COMP/MATH 553 Algorithmic Game Theory Lecture 19& 20: Revenue Maximization in Multi-item Settings. Nov 10, Yang Cai

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1 COMP/MATH 553 Algorithmic Game Theory Lecture 19& 20: Revenue Maximization in Multi-item Settings Nov 10, 2016 Yang Cai

2 Menu Recap: Challenges for Revenue Maximization in Multi-item Settings Duality and Upper Bound of the Optimal Revenue SREV and BREV

3 Optimal Multi-Item Auctions Large body of work in the literature : e.g. [Laffont-Maskin-Rochet 87], [McAfee-McMillan 88], [Wilson 93], [Armstrong 96], [Rochet-Chone 98], [Armstrong 99],[Zheng 00], [Basov 01], [Kazumori 01], [Thanassoulis 04],[Vincent-Manelli 06, 07], [Figalli-Kim-McCann 10], [Pavlov 11], [Hart-Nisan 12], No general approach. Challenge already with selling 2 items to 1 bidder: Simple and closed-form solution seems unlikely to exist in general. Simple and Approximately Optimal Auctions.

4 Selling Separately and Grand Bundling Theorem: For a single additive bidder, either selling separately or grand bundling is a 6-approximation [Babaioff et. al. 14]. Selling separately: post a price for each item and let the bidder choose whatever he wants. Let SREV be the optimal revenue one can generate from this mechanism. Grand bundling: bundle all the items together and sell the bundle. Let BREV be the optimal revenue one can generate from this mechanism. We will show that Optimal Revenue 2BREV + 4SREV.

5 UpperBound for the Optimal Revenue Social Welfare is an upper bound for revenue. Unfortunately, could be arbitrarily bad. Consider the following 1 item 1 bidder case, and suppose the bidder s value is drawn from the eual revenue distribution, e.g., v 1, +, f v = + and F v = 1 +., -, The optimal revenue = 1. What is the optimal social welfare?

6 UpperBound for the Optimal Revenue Suppose we have 2 items for sale. r + is the optimal revenue for selling the first item and r 1 is the optimal revenue for selling the second item. Is the optimal revenue upper bounded by r + + r 1? NO We have seen an example. What is a goodupper boundfor the optimal revenue, i.e., within a constant factor?

7 Upper Bound of the Optimal Revenue via Duality

8 Multi-item Auction: Set Up Items Bidder Auctioneer 1 v~d j Bidder: Valuation aka type v~d. Let V be the support of D. Additive and uasi-linear utility: v = (v +,v 1,, v 8 ) and v S = v < < = for any set S. Goal: Optimize Revenue! Independent items: v = (v +, v 1,, v 8 ) is sampled from D = < D <. m

9 Our Duality (Single Bidder) Primal LP (Revenue Maximization for 1 bidder) Variables: x j v : the prob. for receiving item j when reporting v. p(v): the price to pay when reporting v. Constraints: v x v p v v x v F p v F, v V, v F V { } (BIC & IRConstraints) x v P = 0,1 8, v V (Feasibility Constraints) Objective: maxq f(v)p v v

10 Partial Lagrangian Primal LP: max Q f(v)p v v s.t. v x v p v v x v F p v F, v V, v F V { } (BIC & IR Constraints) x v P = 0,1 8, v V (Feasibility Constraints) where Partial Lagrangian (Lagrangify only the truthfulness constraints): min max L(,x,p) >0 x2p,p L(,x,p)= X v f(v)p(v)+ X v,v 0 (v, v 0 ) (v (x(v) x(v 0 )) (p(v) p(v 0 )) Strong Duality: Opt Rev = max min L(λ, x, p) = min max L(λ, x, p). X Y,Z ]^_ ]^_ X Y,Z Weak Duality: Opt Rev max L(λ, x, p) for all λ 0. X Y,Z Proof: On the board.

11 Partial Lagrangian Primal LP: max Q f(v)p v v s.t. v x v p v v x v F p v F, v V, v F V { } (BIC & IR Constraints) x v P = 0,1 8, v V (Feasibility Constraints) where Partial Lagrangian (Lagrangify only the truthfulness constraints): L(,x,p)= X v = X v + X v p(v) f(v)+ X v 0 (v 0,v) x(v) min max L(,x,p) >0 x2p,p f(v)p(v)+ X v,v 0 (v, v 0 ) (v (x(v) x(v 0 )) (p(v) p(v 0 )) v X v 0 (v, v 0 ) X v X (v, v 0 )! v 0 v 0 (v 0,v)!! Better be 0, o.w. dual = +

12 The Dual Variables as a Flow Observation: If the dual is finite, for every v V f v + vf λ(v, v) λ(v, v ) vf =0 This means λ is a flow on the following graph: There is a super source s, a super sink (IR type) and a node for each v V. f(v) flow from s to v for all v V. λ(v, v ) flow from v to v, for all v V and v F V { }. f(v) v λ(v, ) S f(v ) λ(v, v) v λ(v, v ) λ(v, ) Suffice to only consider λ that corresponds to a flow!

13 Duality: Interpretation Partial Lagrangian Dual (after simplification) where virtual welfare of allocation x w.r.t. Φ ] ( ) L(,x,p)= X v Note: every flow λ corresponds to a virtual value function Φ ] ( ) Primal = X v min max L(λ, x, p) flow λ x Y f(v) x(v) Dual virtual valuation of v (m-dimensional vector) w.r.t. λ Optimal Revenue Optimal Virtual Welfare w.r.t. any λ (Weak Duality) v f(v) X x j (v) j 1 f(v) Optimal Revenue = Optimal Virtual Welfare w.r.t. to optimal λ (Strong Duality) X ( ) j (v) v 0 (v 0,v)(v 0 v) Φ λ v = v 1 Q λ v, v f v vf v v k where Φ (]) l v = v < + λ v, v v m v v k < F v <!

