Micro II , SS 2014

Micro II , SS 2014

B. Moldovanu

April 2014

Quasi-Linear Utility

x = (k,t₁, ..,t_I)^,where: k 2K (physical outcomes, "projects"), t_i 2^R (money)

u_i(x,θi) =v_i(k,θi) +t_i

f(θ) =f(θ1, ..θI) = (k(θ)^,t₁(θ)^{, ..,}t_I(θ))

De…nition

E¢cient SCF f (θ) = (k (θ)^,t₁(θ)^{, ..,}t_I (θ))^:

1 Value maximization: 8θ, k (θ)2arg maxk∑_iv_i(k,θi)

2 Budget Balance:8θ, ∑_it_i(θ) =0

Example: Allocation of indivisble good

Indivisible good owned by seller I buyers

k = (y₁, ..,y_I)where yi 2 f0,1gand∑yi 1 vi(k,θi) =vi((y₁, ..,yI)^,θi) =yiθi

E¢cient allocation: y_i =1 if θi 2arg maxjθj ; all monetary transfers from buyers go to seller

The Vickrey-Clarke-Groves (VCG) Mechanism

Direct Revelation Mechanism k(θ) =k (θ) (value maximization)

t_i (θ) =∑_j6=iv_j(k (θ)^,θj) +h_i(θ i)^,whereh_i is arbitrary

Theorem

The VCG mechanism truthfully implements the value maximizing SCF in dominant strategies

The Pivot Mechanism

Problem

VCG mechanism requires huge transfers to the agents.

Solution

Appropriate de…nition of the hi functions

Denote byk _i(θ i)the value maximizing project in the absence of i De…ne

t_i(θ) =

∑

j6=i

v_j(k (θ)^,θj) +h_i(θ i)

=

∑

j6=i

v_j(k (θ)^,θj)

∑

j6=i

v_j(k _i(θ i)^,θj)

Example: Allocation of Indivisble Object

E¢cient allocation: yi =1 if θi 2arg maxjθj

In VCG mechanism:

t_i(θ) = ⁰+h_i(θ i)^,ifi =arg maxθj

arg maxθj +h_i(θ i)^,otherwise In pivot mechanism:

t_i(θ) = ^{arg maxj6=iθj,ifi} =arg maxθj

0,otherwise Second price auction !

The Green-La¤ont Theorem for Bilateral Bargaining I

Agent 1 is seller, owns indivisble object , value for object θ1

Agent 2 is buyer, value for object θ2

Values are distributed independently on interval[0,1] according to densitiesφ₁,φ₂.

VCG Mechanism:

k (θ) =k (θ1,θ2) = ^{1, if θ1 θ2} 2, otherwise

t₁(θ) = ⁰+h₁(θ2)^,if θ1 θ2

θ2+h₁(θ2)^,otherwise t₂(θ) = ^θ1+h₂(θ1)^,if θ1 θ2

The Green-La¤ont Theorem for Bilateral Bargaining II

Budget Balance:

t₁(θ) +t₂(θ) = 0)

Z ₁

0

Z ₁

0 [t₁(θ) +t₂(θ)]φ₁(θ1)φ₂(θ2)dθ1dθ2 = 0) H₁+H₂+

Z 1

0

Z 1

0 max[θ1,θ2]φ₁(θ1)φ₂(θ2)dθ1dθ2 = 0 whereHi =E_{θ i}hi.Noting thatθ1 <_max[θ1,θ2]a.e., this yields:

H₁+H₂ < _Eθ1

With positive

probability

The Green-La¤ont Theorem for Bilateral Bargaining III

Participation Constraints:

Highest Seller Type : 1+H₁ 1)H₁ 0 Lowest Buyer Type : Eθ1+H₂ 0)H₂ Eθ1

This yields:

H₁+H₂ Eθ1

a contradiction !

Equivalent Mechanisms

De…nition

1 Two mechanisms MandMe are P-equivalent if, for eachi,k andx_i, it holds that Q_i^k(xi) =Q^e_i^k(xi)^,whereQ_i^k andQe_i^k are the conditional expected probabilities associated with MandMe,respectively.

2 Two mechanisms MandMe are U-equivalent if they provide the same interim utilities for each agent i and each typexi of agenti.

For each agenti,interim utility is obtained (up to a constant) by integrating the function ∑^K_k=1a^k_iQ_i^k(xi)with respect toxi - this is the Payo¤ Equivalence Theorem. ThusP-equivalence implies

U-equivalence.