Price Customization and Targeting in Many-to-Many Matching Markets

Transcription

1 Price Customization and Targeting in Many-to-Many Matching Markets Renato Gomes Alessandro Pavan February 2, 2018

2 Motivation Mediated (many-to-many) matching ad exchanges B2B platforms Media platforms

3 Targeting and Price Customization Technological progress: targeting Concerns that accompanying price customization detrimental to consumer surplus Proposal: uniform-pricing obligations Policy debate lacks formal framework third-degree + second-degree price discrimination Unclear how firms will respond to uniform-pricing obligations More generally, distortions in matching mkts not well understood

4 This Paper Tractable, yer rich model, of mediated many-to-many matching third-degree price discrimination second degree price discrimination Menus of Matching Plans baseline configuration baseline price customizing tariffs Targeting (structural elasticities) Mkt power distortions Uniform-pricing regulations Centralized vs decentralized markets

5 Related literature One-to-One Matching Design: Damiano and Li (2007), Johnson (2013)... Many-to-Many Matching Design: Gomes and Pavan (2016), Jeon, Kim and Menicucci (2017), Fershtman and Pavan (2016)... Two-Sided Markets: Rochet and Tirole (2006), Armstrong (2006), Weyl (2010), Jullien and Pavan (2017), Rysman (2009), Tan and Zhou (2017)... Price Discrimination: Mussa and Rosen (1978), Myerson (1981), Maskin and Riley (1984), Armstrong (1996), Wilson (1998), Aguirre et al. (2010), Bergemann, Brooks, and Morris (2015)... Bundling: Armstrong (2013), Hart and Reny (2015)... Targeting: Bergemann and Bonatti (2011)...

6 Plan Model Customized tariffs Uniform-pricing regulations Targeting Welfare under customized and uniform pricing Centralized vs decentralized markets

7 Model Monopolistic platform Two sides k {a,b} Each side: unit-mass continuum of agents i [0,1] Type of agent i from side k: θk i = (v k i,xi k ) vk i V k = [v k,v k ] : vertical dimension x k i [0,1] : horizontal dimension (location) Each θ i k drawn independently from cdf F k with support Θ V k [0,1]

8 Model Type profile θ (θ i k )i [0,1] k=a,b Utility of agent i from side k from being matched to agent j from side l k: u k (v i k, xi k xj l ) increasing in v i k decreasing in x i k xj l Total utility from being matched, at a price p, to set s Θ l ( πk i (s,p;θ) = u k vk i, xi k xj l )df l (θ l ) p. s

9 Model

10 Examples Example (ad exchange) Platform matches advertisers (side a) with publishers (side b). Expected payoff u a (v a, x a x b ) = v a φ ( x a x b ) of advertiser of type θ a = (v a,x a ) from impression at publisher of type θ b = (v b,x b ), where v a : markup φ: conversion probability Publishers indifferent wrt type of advertisement displayed (first approximation) Heterogeneity in publishers payoffs reflects differences in opportunity costs u b (x b, x a x b ) = v b 0

11 Examples Example (media platform) Media outlet (e.g., cable TV provider) matching viewers (side a) with content providers (side b). Utility viewer of type θ a = (v a,x a ) derives from content provider of type θ b = (v b,x b ): [ ] u a (v a, x a x b ) = α (v a ) δ + (1 α) φ ( x a x b ) δ 1 δ Providers indifferent wrt type of viewers. Providers payoff u b (v b, x b x a ) = v b may be positive (advertising) or negative (royalties)

12 Information Vertical types are private information Horizontal types (i.e., locations) Scenario (i): Locations public on both sides Scenario (ii): Locations private on side a, public on side b Scenario (iii): Locations public on side a, private on side b Scenario (iv): locations private on both sides.

13 Reciprocity Given tariff T k, matching demand of type θ k = (v k,x k ) given by matching set { } ŝ k (θ k ;T k ) arg max u k (v k, x k x l )df l (θ l ) T k (s k ). s k Θ l Θ l Definition Tariffs T k, k = a,b, are feasible if, for all (θ k,θ l ) Θ k Θ l, k,l {a,b}, l k, θ l ŝ k (θ k ;T k ) θ k ŝ l (θ l ;T l ).

14 Customized tariffs Definition Tariff T k is customized if there exists collection of matching plans {(s k (x k ),T k (x k ),ρ k ( ;x k ),S k (x k )) : x k [0,1]}, s k (x k ): baseline configuration T k (x k ): baseline price S k (x k ): set of possible customizations ρ k ( ;x k ): customizing price schedules s.t. each side-k agent selecting plan x k and customization s k S k (x k ) is charged 1 T k (x k ) + ρ k (q xl (s k ) x l ;x k )dx l, 0 where q xl (s k ) is total mass of x l -agents under customization s k.

