Coarse Matching and Price Discrimination H. Hoppe, B. Moldovanu, and E. Ozdenoren Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
introduction Research Question Two kinds of agents (“men”, “women”) look for a match. An intermediary can extract transfers and match agents based on their reported types. ▶ randomly? ▶ coarsely? ▶ assortatively? RQ: How good is coarse matching with two categories for each kind of agent, relative to efficient matching or random matching? ▶ total surplus ▶ agents’ utility ▶ matchmaker’s revenue Look for lower bounds . Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
introduction Motivation Extend McAfee (2002) ▶ Obtain lower bounds on surplus in more environments. ▶ Private types mean that the matching must be incentive compatible. Authors’ motivation: If coarse matching is “pretty good” in the worst case, then (unmodeled) costs of using a finer scheme may offset the benefits. Why do firms offer a “small” menu of qualities? ▶ One reason: a price-discriminating monopolist can get “close” to maximum revenue with two quality levels. Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Model Model Men: x ∼ F ( x ) on [0 , 휏 F ]. Women: y ∼ G ( y ) on [0 , 휏 G ]. ▶ Assume densities f ( x ) , g ( y ) > 0, and measure 1 of each type. ▶ x , y private information. Intermediary chooses ▶ Matching rule 휙 : [0 , 휏 F ] ⇉ [0 , 휏 G ] That is, 휙 ( x ) ⊆ [0 , 휏 G ]. ▶ Price schedules p m : [0 , 휏 F ] → ℝ , p w : [0 , 휏 G ] → ℝ . ▶ Implicitly restricts attention to direct mechanisms. Surplus: ▶ Total surplus xy . ▶ Fixed sharing rule 훼 ∈ [0 , 1]. ▶ If man x and woman y match, man gets 훼 xy and woman gets (1 − 훼 ) xy before transfers to the intermediary IR : Agents who do not use the intermediary are matched to each other randomly. (Q: what happens to deviators when everyone uses the intermediary?) Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Model What do matchings look like? Let 휈 M ( A ) be the measure of men announcing types in A . Define 휈 W ( . ) similarly. A matching 휙 is feasible if 휈 M ( A ) = 휈 W ( 휙 ( A )) for all (measurable) A ⊆ [0 , 휏 G ] Damiano and Li (2005): Incentive-compatible and feasible matchings partition each group into n bins, match the bins assortatively, and match randomly within bins. Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Assortative Matching Assortative Matching Match each man x with the woman 휓 ( x ), where 휓 ( x ) solves F ( x ) = G ( 휓 ( x )) Incentive compatibility: x ∈ arg max 훼 x 휓 (ˆ x ) − p m (ˆ x ) (1) x ˆ (1 − 훼 ) 휓 − 1 (ˆ y ∈ arg max y )( y ) − p w (ˆ y ) (2) ˆ y Take FOC: (Can plug in solution and show SOC holds) 훼 x 휓 ′ ( x ) − ∂ p m ( x ) = 0 ∂ x Lowest pair generates 0 surplus. Hence p m (0) = 0. Therefore, ∫ x p m ( x ) = 훼 z 휓 ′ ( z ) dz (3) 0 Similarly, letting 휑 = 휓 − 1 , ∫ y (1 − 훼 ) z 휑 ′ ( z ) dz p w ( y ) = (4) 0 Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Assortative Matching Stability Another solution concept in matching models is stability . In this setting: pick a matching rule 휙 ( . ), and let man x get surplus 훿 ( x ), and woman y get surplus 휌 ( y ). The sharing rule is called “stable” if ∀ x , 훿 ( x ) + 휌 ( 휙 ( x )) = x 휙 ( x ) (5) ∀ x , y , 훿 ( x ) + 휌 ( y ) ≥ xy (6) Typically, stability means IR and no blocking pairs . Here, stable sharing implies stability. Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Assortative Matching Stability and efficiency Claim: the stable matching must be assortative. First, show a stable matching must be monotone increasing. Take x ′ > x , y ′ > y , and for a contradiction, suppose a stable sharing rule assigns x ↔ y ′ and x ′ ↔ y . 