Leonardo Felli
29 November, 2002
Advanced Economic Theory
Lecture 8
Multiple Agents Screening:
We consider now the screening problem when there
are at least two agents.
The key consideration that differentiate this problem
from the simple principal-agent one is the competition among agents.
We analyze a simple environment characterized by a
seller (the principal) that wants to sell one indivisible
unit of a commodity to I buyers (the agents).
1
Advanced Economic Theory
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Optimal Auctions
Assume that the agents are indexed by
i ∈ {1, . . . , I}
Both the principal and the agents are risk neutral.
Assume also that each agent’s preferences are characterized by a parameter θi that denotes the agent’s
valuation for the good.
This valuation is assumed to be private information
to each agent.
The principal and the other agent believe that the
valuations θi are i.i.d. draws from a density p(θi)
with support θi ∈ [θ, θ].
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A contract:
(φ, t)
where
• t is the bid (a transfer from the agent to the principal),
• φ is the probability of getting the unit of the good.
The payoff to each agent is then:
Ui = φi θi − ti
All agents require a non-negative payoff to participate: U i ≥ 0
No loss in generality in assuming that t is not contingent on getting the unit of the good.
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This is an independent private values setup: each
agent’s payoff depends only on the agent’s valuation.
Revelation principle allows us to restrict attention to
direct revelation mechanisms:
(φi(θ̂), ti(θ̂))
where
θ̂ = (θ̂1, . . . , θ̂I )
and in equilibrium every agent reports the truth:
θ̂ = θ
The agent’s indirect utility is then:
Ui(θ̂|θi) = φi(θ̂) θi − ti(θ̂)
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We also use the notation:
φi(θi) = Eθ−i [φi(θ)] ,
ti(θi) = Eθ−i [ti(θ)]
In a truth-telling BNE we have:
h
i
Ui(θ̂i|θi) = Eθ−i Ui(θ̂i, θ−i|θi) = φi(θ̂i) θi − ti(θ̂i)
Result: (Myerson, 1981) A general auction mechanism with a twice continuously differentiable φ(θi)
is incentive compatible if and only if
∂Ui(θi|θi)
= φi(θi)
∂θi
and
dφi(θi)
≥0
dθi
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Proof: The incentive compatibility constraint is:
θi ∈ argmax Ui(θ̂i|θi)
θ̂i
Agent i’s payoff when the (IC) constraint is satisfied
is then:
Ui(θi|θi) = φi(θi) θi − ti(θi)
(1)
Envelope theorem then implies:
∂Ui(θi|θi)
= φi(θi)
∂θi
The FOC conditions of the (IC) problem are:
∂Ui(θi|θi)
∂ θ̂i
∀θi
= 0,
(2)
The SOC are:
∂ 2Ui(θi|θi)
2
∂ θ̂i
≤0
(3)
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Differentiating (2) with respect to θi we get:
∂ 2Ui(θi|θi)
2
∂ θ̂i
=−
∂ 2Ui(θi|θi)
∂θi∂ θ̂i
Using (1) we then conclude:
∂ 2Ui(θi|θi)
∂θi∂ θ̂i
dφi(θi)
≥0
=
dθi
The remaining step requires to show that local (IC)
constraints imply global (IC). This analogous to the
similar proof that was presented before.
Denote now
Ui(θ) = Ui(θ|θi),
U (θ) = (U1(θ), . . . , UI (θ))
and
φ(θ) = (φ1(θ), . . . , φI (θ))
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The principal’s profits from the mechanism are then:
I
X
Π(φ, U (θ)) =
1−
φi(θ) θ0 +
i=1
+
I
X
φi(θ)θi −
i=1
I
X
Ui(θ)
i=1
Where θ0 is the principal’s valuation for the good.
The principal’s choice problem is then:
max Eθ [Π(φ, U (θ))]
φi ,Ui
s.t.
∂Ui(θi|θi)
= φi(θi)
∂θi
dφi(θi)
≥0
Ui(θ) ≥ 0
dθi
I
X
i=1
φi ≤ 1
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Integrating (1) we then get:
Z θi
Ui(θi|θi) =
φi(τ )dτ + Ui(θ|θ)
θ
Recall that, given a general function F (θ), the law of
iterated expectations implies
Eθ [F (θ)] = Eθi
Eθ−i [F (θ)|θi]
Therefore substituting U (θi | θi) into the principal’s
expected profit we obtain that the last part of the
principal’s objective function is:
!
