Micro II

Micro II

Benny Moldovanu

University of Bonn

Summer Term 2015

Quasi-Linear Utility

x = (k , t ₁ , .., t _I ), where: k ∈ K (physical outcomes, ”projects”), t i ∈ R (money)

u i (x, θ i ) = v i (k, θ i ) + t i

f (θ) = f (θ ₁ , ..θ _I ) = (k (θ), t ₁ (θ), .., t _I (θ)) Definition

Efficient SCF f ^∗ (θ) = (k ^∗ (θ), t ₁ ^∗ (θ), .., t _I ^∗ (θ)) :

1

Value maximization: ∀θ, k ^∗ (θ) ∈ arg max _k P

i v _i (k , θ _i )

2

Budget Balance: ∀θ, P

i t _i ^∗ (θ) = 0

Example: Allocation of indivisble good

Indivisible good owned by seller I buyers

k = (y ₁ , .., y _I ) where y _i ∈ {0, 1} and P y _i ≤ 1 v _i (k, θ _i ) = v _i ((y 1 , .., y _I ), θ _i ) = y _i θ _i

Efficient allocation: y i = 1 if θ i ∈ arg max j θ j ; 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) = P

j 6=i v _j (k ^∗ (θ), θ _j ) + h _i (θ −i ), where h _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 definition of the h _i functions

Denote by k _−i ^∗ (θ −i ) the value maximizing project in the absence of i Define

t _i ^∗ (θ) = X

j 6=i

v _j (k ^∗ (θ), θ _j ) + h _i (θ −i )

= X

j 6=i

v _j (k ^∗ (θ), θ _j ) − X

j 6=i

v _j (k _−i ^∗ (θ −i ), θ _j ) Exercise: Prove that : ∀θ, P

i t _i ^∗ (θ) ≤ 0

Example: Allocation of Indivisible Object

Efficient allocation: y _i = 1 if θ _i ∈ arg max _j θ _j In VCG mechanism:

t _i ^∗ (θ) =

0 + h i (θ −i ), if i = arg max θ j

arg max θ _j + h _i (θ −i ), otherwise

In pivot mechanism:

t _i ^∗ (θ) =

− arg max _j 6=i θ j , if i = arg max θ j

0, otherwise

Second price auction !

The Green-Laffont Theorem for Bilateral Bargaining I

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

Agent 2 is buyer, value for object θ ₂

Values are distributed independently on interval [0, 1] according to densities φ ₁ , φ ₂ .

VCG Mechanism:

k ^∗ (θ) = k ^∗ (θ 1 , θ 2 ) =

1, if θ ₁ ≥ θ ₂ 2, otherwise

t ₁ ^∗ (θ) =

0 + h ₁ (θ ₂ ), if θ ₁ ≥ θ ₂ θ ₂ + h ₁ (θ ₂ ), otherwise

t ₂ ^∗ (θ) =

θ ₁ + h ₂ (θ ₁ ), if θ ₁ ≥ θ ₂ 0 + h ₂ (θ ₁ ), otherwise

The Green-La¤ont Theorem for Bilateral Bargaining II

Budget Balance:

t ₁ ( θ ) + t ₂ ( θ ) = 0 )

Z ₁

0

Z ₁

0 [ t ₁ ( θ ) + t ₂ ( θ )] φ ₁ ( θ 1 ) φ ₂ ( θ 2 ) d θ 1 d θ 2 = 0 ) H ₁ + H ₂ +

Z 1

0

Z 1

0 max [ θ 1 , θ 2 ] φ ₁ ( θ 1 ) φ ₂ ( θ 2 ) d θ 1 d θ 2 = 0 where H i = E _θ

h i . Noting that θ 1 < _max [ θ 1 , θ 2 ] a.e., this yields:

H ₁ + H ₂ < _{E θ 1}

With positive

probability

The Green-Laffont Theorem for Bilateral Bargaining III

Participation Constraints:

Highest Seller Type : 1 + H ₁ ≥ 1 ⇒ H ₁ ≥ 0 Lowest Buyer Type : Eθ 1 + H 2 ≥ 0 ⇒ H 2 ≥ −Eθ ₁ This yields:

H 1 + H 2 ≥ −E θ 1

a contradiction !

