Coarse Matching and Price Discrimination
H. Hoppe, B. Moldovanu, and E. Ozdenoren
Slides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 1
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introduction
Research Question
Two kinds of agents (“men”, “women”) look for a match.
An intermediary can extract transfers and match agents based on theirreported 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.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 2
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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.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 3
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Model
Model
Men: x ∼ F (x) on [0, �F ]. Women: y ∼ G (y) on [0, �G ].▶ Assume densities f (x), g(y) > 0, and measure 1 of each type.▶ x , y private information.
Intermediary chooses▶ Matching rule � : [0, �F ] ⇉ [0, �G ] That is, �(x) ⊆ [0, �G ].▶ Price schedules pm : [0, �F ]→ ℝ, pw : [0, �G ]→ ℝ.▶ Implicitly restricts attention to direct mechanisms.
Surplus:▶ Total surplus xy .▶ Fixed sharing rule � ∈ [0, 1].▶ If man x and woman y match, man gets �xy and woman gets (1− �)xy
before transfers to the intermediary
IR: Agents who do not use the intermediary are matched to each otherrandomly. (Q: what happens to deviators when everyone uses theintermediary?)
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 4
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Model
What do matchings look like?
Let �M(A) be the measure of men announcing types in A. Define �W (.)similarly.
A matching � is feasible if �M(A) = �W (�(A)) for all (measurable)A ⊆ [0, �G ]
Damiano and Li (2005): Incentive-compatible and feasible matchingspartition each group into n bins, match the bins assortatively, and matchrandomly within bins.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 5
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Assortative Matching
Assortative MatchingMatch each man x with the woman (x), where (x) solves
F (x) = G ( (x))
Incentive compatibility:
x ∈ arg maxx
�x (x)− pm(x) (1)
y ∈ arg maxy
(1− �) −1(y)(y)− pw (y) (2)
Take FOC: (Can plug in solution and show SOC holds)
�x ′(x)− ∂pm(x)
∂x= 0
Lowest pair generates 0 surplus. Hence pm(0) = 0. Therefore,
pm(x) =
∫ x
0
�z ′(z)dz (3)
Similarly, letting ' = −1,
pw (y) =
∫ y
0
(1− �)z'′(z)dz (4)
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 6
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Assortative Matching
Stability
Another solution concept in matching models is stability. In this setting: pick amatching rule �(.), and let man x get surplus �(x), and woman y get surplus�(y). The sharing rule is called “stable” if
∀x , �(x) + �(�(x)) = x�(x) (5)
∀x , y , �(x) + �(y) ≥ xy (6)
Typically, stability means IR and no blocking pairs. Here, stable sharing impliesstability.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 7
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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 .
�(x) + �(y ′) = xy ′
�(x ′) + �(y) = x ′y
=⇒ �(x) + �(y) + �(x ′) + �(y ′) = x ′y + xy ′
xy + x ′y ′ ≤ x ′y + xy ′
Hence (x ′ − x)(y ′ − y) ≤ 0
If x ∕↔ (x), then wlog say x ↔ y > (x). Then, since g(.) > 0, we haveG (y) = G ( (x)) + � for some � > 0. If the matching is monotone increasing,then �m([x , �F ]) ≥ �w (�([x , �F ])) + �.
To summarize, if a matching rule is stable, either it is infeasible or it is assortative.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 8
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Assortative Matching
Stable shares
Differentiate �(x) + �( (x)) = x (x). Obtain
�′(x) + �′( (x)) ′(x) = (x) + x ′(x)
Matching coefficients, �′(x) = (x), and �′(�(x)) = x .
We know that �(0) = �(0) = 0.
Hence, by the FTC the stable shares are
�(x) =
∫ x
0
(z)dz (7)
�(y) =
∫ y
0
'(z)dz (8)
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 9
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Assortative Matching
Connection to assortative matching
Proposition
The IC price schedules satisfy
pm(x) = ��( (x)) (9)
pw (y) = (1− �)�('(y)) (10)
The net utilities of x and y are ��(x) and (1− �)�(y).