Z θ
Z θi
−
φi(τ )dτ dPi(θi) − Ui(θ|θ)
θi =θ
θ
or integrating by parts:
Z θ
1 − Pi(θi)
φi(θi)
−
dPi(θi) − Ui(θ|θ)
p
(θ
)
i i
θi =θ
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In other words we get:
I
X
1 − Pi(θi)
max
Eθi φi(θi) θi −
− θ0 −
p
(θ
)
φi ,Ui (θ|θ)
i i
i=1
−
I
X
Ui(θ|θ) + θ0
i=1
dφ(θi)
s.t.
≥ 0 Ui(θi|θi) ≥ 0
dθi
I
X
φi ≤ 1
i=1
The monotonicity of φi(θi) together with φi(θi) ≥ 0
and Ui(θ|θ) ≥ 0 imply Ui(θi|θi) ≥ 0. In other words
the principal’s problem is:
I
X
1 − Pi(θi)
max
Eθi φi(θi) θi −
− θ0 −
p
(θ
)
φi ,Ui (θ|θ)
i i
i=1
−
I
X
Ui(θ|θ) + θ0
i=1
dφ(θi)
s.t.
≥ 0 Ui(θ|θ) ≥ 0
dθi
I
X
i=1
φi ≤ 1
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Notice that the expected value of the optimal mechanism is completely determined by the probabilities
φi and by the utilities Ui(θ|θ).
Revenue Equivalence Theorem: The seller’s
expected utility from the auction is invariant to
the particular rule disciplining the auction provided that this rule leads to the same probability
functions φi and values Ui(θ|θ).
Proof: By inspection.
Notice now that the principal’s problem is a linear
programming problem.
Define the following virtual types:
1 − Pi(θi)
Ji(θi) = θi −
pi(θi)
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The principal’s problem is then:
" I
#
X
max Eθi
φi(θi) (Ji(θi) − θ0) − Ui(θ|θ)
φi ,Ui (θ|θ)
i=1
dφ(θi)
s.t.
≥ 0,
dθi
Ui(θ|θ) ≥ 0,
I
X
φi ≤ 1
i=1
Result: (Myerson 1981) If Pi satisfied the MHRP,
the optimal auction is characterized by the following probabilities φi:


1









φi(θ) = ∈ [0, 1]









0
if Ji(θi) > maxk6=i Jk (θk )
and Ji(θi) ≥ θ0
if Ji(θi) = maxk6=i Jk (θk )
and Ji(θi) ≥ θ0
otherwise
Moreover Ui(θ|θ) = 0 ∀i.
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Proof: The choice of φi is clearly the one that solves
the principal’s linear programming problem.
The choice of Ui(θ|θ) = 0 is the one compatible
with the objective function monotonic decreasing in
Ui(θ|θ) and the (IR) constraint monotonic increasing.
Monotonicity obtains from φ(θ) weakly increasing in
Ji(θi) and from MHRP that implies Ji(θi) increasing
in θi.
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There are two distortions with respect to the efficient
allocation.
Under-consumption:
max Ji(θi) < θ0 < max θi
i
i
Mis-allocation:
argmax Ji(θi) 6= argmax θi
i
i
These inefficiencies are introduced by the (IC) constraint, the source of the virtual valuation.
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Independent private value optimal auction
Consider the following simpler setting of the one previously analyzed.
Assume there are only two bidders, i ∈ {1, 2}.
Assume that
θ0 = 0
Let
v i ∈ {vL, vH },
0 < vL < vH
and
Pr{v i = vH } = π
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Clearly:
vH
vH
vL
π2
π (1 − π)
vL (1 − π) π (1 − π)2
The total ex-ante surplus present among the bidders
is:
2
π + 2π (1 − π) vH + (1 − π)2 vL
Since both buyers are ex-ante identical wlog we restrict attention to symmetric mechanisms.
The direct revelation mechanism is
(φk,j , tk,j )
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where:
• φk,j = the probability that the buyer announcing
vk gets the good;
• tkj = the transfer from the buyer to the seller
contingent on the announcement (vk , vj ).
The restrictions on this mechanism are:
feasibility:
2 φHH ≤ 1,
2 φLL ≤ 1,
φLH + φHL ≤ 1
(IR) constraints:
π (φHH vH − tHH ) + (1 − π) (φHL vH − tHL) ≥ 0
π (φLH vL − tLH ) + (1 − π) (φLL vL − tLL) ≥ 0
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(IC) constraints:
π (φHH vH −tHH )+(1−π) (φHL vH −tHL) ≥
≥ π (φLH vH − tLH ) + (1 − π) (φLL vH − tLL)
π (φLH vL−tLH )+(1−π) (φLL vL−tLL) ≥
≥ π (φHH vL − tHH ) + (1 − π) (φHL vL − tHL)
Since all parties are risk neutral we can only solve for
the expected transfers from the buyer to the seller:
t̄L = π tLH +(1−π) tLL,
t̄H = π tHH +(1−π) tHL
The seller’s problem can now be simplified by dropping the (IRH ) and the (ICL) constraints.