Bayesian Implementation and Payoff Equivalence

Benny Moldovanu

University of Bonn

Summer Term 2015

First-Price Auction

n bidders; i.i.d types F (θ _i ), ϕ(θ _i ) > 0

Symmetric Equilibrium b i (θ i ) = b(θ i ) increasing and differentiable.

First-Price Auction

U _i (θ _i , θ ˆ _i ) = [θ _i − b(ˆ θ _i )] · F (ˆ θ _i ) ⁿ⁻¹

FOC: −b ⁰ (ˆ θ _i )F (ˆ θ _i ) ⁿ⁻¹ + (n − 1)(θ _i − b(ˆ θ _i ))F (ˆ θ _i ) ⁿ⁻² ϕ(ˆ θ _i ) = 0 b ⁰ (θ _i ) = (n − 1)(θ _i − b(θ _i ) ^ϕ(θ _F(θ

⁾

i

)

b(θ _i ) = θ _i

First-Price Auction

Example

n = 2, θ i ∼ U[0, 1]

b ⁰ (θ _i ) = _θ ¹

i

(θ _i − b(θ _i )); b(0) = 0 Solution: b(θ _i ) = ¹ ₂ θ _i

Expected Utility: (θ _i − ¹ ₂ θ _i ) · θ _i = ¹ ₂ θ ² _i

Second Price: (θ _i − E[θ _j |θ _j ≤ θ _i ])θ _i = ¹ ₂ θ _i ²

Revenue= ¹ ₃

Myerson’s Auction Model

1 object, n bidders

independent, private values θ _i distributed: F _i (θ _i ), ϕ _i (θ _i ) > 0

Revelation Mechanism p i (θ) ∈ [0, 1], ∀i ; P n

i =1 p i (θ) ≤ 1

t i (θ) ∈ R, ∀i

Myerson’s Auction Model

q _i (θ _i ) = E _θ

p _i (θ _i , θ −i ) T _i (θ _i ) = E _θ

t _i (θ _i , θ −i ) U _i (θ _i , θ ˆ _i ) = θ _i q _i ( ˆ θ _i ) + T _i ( ˆ θ _i ) Bidder i’s problem:

max _{θ ˆ}

i

U _i (θ _i , θ ˆ _i ) = U _i (θ _i , θ _i )

Truthtelling condition!

Denote ¯ U _i (θ _i ) = U _i (θ _i , θ _i )

Myerson’s Auction Model

Theorem (Myerson)

A mechanism {p _i , t i } ⁿ _i=1 , is Bayesian incentive compatible if and only if:

1

q i is increasing in θ i

2

U ¯ _i (θ _i ) = ¯ U _i (θ _i ) + R θ

θ

q _i (s)ds, where U ¯ _i (θ _i ) = U _i (θ _i , θ _i )

Myerson’s Auction Model

Proof.

1) → θ ˆ _i > θ _i

θ ˆ _i q _i (ˆ θ _i ) + T _i (ˆ θ _i ) ≥ θ ˆ _i q(θ _i ) + T _i (θ _i ) θ _i q _i (θ _i ) + T _i (θ _i ) ≥ θ _i q _i (ˆ θ _i ) + T _i (ˆ θ _i )

(ˆ θ _i − θ _i )q _i (ˆ θ _i ) ≥

(ˆ θ _i − θ _i )q _i (θ _i ) U ¯ i (θ i ) is maximum of affine functions

→ convex, equal integral of its derivative.

Myerson’s Auction Model

Proof. (cont.)

2)← assume w.l.o.g. θ _i > θ ˆ _i

U ¯ _i (θ _i ) − U _i (θ _i , θ ˆ _i )

= ¯ U _i (θ _i ) − U ¯ _i (ˆ θ _i ) − q(ˆ θ _i )[θ _i − θ ˆ _i ]

= R θ

θ ˆ

q(s)ds − q(ˆ θ _i )[θ _i − θ ˆ _i ]

= R θ

θ ˆ

[q(s) − q(ˆ θ i )]ds ≥ 0

Myerson’s Auction Model

Consequence:

U ¯ i (θ i ) = ¯ U i (θ _i ) + R θ

θ q(s)ds

= θ _i q _i (θ _i ) + T _i (θ _i )

⇒ T _i (θ _i ) = −θ _i q _i (θ _i ) + R θ

θ q _i (s)ds + ¯ U _i (θ _i )

Payoff and revenue equivalence!

