P = N P .Proof.
The proof is directly from [1].
Theorem 1.
There is no polynomial-time c -approximation algorithm for welfare-maximization with known valuation and K-anonymous constraint, for any con-stant c < ee − , unless P = N P .Proof.
Prove by contradiction. Assume there is a c -approximation algorithmfor c < ee − .Given any signal number-constraint problem instance I with input m, S, V ij , m is the number of items, S is signal number constraint, V ij = v i,j p j . W.l.o.g,we assume V ij are all integer values by scaling (for the proof of [1] still applieswhen v i,j p j are rational number). We denote OP T as the optimal solution. Wecan assume there are exactly S bundles(signals) in OP T as b , b , ..., b S sincewe can split bundles without decreasing welfare if bundle (signal) number is lessthan S .We create a new instance for K-anonymous signaling I K . Set K = (cid:100) m − S +1 (cid:101) ,items are m old items plus K · S − m + K − n oldbidders plus K · S − m + K − i th new bidder only positivelyvalues the i th new item with S , and values the other items zero.Notice K · S − m + K − K · ( S + 1) − m − S > K · ( S + 1) − m − ≥ ( m − S S + 1) − m − ≥ m − S + ( S + 1) − m − OP T (cid:48) as the optimal solution of instance I K . Lemma 1. (cid:80) b i ∈ OP T ( b i − K ) + < K Proof.
Assume left-hand side ≥ K , let c the number of bundles that exceeds Kitems. Then there are S − c bundles with at least one item but less than K + 1items. 3hen c = 0 the lemma holds obviously, when c > m ≥ c · K + K + S − c (1) ≥ c · m − S m − S c + 1 + s − c (2) ≥ m − S + c + 1 + s − c = m + 1 (3)Contradiction. The second inequality holds by substituting K = m − S + 1, thethird inequality holds for c > Lemma 2.
OP T (cid:48) ≥ OP T
Proof.
Since the total number of items are K · S + K − (cid:80) b i ∈ OP T ( b i − K ) + OP T (cid:48) into 2 kinds of bundles A = { a , ..., a t } , C = { c , ..., c t } .A bundle is in C if and only if all of its items are new items. t + t ≤ S sincethe total number of items is K · S + K − 1. Winners in A are all old bidderssince new bidder can value a bundle at most 1 /S but at least one old biddervalue the bundle ≥ 1. On the other hand, each bundle in C is valued at most1 /S , and t < S , thus sum of welfare in C is less than ( S − /S .Remove all new items in OP T (cid:48) , we will obtain a feasible solution OP T (cid:48)(cid:48) for I .We only lose welfare in C which is at most ( S − /S .Thus OP T (cid:48)(cid:48) > OP T − V ij are all integers, we have OP T (cid:48)(cid:48) = OP T . Therefore, we obtain a c -approximation algorithm for welfare-maximization with signal number con-straint, for c < e/ ( e − e/ ( e − Approximation Algorithm We provide a 2 e/ ( e − 1) Approximation Algorithm by incorporate result from[1] Proposition 2. [1] For cardinality constrained signaling with known valuations,there is a randomized, polynomial-time, e/ ( e − -approximation algorithm forcomputing the welfare-maximizing signaling scheme.Proof. The proof is directly from [1]. Theorem 2. For K-anonymous signaling with known valuations, there is arandomized, polynomial-time, e/ ( e − -approximation algorithm for computingthe welfare-maximizing signaling scheme.Proof. Given a K-anonymous signaling instance I K , we set S as (cid:98) m/K (cid:99) andsolve cardinality constrained signaling problem, without K-anonymous con-straint. 4fter obtain the solution ALG , assume set A has t signals, each of whichhas less than K items: A = { a , a , ..., a t } , set B has (cid:98) m/K (cid:99) − t signals, eachof which has no less than K items: B = (cid:8) b , b , ..., b (cid:98) m/K (cid:99)− t (cid:9) .If welfare of B is no less than half of the total welfare, then we are done.Otherwise, A ’s welfare will be larger than half of the total value. We can filleach a by B ’s items, since items left is no less than m − ( a + ... + a t ) ≥ k · t − ( a + ... + a t ). Then we can obtain A ’s welfare in a feasible solution, it isalso a 2 e/ ( e − 1) approximation.We show the solution of algorithm above can be only half of the optimum. Example 1. There are K + K items and K + 2 bidders. Bidder A values eachof the first K items for /K . And bidder B i values the K + i th item for 1and others zero. Bidder C value each of the last K items for − (cid:15)K .The optimal solution’s welfare is K − (cid:15) , by giving first K items to A andlast K items to C . But the algorithm’s best welfare can be at most K + 1 , since B i will win in ALG . The gap approaches to 2 as K goes to infinity. Given some extra assumption, we can calculate the optimal solution for welfaremaximization in polynomial time. We constrain the number of signals being used as a constant, and propose analgorithm that optimally solve the problem in polynomial time. It consists of 3steps as below:1. Enumerate signals number and signals’ winners. This only cost O ( n c )time.2. Remove bidders who won no signal, making each item chosen by bidderwho favors it most. After this step, the i th bidder get k i items. Thisscheme achieve the best social welfare, yet some winner may win less thank items.3. Use cost flow technique to find the best allocation which satisfy the K-anonymous constraint. The graph is shown in the figure 1, arcs are de-scribed as ( l, u, c ), l is the lower bound of flow, u is the upper bound, c isthe cost per flow. w i and w i (cid:48) correspond to the i th bidder, item j corre-sponds to the j th