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The seller’s problem is then:
max
φk,j ,t̄k
2 [π t̄H + (1 − π) t̄L]
s.t. (π φHH + (1 − π) φHL) vH − t̄H ≥
≥ (π φLH + (1 − π) φLL) vH − t̄L
(π φLH + (1 − π) φLL) vL − t̄L ≥ 0
2 φHH ≤ 1, 2 φLL ≤ 1, φLH + φHL ≤ 1
Since both constraint are shown to be binding we get
that:
n
max 2 π (π φHH + (1 − π) φHL) vH −
φk,j
o
− (π φLH + (1 − π) φLL)(vH − vL) +
+ 2 (1 − π) (π φLH + (1 − π) φLL) vL
s.t. 2 φHH ≤ 1, 2 φLL ≤ 1, φLH + φHL ≤ 1
This is a linear programming problem.
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Consider first the coefficient of φHH :
2 π 2 vH > 0
hence
φHH =
1
2
The coefficient of φHL is:
2 π(1 − π) vH > 0
and it is always bigger than the coefficient of φLH
hence from the constraint:
φHL = 1,
φLH = 0
Finally the coefficient of φLL: is
2
2 −π(1 − π)(vH − vL) + (1 − π) vL
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Therefore, if (1−π) vL ≥ π (vH −vL) then φLL = 1/2
and the seller’s expected profit is:
π vH + (1 − π) vL
while the rents left to the high value bidder are:
π (1 − π)(vH − vL) > 0
The total surplus generated is efficient:
2
π + 2π(1 − π) vH + (1 − π)2 vL
If instead (1 − π) vL < π (vH − vL) then: φLL = 0
and the seller’s expected profit is:
2
π + 2π (1 − π) vH
while the resources waisted (inefficiency) are:
(1 − π)2 vL > 0
Advanced Economic Theory
Correlated values optimal auction
Consider now an environment such that:
Pr{v 1 = vk , v 2 = vj } = π k,j
In other words:
vH
vL
vH π HH π HL
vL π LH π LL
Where we assume that:
π HH π LL − π HL π LH 6= 0
The total surplus is then:
π LL vL + (1 − π LL) vH
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The optimal auction is dramatically different.
Result: (Crémer and McLean 1988) The seller
can extract all the surplus from the buyer:
π LL vL + (1 − π LL) vH
Proof: Consider a general contract that specifies:
• a payment wkj from buyer 1 when his value is vk
and buyer 2’s value is vj ; (wjk then specifies the
payment from buyer 2);
• an efficient allocation of the good φkj :
1
φHH = φLL = ,
2
φHL = 1,
φLH = 0
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This contract extract all the surplus if it is incentive
compatible:
v
L
π LL
− wLL − π LH wLH ≥
2
≥ π LL (vL − wHL) + π LH
π HH
v
H
2
v
L
2
− wHH
− wHH −π HL (vH −wHL) ≥
v
H
≥ −π HH wLH + π HL
− wLL
2
and the buyers’ (IR) constraints are binding for all
types:
π LL
π HH
v
v
H
2
L
2
− wLL − π LH wLH = 0
− wHH − π HL (vH − wHL) = 0
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We can now find wkj such that these four conditions
are satisfied.
Rewrite the (IR) constraints as:
π LL vL
wLH =
− wLL
π LH 2
wHL
π HH vH
− wHH
= vH +
π HL 2
Define now
π HH π LL
β=
6= 1
π HL π LH
Using these equations and the fact that each buyer’s
surplus is 0 the (IC) constraints become:
0≥
π LH
(vL−β vH )−π LL (vH −vL)+π LH (β−1) wHH
2
π HL
0≥
(vH − β vL) + π HL (β − 1)wLL
2
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If β 6= 1 then the latter two inequalities can be satisfied.
In particular:
• If β > 1 then one needs to set wHH and wLL low
enough.
• The H buyer is deterred from misreporting by
imposing a low enough wLL and by (IRL) a high
enough wLH .
• Notice that in this case (ICH ) is easier to meet: if
a buyer’s type is vH the probability that the other
buyer is vH as well is higher than the probability
that is vL and hence paying wLH is more likely
than wLL.
• A similar argument applies to the buyer whose
type is vL.
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Of course if β < 1 then the same outcome can be
accomplished by setting wHH and wLL high enough.
In other words, the idea is to set the high payments
in the states of nature that are less likely under truthtelling than under deviation from truth-telling.
Notice that the following assumptions are critical:
• the buyers are risk neutral,
• the buyers are not resource constrained.
Indeed, payments vary a lot as β → 1.