The Optimal Auction

max _{p

_,t

_}

i=1

E _θ

n

X

i=1

T _i (θ _i )

!

1

p i (θ) ∈ [0, 1]; P n

i=1 p i (θ) ≤ 1, ∀θ

2

q _i (θ _i ) is monotone

Observation: ¯ U i (θ _i ) = 0 is optimal

The Optimal Auction

− R θ ^¯

θ

T i (θ i )ϕ i (θ i )d θ i

= R θ ^¯

θ

[θ i q i (θ i ) − R θ

θ

q i (s )ds]ϕ i (θ i )d θ i

= E _θ [θ _i p _i (θ)] − R θ ^¯

θ

[ R θ

θ

q _i (s )ds]ϕ _i (θ _i )d θ _i

The Optimal Auction

R θ ^¯

θ

[ R θ

θ

q i (s )ds]ϕ i (θ i )d θ i

= R θ ^¯

θ

q i (s )ds − R θ ^¯

θ

q i (θ i )F i (θ i )d θ i

= R θ ^¯

θ

q _i (θ _i )[ ^1−F _ϕ

^(θ

⁾

i

(θ

) ]ϕ _i (θ _i )d θ _i

The Optimal Auction

= E θ (p i (θ)[ ^1−F _ϕ(θ

^(θ

⁾

i

) ])

to conclude:

−E _θ (T i (θ i )) = E θ [p i (θ i )(θ i − ^1−F _ϕ

^(θ

⁾

i

(θ

) )]

E _θ ( P n

i =1 −T _i (θ _i )) = E _θ [ P n

i =1 p _i (θ)(θ _i − ^1−F _ϕ(θ

^(θ

⁾

i

) )]

The Optimal Auction

Assumption: J _i (θ _i ) = θ _i − ^1−F _ϕ

^(θ

⁾

i

(θ

) increasing Revenue Maximization:

p i (θ) =







1, {i ∈ arg max

j

J j (θ j )} ∧ {J _i (θ i ) ≥ 0}

0, otherwise

Satisfies monotonicity constraint!

The Optimal Auction

Assumption: F _i = F , ∀i

Result: second-price auction with reservation price

that satisfies R ^∗ − ^1−F(R _ϕ(R

)

⁾ = 0 is optimal!

Equivalence between Bayesian and Dominant Strategy Incentive Compatible Mechanisms

Benny Moldovanu

University of Bonn

Summer Term 2015

A Problem in Discrete Tomography

Problem

When does a 0 1 matrix with given row and column sums exist ? Consider row sum ( 3 , 2 , 2 , 1 , 1 ) and two di¤erent column sums:

1 1 0 0 1 3

1 1 0 0 0 2

1 1 0 0 0 2

1 0 0 0 0 1

0 1 0 0 0 1

4 4 0 0 1

1 2! 0 0 0 3

1 1 0 0 0 2

1 1 0 0 0 2

1 0 0 0 0 1

1 0 0 0 0 1

5 4 0 0 0

Matrix exist if the vector of column sums is "less diverse" than the vector ( 5 , 3 , 1 , 0 , 0 ) ^.

See Gale (1957), and Ryser (1957) for the general result. Variations

(continuous case, densities) are in Kellerer (1961) and Strassen

(1965).

The Monotone Lift

Problem

When unique reconstruction is not possible, are there solutions with special properties ?

Theorem (Gutmann et al. (1991))

Let φ = φ ( x ₁ , x ₂ , .. x _n ) be measurable on [ 0 , 1 ] ⁿ with 0 φ 1 . Assume that the one-dimensional marginals

Φ i ( x _i ) =

Z

φ ( x ₁ , x ₂ , .. x _n ) dx _i

are non-decreasing in x i , i = 1 , 2 , .. n . Then there exists ψ measurable on

[ 0 , 1 ] ⁿ such that 0 ψ 1 , ψ has the same marginals as φ , and

moreover, ψ is non-decreasing in each coordinate.

Monotone Lift: Example

Example

φ =

2 4 4 10

4 2 6 12

4 6 4 14

10 12 14

= ) _ψ =

2 4 4 10

4 4 4 12

4 4 6 14

10 12 14

Note that ∑ i , j ( ψ _ij ) ² _{∑ i} , j ( φ _ij ) ² .

