Strong Duality for a Multiple Good Monopolist

Transcription

1 Strong Duality for a Multiple Good Monopolist Constantinos Daskalakis EECS, MIT joint work with Alan Deckelbaum (Renaissance) and Christos Tzamos (MIT) - See my survey in SIGECOM exchanges: July 2015 volume (or my webpage) - Forthcoming Econometrica: Strong Duality for a Multiple Good Monopolist

2 Optimal Mechanism Design Of all possible mechanisms, which one optimizes the seller s revenue? Focus: 1 seller (with one or multiple items), 1 buyer

3 Single-Item Mechanisms z F [Riley-Zeckhauser 81, Myerson 81]: The optimal mechanism is a takeit-or-leave-it offer of the item at price: p arg max z 1 F z. Resulting bidder utility: u z = max{0, z p } u(z) p z

4 Optimal Multi-Item Mechanisms? 1 z F n Large body of work in Economics 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], [Hart-Reny 12], Progress slow Challenge already with 2 items

5 Example 0: One Uniform Item z U({1,2}) Optimal Mechanism? Find price p maximizing: p Pr [z p]. p = 1 (p = 2 also works) Expected Revenue is 1

6 Example 1: Two Uniform Items z 1 U({1,2}) additive z 2 U({1,2}) Optimal Mechanism? Van Gogh doesn t affect value for Picasso, and vice versa Value for each painting drawn independently. No interaction whatsoever between items. So sell each separately at price 1. Make revenue of 1 per item. 2 total. Not optimal: Set price of 3 for bundle of both paintings. Sells with probability ¾. Expected revenue 9/4 > 2. \ Bundling may be necessary

7 Example 2: Two Non-IID Uniform Items z 1 U({1,2}) (z 1, z 2 ) correlated distribution z 2 U({1,3}) This item with Optimal Mechanism? probability ½ [Daskalakis-Deckelbaum-Tzamos 14]: Unique optimal mechanism offers following menu: \ Randomization may be necessary $4 $2.50 [Briest-Chawla-Kleinberg-Weinberg 10, Hart-Nisan 13]: If values are correlated, gap between deterministic and randomized can be infinite.

8 Example 3: Two Beta Distributions f G z G z G I 1 z G I f I z I z I I 1 z I J [D-Deckelbaum-Tzamos EC 13]: The optimal mechanism offers un-countably many randomized bundles. \ Menu representation not always a good idea

9 Example 4: Non-monotonicity z G D vs. F z: F z D( z) Optimal Mechanism? Rev(F) Rev(D) always. Proof: For a fixed price mechanism, proof immediate by stochastic dominance.

10 Example 4: Non-monotonicity z G D vs. F z: F z D( z) z I D vs. F Optimal Mechanism? Rev(F F) Rev(D D) always? [Hart-Reny 12]: No! Strictly better markets may yield strictly less revenue.

11 Example 5: Several IID Items z 1 U([c,c+1]) z n U([c,c+1]) Optimal mechanism? [Daskalakis-Deckelbaum-Tzamos 15]: - For all n, exists large enough c 0 such that if c c 0 then offering all items in one bundle is optimal. - extends [Pavlov 11] from n=2 to arbitrary n - For all c, exists large enough n 0 such that if n n 0 then offering all items in one bundle is not optimal.

12 Multi-Item Mechanisms (summary) Optimal Multi-Item Mechanism may involve bundling. may require randomization. may satisfy unintuitive properties such as non-monotonicity. [Alaei et al 12, Cai-Daskalakis-Weinberg 12-14]: Algorithms for computing multi-item multi-bidder auctions in exceedingly more general settings. Unclear how to extend single-item characterization Unclear if there is a one-fits-all multi-item mechanism Algorithms Analytical Characterizations optimal mechanism computation on an instance-to-instance basis universal claims about optimal mechanism s structure (a la Myerson)

13 Characterization Front Optimal Multi-Item Mechanism may involve bundling. may require randomization. may satisfy unintuitive properties such as non-monotonicity. Unclear how to extend single-item characterization Unclear if there is a one-fits-all multi-item mechanism Main Analytical Difficulty: Lack of tight revenue benchmark: given description of the setting, how much revenue to expect? e.g. E V z V is a terrible benchmark No tight virtual welfare benchmark even given a conjectured mechanism, it is unclear how to certify its optimality z F 1 n

14 The Menu Multi-Item mechanism Background Results Proof Ideas

15 The Menu Multi-Item mechanism Background Results Proof Ideas

16 1 seller, 1 buyer, n items Setting, Notation Seller: wants to optimize revenue from selling items Buyer is: additive, i.e. characterized by vector z = z 1,, z n R ] s.t. v T = z V V ` linear, i.e. paying price p for lottery q [0,1] n gives her utility: z q p Bayesian assumption: z F bidder s type How to optimize revenue if only know F? F: differentiable w/ bounded partial derivatives Main Handicap: lack of tight revenue benchmark (a la virtual welfare) even given a conjectured mechanism, it is unclear how to certify its optimality