훿 ( x ) + 휌 ( y ′ ) = xy ′ 훿 ( x ′ ) + 휌 ( y ) = x ′ y ⇒ 훿 ( x ) + 휌 ( y ) + 훿 ( x ′ ) + 휌 ( y ′ ) = x ′ y + xy ′ = xy + x ′ y ′ ≤ x ′ y + xy ′ Hence ( x ′ − x )( y ′ − y ) ≤ 0 If x ∕↔ 휓 ( x ), then wlog say x ↔ y > 휓 ( x ). Then, since g ( . ) > 0, we have G ( y ) = G ( 휓 ( x )) + 휖 for some 휖 > 0. If the matching is monotone increasing, then 휈 m ([ x , 휏 F ]) ≥ 휈 w ( 휙 ([ x , 휏 F ])) + 휖 . To summarize, if a matching rule is stable, either it is infeasible or it is assortative. Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Assortative Matching Stable shares Differentiate 훿 ( x ) + 휌 ( 휓 ( x )) = x 휓 ( x ). Obtain 훿 ′ ( x ) + 휌 ′ ( 휓 ( x )) 휓 ′ ( x ) = 휓 ( x ) + x 휓 ′ ( x ) Matching coefficients, 훿 ′ ( x ) = 휓 ( x ), and 휌 ′ ( 훿 ( x )) = x . We know that 훿 (0) = 휌 (0) = 0. Hence, by the FTC the stable shares are ∫ x 훿 ( x ) = 휓 ( z ) dz (7) 0 ∫ y 휌 ( y ) = 휑 ( z ) dz (8) 0 Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Assortative Matching Connection to assortative matching Proposition The IC price schedules satisfy p m ( x ) = 훼휌 ( 휓 ( x )) (9) p w ( y ) = (1 − 훼 ) 훿 ( 휑 ( y )) (10) The net utilities of x and y are 훼훿 ( x ) and (1 − 훼 ) 휌 ( y ) . The intermediary’s revenue satisfies min( 훼, 1 − 훼 ) x 휓 ( x ) ≤ p m ( x ) + p w ( 휓 ( x )) ≤ max( 훼, 1 − 훼 ) x 휓 ( x ) Hence the intermediary extracts half the total surplus if 훼 = 1 / 2 . Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Assortative Matching Proof of prop 1-1 Total surplus from a pair is u ( x , 휓 ( x )) = x 휓 ( x ). Totally differentiate: d dx u ( x , 휓 ( x )) = 휓 ( x ) + x 휓 ′ ( x ) ∫ x ∫ x By the FTC, u ( x , 휓 ( x )) = 0 휓 ( z ) dz + 0 z 휓 ′ ( z ) dz . Hence ∫ x ∫ x z 휓 ′ ( z ) dz 훼 u ( x , 휓 ( x )) = 훼 휓 ( z ) dz + 훼 0 0 � �� � � �� � 훼훿 ( x ) p m ( x ) Also, using a change of variables w = 휓 ( z ), we have ∫ x ∫ 휓 ( x ) 1 0 z 휓 ′ ( z ) dz = 훼 p m ( x ) = 휑 ( w ) dw = 휌 ( 휓 ( x )). 0 Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Assortative Matching Total Revenue and Surplus ∫ x Man x pays P m ( x ) = 0 훼 z 휓 ′ ( z ) dz . After some computation (see appendix), write R a 훼 = R a men + R a women , where ∫ 휏 F [ ] x − 1 − F ( x ) r a men = 훼 휓 ( x ) f ( x ) dx f ( x ) 0 ∫ 휏 F [ ] 휓 ( x ) − 휓 ′ ( x )1 − F ( x ) R a women = (1 − 훼 ) x f ( x ) dx f ( x ) 0 Additionally, the total surplus is ∫ 휏 F U a = 피 ( x 휓 ( x )) = x 휓 ( x ) dF ( x ) 0 Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Coarse Matching Why Coarse Matching? Perfect (Assortative) matching incurs various transaction costs: ▶ Intermediary: communication (decoding) cost ▶ Agents: evaluation (coding) cost Agents only need to reveal partial information. In terms of total surplus, the intermediary’s revenue and agents’s welfare: ▶ It is significantly higher than completely random matching. ▶ It may achieve a large proportion of assortative matching. Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
Coarse Matching Coarse Matching Model ∫ ˆ ∫ 휏 G y ˆ xy xy ˆ y ) dG ( y ) − p c 훼 y ) dG ( y ) = 훼 (11) m G (ˆ 1 − G (ˆ 0 y ˆ ∫ ˆ ∫ 휏 F x x ˆ y x ˆ y x ) dG ( y ) − p c (1 − 훼 ) x ) dF ( x ) = (1 − 훼 ) (12) w F (ˆ 1 − F (ˆ 0 ˆ x y = 휓 (ˆ ˆ x ) (13) two classes: willing to pay and not willing to pay y ) the lowest type of men (women) who is willing to pay p c m ( p c x (ˆ ˆ w ). such pricing scheme is incentive compatible. Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, Ap H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price Discrimination / 32
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