The Independent Private Values Model with Linear Utility

K social alternatives and N agents. The utility of agent i in

alternative k is given by a ^k _i x i + c _i ^k + t i where x i 2 [ 0 , 1 ] is agent i ’s private type, where a ^k _i , c _i ^k 2 R with a ^k _i 0 , and where t _i 2 R is a monetary transfer.

Types are drawn independently of each other, according to strictly

increasing distributions F _i . Type x _i is private information of agent i .

Manelli and Vincent assume: K = N ; a ⁱ _i = 1 , a ^j _i = 0 for any j 6= i ;

c _i ^k = 0 for any i , k .

Incentive Compatible Mechanisms I

De…nition

A direct revelation mechanism (DRM) M is given by K functions

q ^k : [ 0 , 1 ] ^N ! [ 0 , 1 ] and N functions t _i : [ 0 , 1 ] ^N ! R where q ^k ( x ₁ , ..., x _N ) is the probability with which alternative k is chosen, and t i ( x 1 , ..., x N ) is the transfer to agent i if the agents report types x ₁ , ..., x _N .

De…nition

A DRM M is Dominant-Strategy Incentive Compatible (DIC) if

truth-telling constitutes a dominant strategy equilibrium in the game

de…ned by M and the given utility functions. A DRM M is Bayes-Nash

Incentive Compatible (BIC) if truth-telling constitutes a Bayes-Nash

equilibrium in the game de…ned by M and the given utility functions.

Incentive Compatible Mechanisms II

Fact

A necessary condition for M to be DIC is that, for each agent i , and for any signals of others, the function ∑ ^K k = 1 a ^k _i q ^k ( x ₁ , ..., x _N ) is non-decreasing in x _i . Moreover, any K functions q ^k that satisfy this condition are part of a DIC mechanism.

Fact

A necessary condition for M to be BIC is that, for each agent i , the function ∑ ^K _k = 1 a ^k _i Q _i ^k ( x _i ) is non-decreasing, where

8 i , k , Q _i ^k ( _b x i ) =

Z

[ 0 , 1 ]

q ^k ( x 1 , ..., x i , b x i , x i + 1 , ..., x _N ) dF i ,

is the expected probability that alternative k is chosen if agents j 6= i

report truthfully while agent i reports type b x _i . Moreover any K functions

q ^k that satisfy this condition are part of a BIC mechanism.

Equivalent Mechanisms

De…nition

1

Two mechanisms M and M e are P-equivalent if, for each i , k and x _i , it holds that Q _i ^k ( x _i ) = Q e _i ^k ( x _i ) ^, where Q _i ^k and Q e _i ^k are the conditional expected probabilities associated with M and M e , respectively.

2

Two mechanisms M and M e are U-equivalent if they provide the same interim utilities for each agent i and each type x _i of agent i.

For each agent i , interim utility is obtained (up to a constant) by integrating the function ∑ ^K _k = 1 a ^k _i Q _i ^k ( x _i ) with respect to x _i - this is the Payo¤ Equivalence Theorem. Thus P -equivalence implies

U -equivalence.

P- and U-Equivalence for 2 Alternatives

Since q ² ( x ₁ , ..., x _N ) = 1 q ¹ ( x ₁ , ..., x _N ) ^, we have

∑ 2 k = 1

a ^k _i Q _i ^k ( x _i ) = a ² _i + ( a _i ¹ a ² _i ) Q _i ¹ ( x _i ) ^,

and therefore U-equivalence implies P -equivalence (the two notions coincide).

Theorem

Assume that K = 2 . Then for any BIC mechanism there exists a

P-equivalent (and thus U-equivalent) DIC mechanism.

U-Equivalence for Symmetric Settings

Theorem

Assume that a _i ^k = a ^k _j = a ^k for all k , i , j , and that F _i = F for all i.

Moreover, assume that 0 = a ¹ a ² a ^K = 1. Then for any symmetric, BIC mechanism there exists an U-equivalent symmetric DIC mechanism

Proof shows how to achieve U-equivalence using only the 2

alternatives with highest and lowest slope, respectively. Thus,

U-equivalence does not necessarily ensure that the ex-ante

probabilities of di¤erent alternatives are preserved.