17 Certifying Optimality of Solutions Two Generic Approaches: Concave Optimization, first order conditions Single-item mechanisms [Myerson 81] [Rochet- Chone 98] Duality, complementary slackness with Alan Deckelbaum and Christos Tzamos

18 Results Theorem: Finding the optimal mechanism has a tight dual optimal transportation flow problem. Max mechanisms = Min flows What s the point? Every optimal mechanism has a certificate of optimality in the form of a transportation flow Complementary slackness gives certificates of optimality, and guides us to identify optimal mechanism Corollary: Characterization of optimal multi-item mechanisms. Example: Grand-bundle pricing is optimal Û Two stochastic dominance conditions hold between measures derived from F

19 How to Optimize over Mechanisms? Standard Approach: Optimize over the allocation and price rule of the mechanism: - For each type z R ] maintain as variables: the probabilities that each item is allocated to the bidder if his realized type is z: x G z,, x ] z the price paid by z: p(z) - GOAL: find optimal functions x, p: sup h,i p z df(z) s.t. z x z p z z x z l p z l, z, z - Issue: (x, p) is multi-dimensional object in both input and output Our Approach: Indirect---optimize w.r.t. the utility function induced by optimal mechanism

20 How to Optimize over Mechanisms? - Given a mechanism, every type z R ] enjoys some utility u(z) -[Rochet 85]: Function u is induced by mechanism iff: u(z): u z z u(z): price paid by type z - Expected revenue : Z max u primal u(z): 1-Lipschitz, convex, non-decreasing, non-negative subject to: allocation probabilities to type z (exists a.e.) (ru(z) z u(z))f(z)dz u : convex, continuous, non-negative ru(z) 2 [0, 1] n almost everywhere

21 Massaging the Objective X Integration by Parts + Divergence Thm Expected Revenue = u(z)f(z)(z ˆn)dz Z X u(z)(rf(z) z +(n + 1)f(z))dz outer unit normal field to boundary

22 Massaging the Objective X Integration by Parts + Divergence Thm Expected Revenue = u(z)f(z)(z ˆn)dz Z Riesz Representation u(z)(rf(z) z +(n Thm+ 1)f(z))dz X µ is signed measure

23 Optimal Mechanism Design signed measure of total mass 0 derived from type distribution F Want to pick u so that expected revenue is as high as possible. want u( ) large under constraints: want u( ) small

24 Example 1 item, 1 buyer with value z U 0,1 μ = max Z X udµ

25 Example 2 2 items, 1 buyer with values z U 0,1 I μ = max Z X udµ

26 Dual problem? max Z X udµ Warmup: [DDT 13], [GK 14] s.t. want u( ) large want u( ) small Monge-Kantorovich Duality (continuous analog of mincost matching duality) µ - µ +

27 Example Optimal Transport (1-d) pay traveled distance μ r 0-cost moves μ s Optimal transport of μ r to μ s when cost x, y = max (0, x y)?

28 Dual problem? max Z X udµ [DDT 13], [GK 14] s.t. want u( ) large want u( ) small Monge-Kantorovich Duality (continuous analog of mincost matching duality) µ - µ +

29 Actual Dual [DDT 15] s.t. max Z X udµ home-brewed version of Monge- Kantorovich Duality Same as before except: without incurring any cost can choose any μ l such that: µ 0 cvx µ minimize transportation cost from μ l r to μ l s, where μ l r μl s = μ want u( ) large µ + u dμ l u dμ non-decreasing convex u want u( ) small µ -

30 visualizing µ 0 cvx µ µ can be obtained from µ via a sequence of moves: (center of mass weakly increases) +1 similar to sweeping operation of [Rochet-Chone 98] and the complements of these moves (contracting negatives and shifting them down)

31 Actual Dual [DDT 15] s.t. max Z X udµ home-brewed version of Monge- Kantorovich Duality Same as before except: without incurring any cost can choose any μ l such that: µ 0 cvx µ minimize transportation cost from μ r l to μ s l, where μ r l μ s l = μ want u( ) large µ + want u( ) small µ -

32 Example Optimal Transport (1-d) μ r μ s 0 cost, since μ s }~h μ r Given μ r and μ s, choose any μ l r μl s cvx μ r μ s so as to minimize transport cost from μ r to μl s when cost x, y = max (0, x y)?