The intermediary’s revenue satisfiesmin(�, 1− �)x (x) ≤ pm(x) + pw ( (x)) ≤ max(�, 1− �)x (x)
Hence the intermediary extracts half the total surplus if � = 1/2.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 10
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Assortative Matching
Proof of prop 1-1
Total surplus from a pair is u(x , (x)) = x (x). Totally differentiate:
d
dxu(x , (x)) = (x) + x ′(x)
By the FTC, u(x , (x)) =∫ x
0 (z)dz +
∫ x
0z ′(z)dz . Hence
�u(x , (x)) = �
∫ x
0
(z)dz︸ ︷︷ ︸��(x)
+�
∫ x
0
z ′(z)dz︸ ︷︷ ︸pm(x)
Also, using a change of variables w = (z), we have1�pm(x) =
∫ x
0z ′(z)dz =
∫ (x)
0'(w)dw = �( (x)).
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 11
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Assortative Matching
Total Revenue and Surplus
Man x pays Pm(x) =∫ x
0�z ′(z)dz . After some computation (see appendix),
write Ra� = Ra
men + Rawomen, where
r amen = �
∫ �F
0
(x)
[x − 1− F (x)
f (x)
]f (x)dx
Rawomen = (1− �)
∫ �F
0
x
[ (x)− ′(x)
1− F (x)
f (x)
]f (x)dx
Additionally, the total surplus is
Ua = E(x (x)) =
∫ �F
0
x (x)dF (x)
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 12
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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.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 13
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Coarse Matching
Coarse Matching Model
�
∫ y
0
xy
G (y)dG (y) = �
∫ �G
y
xy
1− G (y)dG (y)− pc
m (11)
(1− �)
∫ x
0
xy
F (x)dF (x) = (1− �)
∫ �F
x
xy
1− F (x)dG (y)− pc
w (12)
y = (x) (13)
two classes: willing to pay and not willing to pay
x(y) the lowest type of men (women) who is willing to pay pcm(pc
w ).
such pricing scheme is incentive compatible.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 14
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Coarse Matching
Coarse Matching Model
cutoff point: x = EX , and y = (Ex)
EXL = EX −∫ EX
0F (x)dx
F (EX ),EYL = (EX )−
∫ (EX )
0G (x)dx
G ( (EX ))
total surplus: UEX
=
∫ EX
0
∫ (EX )
0
xy
F (EX )dG (y)dF (x) +
∫ �F
EX
∫ �G
(EX )
xy
1− F (EX )dG (y)dF (x)
= EXEY +F (EX )
1− F (EX )(EX − EXL)(EY − EYL)
intermediary’s revenue (fix � = 1/2):
REX = [1− F (EX )]pcw + [1− G ( (EX ))]pc
m
=1
2[EX (EY − EYL) + (EX )(EX − EXL)]
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 15
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Coarse Matching
Total Surplus
Note definition 2:Ua = (1 + CCV 2(x , (x)))U r
If the distributions F and G both satisfy:1 F and G are both log-concave (F ,G DRFR)1
2 (1-F) and (1-G) are both log-concave (F ,G IFR)2
UEX ≥ Ua + U r
2⇒ UEX ≥ 3
4Ua,UEX ≥ U r
If F and G are both concave and is convex, then:
UEX ≥ 5
4U r
1decreasing reversed failure rate f (t)F (t)
2increasing failure (hazard) rate f (t)1−F (t)
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 16
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Coarse Matching
Total Surplus
Proof:
F (EX ) ≥ E (F (X )) = 1/2⇒ F (EX )
1− F (EX )≥ 1
Fconcave ⇒∫ EX
0F (x)dx
EXF (EX )≥ 1/2⇒ EX − EXL ≥
1
2EX
G concave and convex ⇒
EY − EYL = EY − (EX ) +
∫ (EX )
0G (x)dx
G ( (EX ))≥ EY − 1
2 (EX )
EYL ≤1
2 (EX ) ≤ 1
2E ( (X ))⇒ EY − EYL ≥
1
2EY
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 17
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Coarse Matching
Intermediary’s Revenue
If the distributions F and G both satisfy:1 F and G are both concave2 (1-F) and (1-G) are both log-concave (F ,G IFR)
REX ≥ 1
2Ra
Intuition: As the distribution of types on the other market side becomes moreconcave, the mass of potential partners with very low type gets larger,leading to a higher revenue since agents in high class are willing to pay more.
If EX ≥ EY and is convex, then REXm ≥ REX
w
Intuition: If F and G have the same mean, but G has a higher variance, thechances for men to match with a lower type of women are higher thanwomen, thus men in higher class are willing to pay more.