Multidimensional Types

Benny Moldovanu

University of Bonn

Summer Term 2015

Multidimensional Types

v i (k, θ i ) = θ ^k _i Types - ind. distributed Vector Field: q _i ^k (θ i ) = E _θ

[p _i ^k (θ)]

T _i (θ _i ) = E _θ

[t _i (θ)]

U i (θ i , θ ˆ i ) = θ i ∗ q i (ˆ θ i ) + T i (ˆ θ i )

Multidimensional Types + Bayesian Implementation

Theorem (Jehiel, Moldovanu, Stacchetti, JET (1999)) The mechanism

{p ^k _i } _k∈K , t _{i i∈I} is Bayes-Nash incentive compatible if and only if:

1

[q i (θ i ) − q i ( ˆ θ i )] · [θ i − θ ˆ i ] ≥ 0

2

U ˆ _i (θ _i ) = ˆ U _i (θ _i ) + R θ

θ

q _i (s) · ds, ∀i , θ _i

Path independence!

Multidimensional Types

Example two objects

1 buyer, valuations θ A , θ B

Mechanism: P _A .P _B , P _AB < P _A + P _B

Multidimensional Types + Private Values

Observation

k(θ i , θ −i ) = k(ˆ θ i , θ −i )

⇒ t _i (θ _i , θ −i ) = t _i (ˆ θ _i , θ −i ) Lemma

k(θ _i , θ −i ) = l, k(ˆ θ _i , θ −i ) = ˆ l

⇒ θ ˆ ^ˆ _i ^l − θ ^ˆ _i ^l ≥ θ ˆ _i ^l − θ _i ^l ∀θ _i , θ ˆ _i , θ −i

Weak monotonicity!

Multidimensional Types + Private Values

Proof.

θ ^l _i + t i (θ i , θ −i ) ≥ θ ^{ˆ l} _i + t i (ˆ θ i , θ −i ) θ ˆ ^{ˆ l} _i + t _i (ˆ θ _i , θ −i ) ≥ θ ˆ ^l _i + t _i (θ _i , θ −i )

⇒ θ _i ^l + ˆ θ ^ˆ _i ^l ≥ θ ^ˆ _i ^l + ˆ θ _i ^l

⇒ θ ˆ ^ˆ _i ^l − θ ^ˆ _i ^l ≥ θ ˆ _i ^l − θ _i ^l

Multidimensional Types + Private Values

Theorem (Saks-Yu)

Assume types spaces are convex, and consider k(θ). There exist

t 1 (θ), · · · , t _I (θ) such that f (θ) = [k(θ), t 1 (θ), · · · , t _I (θ)] is truthfully

implementable in dominant strategies if and only if k (θ) is weakly

monotone.

Affine Maximizers

α 1 , · · · , α _I ∈ R +

R 1 , · · · , R K ∈ R

k(θ) ∈ arg max

k

" _I X

i=1

α _i V _i (k , θ _i ) + R _k

#

Affine Maximizers

Theorem (Roberts)

Let Θ _i = R ^|K| and |K | ≥ 3. Then f (θ) is truthfully implementable in

dominant strategies if and only if the associated k (θ) is an affine

maximizer.

Interdependent Values

Benny Moldovanu

University of Bonn

Summer Term 2015

Akerlof’s Model

Sellers with cars of quality θ ∈ [θ, θ] ¯

F (θ); F ⁰ (θ) = f (θ) > 0

Utility =

( θ − P, buyer

R(θ) + P , seller

Complete Information

Equilibrium Condition

P (θ) = θ

S (P ) = {θ|R(θ) ≤ θ} = D(P )

EfficientTrade!

Incomplete Information

S (P ) = {θ|R(θ) ≤ P } E (P ) ≡ E[θ|θ ∈ S (P )]

D(P ) =



 

 

0, E (P ) < P

∈ [θ, θ], ¯ E (P ) = P

[θ, θ], ¯ E (P ) > P

Equilibrium : P ^∗ = E (P ^∗ )

Incomplete Information

Example

R(θ) = 2

3 θ; F (θ) = θ on [0, 1]

E [θ| 2

3 θ ≤ P ] = E [θ|θ ≤ 3 2 P ] = 3

4 P P ^∗ = 3

4 P ^∗ ⇒ P ^∗ = 0

No car is sold at all!