item. Arc from st to w i represents the initial numberof items of bidder i is k i . Arc from w (cid:48) i to end represents the final itemsof bidder i should be no less than k . Arc from w (cid:48) i to item j represent i can give its item j to others, leading a welfare decrease of v ij . Arc from item k to w i represents that bidder i can gain item k from others, leading awelfare increase of v ij . By solving the minimum cost feasible flow problem[6], we can obtain final optimal allocation.5igure 1: Cost Flow In this subsection, we propose a setting that the value of bidder i to item j V ij can be expressed as p i · q j + b i . A dynamic programming approach is given thatsolve it in polynomial-time. Lemma 3. Given any instance of K-anonymous signaling problem, there alwaysexists an optimal signaling scheme that, sort signal winners { w , w , ..., w t } by p i in non-decreasing order. ∀ i < j, p i (cid:54) = p j , for any category t i points to w i ’ssignal, and any category t j points to w j ’s signal, q t i ≥ q t j .Proof. Given any optimal signaling scheme, assume there exists i < j, p i (cid:54) = p j , q k > q l such that w i possesses q l and w j possesses q k . After swap q k and q l .The welfare change is p i q k + p j q l − ( p i q l + p j q k ) = p i ( q k − q l ) − p j ( q k − q l ) (4)= ( p i − p j )( q k − q l ) ≥ p i ( q j ). Building state F ( i, j ), which means the best welfare for bidders { , , ..., i } , items { , , ..., j } . F ( i, j ) = max F ( i − , j ) F ( i, j − F ( i − , j − K ) + K · b i + ( (cid:80) jt = j − K +1 q t ) p i (6) K are the enumerated number of items that bidder i will win, K ≥ K . Theoptimal solution is F ( n, m ). The complexity is O ( nm ).6 Revenue Maximization Theorem 3. Even when the signals number is constant, it is still NP-hard tosolve the revenue maximization problem in K-anonymous constraint setting. Lemma 4. It is NP-hard to partition n integers { x , ..., x n } into two setswith size n , such that the two subsets’ element sum are the same.Proof. Without size n constraint, it is a classical NP-Complete problem(SubsetSum [5]) that we want to find a subset whose sum is (cid:80) i x i .Proof by contradiction, assume that the new problem has polynomial timesolution. For any Subset Sum problem instance with n integers, we can add n zeros and transfer it into the new problem. By finding two sets with size n andthe sum are the same, we can remove zeros and obtain a solution for SubsetSum, contradiction. Proof. Assume the revenue optimization can be polynomially solved. For anyinstance of Subset Sum Problem with Size K − K − K − W and raise an auction: There are 3 bidders, 2 K items,signal’s support size should be no less than K . the value matrix shows in tablebelow: Item W Bidder W x x ... x K − Signal number can be at most 2. When signal number is one, the revenuecan be at most W .When signal number is two, if there exists a solution { S , S } of SSPS, wecan bundle item { ∪ S } and { ∪ S } , achieve revenue W . Since any bundlecan achieve value at most W/ 2, this is optimal.When there is no solution for SSPS, given any signaling scheme with 2signals, if items 1 , W . When 1 , s , s ’s revenue is W/ s ’s revenue isstrictly less than W/ 2. Therefore, the optimal revenue is strictly less than W .From all above, the optimal revenue of the auction is W if and only if thereexists a solution of SSPS, Contradict with the fact that Subset Sum problem isNP-hard. Remark 1. This proof ’s rough thought is coincidentally similar to the proof ofNP-Hardness of revenue maximization when K-anonymous constraint does notexists. [2] .2 Approximation Approach [1] independently gives a randomized mechanism for approximating revenue in cardinality constraint setting, we independently give an deterministic approx-imation algorithm for K-anonymous setting with similar yet different analysis(Since the valuation is known in prior, an algorithm is sufficient to extract rev-enue). Theorem 4. We can achieve 3 β -approximation in revenue maximization prob-lem with K-anonymous constraint, when there is a β -approximation for welfaremaximization problem.Proof. We prove that we can transfer welfare maximization result into revenuemaximization.Fix two parameters β and α , we firstly find the β -approximate optimal sig-naling scheme S for welfare maximization in K-anonymous constraint. Denote V i as the i th bidder’s welfare contribution in S . Let V ∗ = max( V i ), OP T W asthe social welfare of scheme S , and OP T as the optimal revenue we can achieve,obviously OP T W ≥ OP T . We first sort bidders according to V i . • If V ∗ ≤ βOP T W , merge the 1st and the 2nd bidders’ signal, 3rd and 4thbidders’ bundles, and so on. We can at least achieve − β OP T revenuewithout violate K-anonymous constraint. • We denote OP T W ( − i ∗ ) as the best social welfare we can achieve whenthe first bidder i ∗ has been removed(signaling scheme is S (cid:48) ). Obviously, OP T W ( − i ∗ ) ≥ OP T . We further denote V (cid:48)∗ as the max welfare anindividual bidder can contribute in OP T W ( − i ∗ ).If V ∗ ≥ βOP T W and V (cid:48)∗ ≤ αOP T W ( − i ∗ ), In signaling scheme S (cid:48) , wemerge the 1st and the 2nd bidders’ signals, 3rd and 4th bidders’ signals,and so on. We can at least achieve − α OP T revenue. • Otherwise, sell all items in one signal, we can at least achieve min( α, β ) OP T revenue.Setting α = β = , we achieve a competitive ratio. Corollary 1.