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 18
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Coarse Matching
Agents’ Welfare
W EX = UEX − REX , W a = Ua − Ra
Fix � = 1/2, if the distributions F and G both satisfy:1 F and G are both convex2 F and G are both log-concave (F ,G DRFR)
W EX ≥W a
Proof
REX =1
2[EX (EY − EYL) + (EX )(EX − EXL)]
≤ 1
2[EX (EY − 1
2 (EX )) + (EX )
1
2EX ] ≤ 1
2U r
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 19
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Coarse Matching
Agents’ Welfare
W EX = UEX − REX
≥ 1
2(Ua + U r )− 1
2U r =
1
2Ua = W a
However,
W r = U r =1
1 + CCV 2(x , (x))
≥ 1
2Ua = W a
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 20
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Price Discrimination
Application
Lemma
is convex (concave) then Ra1 ≥ (≤)Ra
1/2. If the asssortative matching function
is convex (concave) and if EX ≥ (≤)EY then REX1 ≥ (≤)REX
1/2.
Proposition
Let F be IFR, G be IFR and concave. Then REX1 ≥ 1
4 Ra1 . If, in addition, is
concave then REX1 ≥ 1
2 Ra1 .
Proposition
1) Let F and G be IFR and concave, and let be convex. Then W EX1 ≥ 1
2 W a1 .
2)Let F and G be convex and DRFR, and let be convex. Then W EX1 ≥W a
1 .
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 21
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Price Discrimination
Price discrimination with quality costs
Consumers are distributed over [0, 1] according to distribution F withf = F ′ > 0. Each consumer demands one unit of the good.
Utility of consumer of type v from quality q is vq.
Cost of producing y units of quality q is c(q)y .
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 22
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Price Discrimination
Revenue maximizing profit function
By the standard mechanism design argument we know that monopolist’s revenue and profit aregiven by
Ra =
∫ 1
0q(v)
(v −
1− F (v)
f (v)
)f (v)dv
and
�a1 =
∫ 1
0
[q(v)
(v −
1− F (v)
f (v)
)− c(q(v))
]f (v)dv
Let r be such that(r − 1−F (r)
f (r)
)= 0. Solution is,
q(v) = 0 if v ≤ r
c ′(q(v)) = v −1− F (v)
f (v)if v ≥ r
Define G(y) = F (q−1(y)) we get a distribution of quality levels where q(v) = (v) valuation
(men), quality (women).
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 23
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Price Discrimination
Price discrimination example
In paper’s framework: total revenue of the assortative matching is given by(� = 1)
Ra1 =
∫ �F
0
(x)
(x − 1− F (x)
f (x)
)dF (x)
(from computing E(pm))
Coarse matching: provide two qualities QL =∫ EV
0q(z)dF (z)/F (EV ) and
QH =∫ 1
EVq(z)dF (z)/(1− F (EV )) and REV
1 = EV (EQ − QL)
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 24
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Price Discrimination
Example
c(q) = q2 and v ∼ U[0, 1]. In this case, q(v) = 0 if v ≤ 12 and r = 1/2 and
2q(v) = v − 1−v1 ⇐⇒ q(v) = 2v−1
2
G (y) = 1+2y2 for y ∈
[0, 1
2
]which is concave and IFR.
Computation yields QH = 12 , Ra
1 = 112 , REV
1 = 116 that is REV
1 = 34 Ra
1 . (noteProp 9 tells us REX
1 ≥ 12 Ra
1 )
Total profit is given by �a1 = 1
24 and �EV1 = 1
32 and �EV1 = 3
4�a.
W EV = 132 >
148 = W a (note Prop 10 tells us W EV ≥W a).
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 25
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Price Discrimination
Proof of Lemma 4
Lemma
is convex (concave) then Ra1 ≥ (≤)Ra
1/2. If the asssortative matching function
is convex (concave) and if EX ≥ (≤)EY then REX1 ≥ (≤)REX
1/2.
Proof
dRa
d�=
∫ 1
0(x ′(x)− (x))(1− F (x))dx > 0
if convex. That is Ra� increasing in �.
Now, note EX ≥ EY = E (X ) ≥ (EX ) if convex (concave) and (EY − EYL) ≥ (EX − EXL)by Lemma 3.
REX1 = EX (EY − EYL) ≥
1
2[EX (EY − EYL) + (EX )(EX − EXL)] = REX
1/2
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 26
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Price Discrimination
Proof of Proposition 9
Proposition
Let F be IFR, G be IFR and concave. Then REX1 ≥ 1
4Ra
1 . If, in addition, is concave then
REX1 ≥ 1
2Ra
1 .