Interdependent Values

u i (x , θ 1 , · · · , θ J ) = V i (k , θ 1 , · · · , θ J ) + t i

VCG Transfer:

t _i ^∗ (θ) = X

j6=i

V _j (k, θ) + h _i (θ −i ) Problem: t _i ^∗ depends on θ _i !

Solution?

Interdependent Values

Example (1 Object, 2 Bidders)

k = 1, 2

V _i (k , θ) =

( 0, k 6= i aθ _i + bθ −i , k = i where a > b > 0

k ^∗ (θ) =

( 1, θ 1 ≥ θ 2

2, θ ₁ < θ ₂

Interdependent Values

Example (1 Object, 2 Bidders)

Important Condition

∂V i (θ)

∂θ _i > ∂V j (θ)

∂θ _i , ∀i , j Equilibrium Notion

Nash Equilibrium (ex-post)

Bilateral Bargaining

Seller :V _S (θ _S , θ _B ) % θ _S Buyer :V _B (θ _S , θ _B ) % θ _B

∂V _S

∂θ _S > ∂V _B

∂θ _S ; ∂V _B

∂θ _B > ∂V _S

∂θ _B

Bilateral Bargaining

Definition

V _B (θ _S , θ _B ^∗ (θ _S )) = V _S (θ _S , θ _B ^∗ (θ _S )) V _S (θ ^∗ _S (θ _B ), θ _B ) = V _B (θ ^∗ _S (θ _B ), θ _B )

k ^∗ (θ) =

( S, V _S (θ) ≥ V _B (θ)

B, V B (θ) > V S (θ)

Bilateral Bargaining

Definition (cont.)

t _B ^∗ (θ) =

( 0, k ^∗ = B V _S (θ _S , θ ^∗ _B (θ _S ), k ^∗ = S t _S ^∗ (θ) =

( 0, k ^∗ = S

V _B (θ ^∗ _S (θ _S ), θ _B ), k ^∗ = B

Modified VCG payments

Conditions for efficient trade (Fieseler, Kittsteiner, Moldovanu, JET (2003))

E _θ

[V _B (θ _S , θ _B )] ≥ P ≥ E _θ

[V _S (¯ θ _S , θ _B )]

Akerlof’s case:

E _θ

[V _B (θ _S )] ≥ P ≥ V _S (¯ θ _S ) 1

2 < 2

3

Multidimensional Types+ Interdependent Values

Benny Moldovanu

University of Bonn

Summer Term 2015

Multidimensional Types + Interdependent Values

V _i (k, θ 1 , · · · , θ _J ) Example

2 agents, i = 1, 2

2 alternatives, k = A, B

V _i ^k (θ ^k ₁ ), i = 1, 2; k = A, B

(Only agent 1 is informed)

Multidimensional Types + Interdependent Values

Example (cont.)

Value Maximization k(θ 1 ) ∈ arg max

k 2

X

i=1

V i (k, θ ^k ₁ ) Incentives for agent 1:

k (θ ₁ ) ∈ arg max

k

[V ₁ (A, θ ^A ₁ ) + t ₁ ^A , V ₁ (B, θ ₁ ^B ) + t ₁ ^B ]

Multidimensional Types + Interdependent Values

Example (cont.)

Multidimensional Types + Interdependent Values

Example (cont.)

Congruence Condition

∂V

(A,θ

)

∂θ

∂V

(B,θ

)

∂θ

=

∂

∂θ

h P 2

i=1 V _i (A, θ ₁ ^A ) i

∂

∂θ

h P 2

i=1 V _i (B , θ ₁ ^B ) i

Non-generic condition!

Theorem

Theorem (Jehiel et al., Econometrica (2001))

For generic utility functions only constant social choice functions are truthfully implementable in ex-post equilibrium.

Robust implementation is impossible!