33 Strong Duality [DDT 15] = inf µ 0 cvx µ 1 = µ 0 +, 2 = µ 0 Z X X (x y) + 1 d (x, y) s.t. So what s the point? Every optimal mechanism has a certificate proving optimality. Complementary slackness conditions certify the optimality of a primaldual pair (u, (µ, γ) ) of solutions, and guide us to find optimal mechanisms.

34 2 Uniform Items (no convex shuffling needed)

35 Strong Duality [DDT 15] = inf µ 0 cvx µ 1 = µ 0 +, 2 = µ 0 Z X X (x y) + 1 d (x, y) s.t. So what s the point? Every optimal mechanism has a certificate proving optimality. Complementary slackness conditions certify the optimality of a primal-dual pair (u, (µ, γ) ) of solutions, and guide us to find optimal mechanisms. Important Corollary: Characterization of Optimal Mechanisms

36 Corollary: Characterizing Optimal Mechanisms E.g. when is selling the grand bundle at price p * optimal? receive all goods pay p * B T Receive no goods, utility 0 Hyperplane with intercept p * Pricing grand bundle at p * is optimal µ B 4 cvx µ T

37 Example 5: Several IID Items z 1 U([c,c+1]) z n U([c,c+1]) Optimal mechanism? [Daskalakis-Deckelbaum-Tzamos 15]: - For all n, exists large enough c 0 such that if c c 0 then offering all items in one bundle is optimal. - extends [Pavlov 11] from n=2 to arbitrary n - For all c, exists large enough n 0 such that if n n 0 then offering all items in one bundle is not optimal.

38 Characterizing Arbitrary Mechanisms - Each mechanism partitions type space into regions, corresponding to what lottery each type will get. Theorem: Mechanism is optimal iff μ r and μ s satisfy one convex dominance relation per region.

39 Structural Understanding of Mechanisms online marketplaces [D-Deckelbaum-Tzamos 15]: One additive bidder multi-item mechanisms online advertising sponsored search spectrum auctions [Laffont-Maskin-Rochet 87], [McAfee-McMillan 88], [Myerson 81] [Wilson 93], [Armstrong 96], [Rochet-Chone 98], [Armstrong 99],[Zheng 00], [Basov 01], [Kazumori 01], [Thanassoulis 04],[Vincent-Manelli 06, 07], [Pavlov 11], [Hart-Nisan 12],

40 Summary I presented an analytical framework for obtaining closed-form descriptions of revenue-optimal mechanisms using optimal transport theory. Our analytical framework characterizes single-bidder, multi-item mechanisms. Beyond single-bidder settings? additive bidders? Match success of efficient algorithms [Cai-Daskalakis-Weinberg 12-14] Further Reading: My recent survey at SIGECOM exchanges (July 2015 volume) entitled Multi-Item Auctions Defying Intuition? Forthcoming Econometrica Paper: Strong Duality for a Multiple Good Monopolist Thanks for listening! Questions?

41 Example *: Two Exponential Items z 1 F 1 f 1 =Exp( 1 ) z 2 F f 2 =Exp( 2 ) Optimal mechanism [D-Deckelbaum-Tzamos EC 13]: This item with probability 2 / 1 $p 2/ 1 $2/ 1

42 Example *: Two exponential items (revisited) 1 2 f 2 =Exp( 2 ) 2 2 p o get grand bundle pay p get nothing, pay nothing p f 1 =Exp( 1 ) get item 1 with prob. 1, item 2 with prob. 2/ 1 pay 2/ 1 [D-Deckelbaum-Tzamos EC 13]

43 Complementary Slackness of (u, (μ, γ) ) = inf µ 0 cvx µ 1 = µ 0 +, 2 = µ 0 Z X X (x y) + 1 d (x, y) µ µ : x +1 0 y +1 y i >x i =) (ru(x)) i =(ru(y)) i =0 +1 i.e. types x and y get item i with probability 0 γ : µ + (x, y) > 0 =) (x i >y i =) (ru(x)) i =(ru(y)) i = 1) µ - i.e. types x and y get item i with probability 1

44 Algorithms for Multi-Item Mechanisms [Cai-Daskalakis-Weinberg 12-14]: Algorithms for computing multi-item multi-bidder auctions. Input: m bidders, n items, allocation constraints for each bidder i, a multi-dimensional distribution F i from which bidder i s valuation function v i : 2 []] R is sampled F i : finite (or discretized) Output: The description of a revenue-optimal auction Resulting mechanism is a virtual welfare maximizer: Bidders submit types t 1, t 2,, t m These are transformed into virtual types t 1, t 2,, t m Mechanism chooses allocation that maximizes virtual welfare Mappings t V t V are: randomized (as they have to), output by algorithm, so no good understanding of their structure