15 Conditions Condition [R] Regularity: Virtual values ϕ k (θ k,θ l ) u k (v k, x k x l ) 1 F v x k (v k x k ) f v x u k k (v k x k ) v (v k, x k x l ) continuous and non-decreasing in v k. Condition [VD] Virtual values decreasing in distance: ϕ k (θ k,(v l,x l )) non-increasing in x k x l Condition [I k ] Independence on side k: F k (θ k ) = F x k (x k)f v k (v k) Condition [S k ] Symmetry on side k: F k (θ k ) = x k F v k (v k)

16 Profit-maximizing tariffs Proposition In addition to Condition R, suppose one of following holds: Scenario (i) Scenario (ii) along with Conditions VD, I a and S b Scenario (iii) along with Conditions VD, S a and I b Scenario (iv) along with Conditions VD, S a and S b. Then, for k {a,b}: profit-maximizing tariff Tk is customized; matching sets s k exhibit negative assortativeness at the margin: there exist functions tk : Θ k [0,1] V l s.t. s k (θ k) = {(v l,x l ) Θ l : v l > t k (θ k,x l )}

17 Negative assortativeness at the margin

18 Optimal Threshold Let k (θ k,θ l ) ϕ k (θ k,θ l ) + ϕ l (θ l,θ k ) denote marginal effect of linking θ k and θ l. When interior, t k (θ k,x l ) unique solution to k (θ k,(t k (θ k,x l ),x l )) = 0 Conditions in proposition bunching only for capacity reasons (corners) only binding IC constrains are those wrt vertical dimensions

19 Demands For simplicity, locations public on both sides Mass of agents located at x l demanded by θ k : ˆq xl (θ k ) arg max q [0,f x l (x l )] {u k (v k, x k x l ) q ρ k (q x l ;x k )}. Demand for q-th unit of x l -agents by x k -agents, at marginal price dρ k dq : ( ) ( ( ) dρk D k dq x l;x k = 1 F v x dρk k ˆv xl dq x k ) x k ( where ˆv dρk xl dq x k Elasticity: ) is unique solution to u k (v k, x k x l ) = dρ k dq. ( ) ( ) dρk D dρk k ε k dq x dq x l,x k l;x k ( ) dρk dq dρ k dq D k ( dρk dq x l,x k ).

20 Lerner-Wilson formula Proposition Profit-maximizing schedules satisfy following ( structural conditions for all dρ ) (x a,x b ), all (q a,q b ) such that q a = D b b dq (q b x a ;x b ) x a ;x b and q b = D a ( dρ a dq (q a x b ;x a ) x b ;x a ): dρa dq (q 1 a x b ;x a ) 1 ( ) ε dρ a a dq (q a x b ;x a ) x b ;x a }{{} net effect on side-a profits + dρ b dq (q 1 b x a ;x b ) 1 ( dρ ) = 0. ε b b dq (q b x a ;x b ) x a ;x b }{{} net effect on side-b profits

21 Lerner-Wilson formula

22 Distortions and Horizontal Differentiation Let W = u k (v k, x k x l )df j (θ j )df k (θ k ) k=a,b Θ k ŝ k (θ k ;T k ) Tariffs T e k, k = a,b, maximize welfare iff for all θ k = (v k,x k ) and θ l = (v l,x l ), θ l ŝ k (θ k ;T e k ) u k (v k, x k x l ) + u l (v l, x k x l ) 0.

23 Distortions and Horizontal Differentiation Definition Distortions on side k {a, b} decrease (alternatively, increase) with distance iff u l (t k (θ k,x l ), x l x k ) u l (t e k (θ k,x l ), x l x k ) decreases (alternatively, increases) with x k x l.

24 Distortions and Horizontal Differentiation Proposition Suppose Conditions VD and I k hold, k = a,b. 1 If u k is submodular, u l is submodular and concave in v l, and Fl v increasing hazard rate, then distortions on side k decrease with distance has 2 If u k is supermodular, u l is supermodular and convex in v l, and F v l has decreasing hazard rate, then distortions on side k increase with distance

25 Distortions and Horizontal Differentiation

26 Distortions and Horizontal Differentiation

27 Uniform Pricing Growing support for uniform-pricing regulations (both US and EU) Definition Tariff T k consistent with uniform pricing if there exists collection of price schedules p k : [0,1] 2 R s.t. 1 T k (s k ) = p k (q xl (s k ) x l )dx l 0 where q xl (s k ) is mass of x l -agents in the set s k. Key: p k (q x l ) do not depend on side-k agents own profiles!