Proof.
By Lemma 3 EY − EYL ≥ 12EY if G is concave. This yields
REX1 = EX (EY − EYL) ≥
1
2EXEY =
1
2Ur =
1
2
(EX (X )
1 + CCV 2(X , (X ))
)
=1
2
(Ua
1 + CCV 2(X , (X ))
)≥
1
2
(Ra
1
1 + CCV 2(X , (X ))
)CCV 2(X , (X )) ≤ 1 if F and G are both IFR. Ua ≥ Ra
1 in general and if concave Ra1 < Ra
1/2,
and since Ua = 2Ra1/2
, Ua > 2Ra1 .
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 27
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Price Discrimination
Proof of Proposition 10
Proposition
1) Let F and G be IFR and concave, and let be convex. Then W EX1 ≥ 1
2W a
1 . 2)Let F and G
be convex and DRFR, and let be convex. Then W EX1 ≥W a
1 .
Proof 1)
UEX = EXEY +F (EX )
1− F (EX )(EX − EXL)(EY − EYL)
F (X ) ≥ 1/2 by L2
≥ EXEY + (EX − EXL)(EY − EYL)
algebra= REX
1 + EY (EX − EXL) + EYLEXL
EX − EXL ≥ 1/2EX by L3
≥ REX1 +
1
2EXEY + EYLEXL
algebra=
3
2REX
1 +1
2EXEYL + EYLEXL =⇒ UEX >
3
2REX
1
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 28
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Price Discrimination
Proof of Proposition 10 continued
Now,
W EX1 = UEX − REX
1
23UEX > REX
1
≥1
3UEX
McAfee≥
1
3
1
2(Ua + Ur )
Ur ≥ 1/2Ua if F and G IFR
≥1
6(Ua +
1
2Ua) =
1
4Ua
Ua/2 > Ra1 if cvx
≥1
2W a
1
2)
REX1 = EX (EY − EYL)
L3, F and G cvx≤
1
2EXEY =
1
2Ur
This implies that
W EX1 = UEX − REX
1 ≥1
2(Ua + Ur )−
1
2Ur =
1
2Ua
Ua/2 > Ra1 if cvx
≥ W a1
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 29
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Conclusion
Conclusions
Simple matching schemes can work under incomplete information, and arenot too far from optimal, even in the worst case.
▶ But how easy is it to satisfy the statistical assumptions?▶ Horizontally differentiated goods?
... we focus on clearly suboptimal mechanisms, while identifyingsettings where such mechanisms are very effective (and thus maybecome optimal once transaction costs associated with morecomplex mechanisms are taken into account.
Foundations for costs of complexity?
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 30
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Assortative matching revenue: men’s shareMan x pays Pm(x) =
∫ x
0�z ′(z)dz . Integrating by parts,
pm(x) = (x)x −∫ x
0
(z)dz
Take the expectation over all men:
Ramen =
∫ �F
0
Pm(x)f (x)dx
= �
⎛⎝∫ �F
0
(x)xf (x)dx −∫ �F
0
∫ x
0
(z)dzf (x)dx︸ ︷︷ ︸⎞⎠
∫ �F
0
∫ �
z
f (x)dx (z)dz =∫ �F
0
(1− F (z)) (z)dz =
Ramen = �
∫ �F
0
(x)
[x − 1− F (x)
f (x)
]f (x)dx
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 31
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SOC for assortative matchingPlugging in the solution to the FOC, man x solves
arg maxx�
(x (x)−
∫ x
0
z ′(z)dz
)
=�(x − x) (x) + �
⎛⎜⎜⎜⎜⎝x (x)−∫ x
0
z ′(z)dz︸ ︷︷ ︸x (x)−
∫ x0 (z)dz
⎞⎟⎟⎟⎟⎠=�(x − x) (x) + �
∫ x
0
(z)dz
d
dx:�(x − x) ′(x)
d2
dx2: (x − x) ′′(x)︸ ︷︷ ︸
=0 at x=x
− ′(x) < 0
The above shows that at reporting x = x gives a local maximum. Since thesolution to the FOC is unique, we need to rule out only reporting 0 or reporting � .
H. Hoppe, B. Moldovanu, and E. Ozdenoren () Coarse Matching and Price DiscriminationSlides by Adam Kapor, Sofia Moroni and Aiyong Zhu, April 1, 2011 32
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