14 Duality: Implication Strong duality implies Myerson s result in single-item setting. Φ ] v n = Myerson s virtual value. Weak duality: [Cai-Devanur-Weinberg 16]: A canonical way for deriving approximately tight upper bounds for the optimal revenue.

15 Single Bidder Flow

16 Single Bidder: Flow For simplicity, assume V = H 8 Z 8 for some integer H. Divide the bidder s type set into m regions v 1 H R < contains all types that have j as the favorite item. Our Flow: No cross-region flow (λ v F,v = 0 if v, v are not in the same region). for any v F,v R <, λ v F,v > 0 only if F = v u< and v F < = v < + 1. v u< Our flow λ has the following two properties: for all j and v R < Φ u< ] v = v u<. Φ < ] v = φ < (v < ), where φ < ( ) is the Myerson s Virtual Value function for D <. R 1 R + 0 H Virtual Valuation: Φ l (]) v = v j 1 f v Q λ v, v v v k v + j F v j

17 Single Bidder: Flow (cont.) For item j: v < F f(v < F, v u< ) v < + 2 v < + 1 v < f(v < + 2, v u< ) f(v < + 1, v u< ) s Q f v < F,v u<, y k z,y = f u< (v u< ) { 1 F < (v < ) Φ ] < v = v < 1 f v Q f v < F, v u< = v < 1 F <(v < ) f, k < (v < ) y z, y Myerson virtual value function for D <.

18 Intuition behind Our Flow Virtual Valuation: Φ l (]) v v 1 H = v j 1 f v Intuition: Q λ v, v v v k Empty flow è social welfare. j F v j Replace the terms that contribute the most to the social welfare with Myerson s virutal value. R 1 0 H Our flow λ has the following two properties: for all j and v R < Φ u< ] v = v u<. R + v + Φ < ] v = φ < (v < ), where φ < ( ) is the Myerson s Virtual Value function for D <.

19 Upper Bound for a Single Bidder Corollary: Φ < (]) v = v < { I v R < + φ < (v < ) { I[v R < ]. Upper Bound for Revenue (single-bidder): REV max L λ, x, p = Q Q f v x <(v) { (v < { I v R < + φ < (v < ) x Y, < { I[v R < ]) Interpretaion: the optimal attainable revenue is no more than the welfare of all nonfavorite items plus some term related to the Myerson s single item virtual values. Theorem: Selling separately or grand bundling achieves at least 1/6 of the upper bound above. This recovers the result by Babaioff et. al. [BILW 14]. Remark: the same upper bound can be easily extended to unit-demand valuations. Theorem: Posted price mechanism achieves 1/4 of the upper bound above. This recovers the result by Chawla et. al. [CMS 10, 15].

20 SREV and BREV

21 Single Additive Bidder [BILW 14] The optimal revenue of sellingm independentitems to an additive bidder, whose valuation v is drawn from D = < D < is no more than 6 max SREV(D), BREV(D). SREV(D) is the optimal revenue for selling the items separately. Formally, SREV D = < r < = r, where r < = max x { Pr,y v < x. X BREV(D) is the optimal revenue for selling the grand bundle. Formally, BREV D = max X x { Pr v < v < x.

22 Single Additive Bidder Corollary: Φ < (]) v v < { I v R < + φ < (v < ) { I[v R < ]. Goal: upper bound L λ, x, p for any x P using SREV and BREV. L λ, x, p = Q Q f v x < (v) { (v < { I v R < + φ < (v < ), < { I[v R < ]) = Q Q f v x < (v) { v < { I v R < v < + Q Q f v x < (v) { φ < (v < ) { I v R < v < NON-FAVORITE SINGLE

23 Bounding SINGLE SINGLE = v < f v x < (v) { φ < (v < ) { I v R < =, y f < (v < ) { φ < (v < ) {, y f u< v u< { x < (v) I v R < < view as the probability of allocating item j to the bidder when her value for j is v <. For each item j, this is Myerson s virtual welfare r <. SINGLE r

24 NON-FAVORITE: Core-Tail Decomposition NON-FAVORITE = v < f v x < (v) { v < { I v R < Q Q f v { v < { I v R <, < = Q Q f < (v < ) { v < { Pr, y [v <, y R < ] Q Q f < v < { v < { Pr, y [ k j, v š v < ] <, y^ + Q Q f < (v < ) { v < <, y œ TAIL CORE

25 NON-FAVORITE: Bounding the TAIL TAIL = f < v < { v < { Pr, y k j, v š v <, y^ < Sell each item separately at price v < : v < { Pr, y k j, v š v < šž< v < { Pr, v š v < šž< r š r, v < Sell each item separately at price r: TAIL Q Q f < v < { r = Q r { Pr,y [v < r] Q r < r <, y^ < <

26 NON-FAVORITE: Bounding the CORE CORE = <, y Ÿ f < (v < ) { v < = E[v F ] Lemma: Var v < F 2r < { r v < F = v < { I v < r v F = v < < Corollary: Var v = Var v < F < 2r 1 Chebyshev Ineuality: for any random variable X, Pr X E X a -. By Chebyshev Ineuality, Pr [v F < CORE 2r] Var[v F ] 4r Pr < v < CORE 2r 1/2. If selling the grand bundle at price CORE 2r, the bidder will buy it with prob. 1/2. 2BREV+2r CORE

27 Putting Everything Together REV max x Y L(λ, x, p) SINGLE + TAIL + CORE SINGLE r TAIL r CORE 2BREV + 2r [BILW 14] Optimal Revenue 2BREV + 4SREV.