28 Uniform Pricing Aggregate demands under uniform-pricing ( ) dpa D a dq x b Elasticity of aggregate demand ( ) dpa ε a dq x b 1 0 ( ) dpa D a dq x b;x a dx a D a ( dpa ( dpa dq dq x b ) ) dp a dq D a ( dpa dq x b ( ) Let H x a x b, dp a dq be distribution over X a = [0,1] with density ( h x a x b, dp ) a dq ( ) D dpa a dq x b;x a ) ( dpa dq D ( ) dpa a dq x b ) ( dpa dq )

29 Uniform Pricing Proposition Under uniform pricing on side a, profit-maximizing price schedules satisfy dpa u dq (q x 1 b) 1 ( ) ε dp u a a dq (q x b) }{{} net effect on side-a profits +E ) dρb u H ( x a x b, dpa dq dq ( x 1 a) 1 ( dρ u ) ε b b dq ( x a) x a ;x = 0, b }{{} net effect on side-b profits where dρb u ( ( ) dq ( x a) = dρu b dp u D a a dq dq (q x b) x b ;x a ) x a ;x b

30 Uniform Pricing Mechanism design with novel constraint on implementing tariffs Under uniform pricing, types θ a and θ b matched iff [ ) )] ϕa ((ˆvxb (u a (v a, x b x a ) x a ), x a,θb E H( xa x b,u a (v a, x b x a )) +E H( xa x b,u a (v a, x b x a )) [ ϕb ( θb, (ˆv xb (u a (v a, x b x a ) x a ), x a ))] 0 ( ) where ˆv dρk xl dq x k is unique solution to u k (v k, x k x l ) = dρ k dq.

31 Targeting Definition (targeting) Customized pricing (on both sides) leads to more targeting than uniform pricing (on side a) if, for each θ a = (v a,x a ), there exists d(θ a ) (0, 1 2 ) such that { ta(θ a,x b ) ta u < 0 if xa x (θ a,x b ) b < d(θ a ) > 0 if x a x b > d(θ a ). Conversely, uniform pricing on side a leads to more targeting than customized pricing on both sides if, for each θ a = (v a,x a ), there exists d(θ a ) (0, 1 2 ) such that { ta(θ a,x b ) ta u > 0 if xa x (θ a,x b ) b < d(θ a ) < 0 if x a x b > d(θ a ).

32 Targeting Threshold function t a(θ a,x b ) under customized pricing (black solid curve) and uniform pricing t u a (θ a,x b ) (dashed blue curve) when customized pricing leads to more targeting than uniform pricing

33 Targeting Proposition (comparison: targeting) Suppose Condition VD and I a hold. 1. If u a is submodular and concave in v a, F v a has increasing hazard rate, and, for any θ b, ϕ b (θ b,θ a ) invariant in x b x a, then uniform pricing (on side a) leads to more targeting than customized pricing (on both sides). 2. If u a is supermodular and convex in v a, and F v a has a decreasing hazard rate, then customized pricing (on both sides) leads to more targeting than uniform pricing (on side a).

34 Uniform Targeting: Welfare Let ( ) ( ) dpa 2 D dp u a a CD a dq x dq x b;x a b;x a = ( dp u a dq dpu a dq ) ( 2 dp u ) D a a dq x b;x a dp u ) ( a dq denote convexity of x a -agents demand for q-th unit of x b -agents wrt dp a dq. Condition [IR] Increasing Ratio: For any (x a,x b ) [0,1] 2, any q [0,f x b (x b)], nondecreasing in dp a dq. ( ) dpa z a dq x b;x a dp a dq 2 CD a ( dpa dq x b;x a )

35 Proposition (comparison: welfare) 1. Suppose ( targeting ) is higher under uniform pricing and that convexity CD dpa a dq x b;x a of x a -agents demands declines with x a x b. Then welfare of side-a agents higher under uniform pricing. 2. Suppose targeting ( is higher ) under customized pricing and that convexity CD dpa a dq x b;x a of x a -agents demands increases with x a x b. Then welfare of side-a agents higher under uniform pricing. Targeting + monotonicity of convexity of demands in distance permits one to identify which matches correspond to strong mkt in the sense of third-degree price discrimination literature (e.g., Aguirre et al. (2010)).

36 Centralized vs Decentralized Markets Many mkts are transiting from centralized to decentralized structure (sellers post prices) E.g., Cable TV Proposition (centralized vs decentralized markets) Suppose locations are private on side a (buyers) and public on side b (sellers) and Conditions VD, I a, S b, and IR hold. 1. Suppose ( targeting ) is higher under uniform pricing and convexity CD dpa a dq x b;x a of x a -agents demands declines with x a x b. Then welfare of side-a agents higher in decentralized market. 2. Suppose ( targeting ) is higher under customized pricing and convexity CD dpa a dq x b;x a of x a -agents demands increases with x a x b. Then, again, welfare of side-a agents higher in decentralized market.

37 Conclusions Tractable model of mediated many-to-many matching vertically and horizontally differentiated preferences unrestricted pricing algorithms Customized Tariffs matching plans with plan-specific customizing price schedules Testable (structural) predictions for profit-maximizing schedules Lerner-Wilson elasticity formulas (for matching) Primitive conditions for distortions to be monotone in targeting Uniform-pricing obligations Transition to decentralized structure Applications:... 1 ad exchanges 2 cable TV

38 THANKS!