Multidimensional Types + Conservative Vector Fields

Incentive compatibility ⇔ the vector field q _i is 1) monotone

2) conservative

⇓

∂ ² q _i ^k (θ _i )

∂θ _i ^k ∂θ _i ^k

= ∂ ² q _i ^k

(θ _i )

∂θ ^k _i

∂θ ^k _i

Mechanism Design Without Money

Benny Moldovanu

University of Bonn

Summer Term 2015

Mechanism Design Without Money

Usual setting with general utility function U _i (x, θ _i ) - private values let R _i = {≺ _i | ≺ _i =≺ _i (θ _i ) for some θ _i }

ordinal preferences

Mechanism Design Without Money

Definition

L _i (x, ≺ _i ) = {y|y ≺ _i x}

A social choice function f is monotonic if

∀θ, L _i (f (θ), θ _i ) ⊆ L _i (f (θ), θ _i ⁰ )∀i ⇒ f (θ) = f (θ ⁰ )

=

(θ −i , θ _i ⁰ )

Impossibility Theorem

Theorem (Gibbard-Satterthwaite)

Assume that X is finite with |X | ≥ 3, and that ∀i, P = R i , where P is the set of all rational preferences (without indifference).

Let f be onto and dominant strategy incentive compatible; then f is

dictatorial!

Mechanism Design Without Money

Proof GS (Sketch).

1

R _i = P and f DIC ⇒f is monotonic

2

R _i = P and f DIC and onto ⇒f is pareto efficient

3

f is monotonic and efficient

⇒ f is dictatorial

Mechanism Design Without Money

Possibility results:

1

|X | = 2 - simple majority rule

2

single peaked preference

Single Peaked Preferences

Theorem

Let preferences be single-peaked and let the number of agents be odd.

Then the SCF that chooses at each profile the median peak is dominant

strategy incentive compatible.

Single Peaked Preferences

Theorem (Moulin 1980)

Consider a setting with n agents, and let R _i = SP ∀i (with respect to a given order K).

Let f be DIC, anonymous and unanimous. Then there exist n − 1 phantom

voters such that f chooses the median of all voters’ peaks.

Successive Voting With Thresholds

Voting on single alternatives in the SP order (say 1, 2, · · · , K ). k is chosen

if the number of yes votes is at least τ (k ), where τ (k) ≥ τ (k ⁰ ), k ≤ k ⁰ . If

no alternative passes threshold, K is chosen.

Successive Voting With Thresholds

Sincere Strategy for voter with peak on k:

No, No, · · · , No Yes , Yes, · · · , Yes 1, 2, · · · , k − 1 k, k + 1, · · · , K

Monotone strategy:

No, No, · · · , No Yes , Yes, · · · , Yes 1, 2, · · · , l l + 1, · · · , K

Sincere ⇒ monotone

Markov strategies

Successive Voting With Thresholds

Theorem

Assume that all players besides i use monotone strategies. Then it is

optimal for i to use the sincere strategy. In particular, sincere voting is an

ex-post perfect Nash equilibrium.

Successive Voting With Thresholds

Theorem

Equivalence for single-peaked preferences.

Anonymous Succsessive

Unanimous ⇐⇒ voting with DIC mechanisms decreasing

threshold τ (k)

Optimization

How many anonymous, unanimous, DIC mechanisms are there?

Optimization

2 alternatives, A, B

n mechanisms (supermajority) l A - number of phantoms on A l _B = n − 1 − l _A

utility: u ^A (x), u ^B (x), x ∈ [0, 1]

types x i.i.d., distribution F

single crossing:

Optimization

Definition

u _L ^A = E h

u ^A (x)|x ≤ x ^AB i u ^A _H = E

h

u ^A (x)|x ≥ x ^AB i

u _L ^B = E h

u ^B (x)|x ≤ x ^AB i u _H ^B = E

h

u ^B (x)|x ≥ x ^AB

i

Optimization

γ = u _L ^A − u ^B _L

u _L ^A − u _L ^B + u ^B _H − u _H ^A > 0 First order conditions:

changing l A to l A + 1 or l A − 1 should not be beneficial l A = [nγ ]

Two-Sided Matching Gale-Shapley 1961

Benny Moldovanu

University of Bonn

Summer Term 2015

Two-Sided Matching

M = {m ₁ , · · · , m n } W = {w ₁ , · · · , w p } M ∩ W = ∅

P (m _i ) = w _j , w _k , · · · m _i w _l · · ·

P (w _j ) = m _i , m _l , · · · w _j m _n · · ·

Ranked ordinal lists

Two-Sided Matching

Matching:

µ: M ∪ W −→ M ∪ W

1

µ ² (x) = x, ∀x ∈ M ∪ W

2

µ(m) ∈ W ∪ {m}

3

µ(w ) ∈ M ∪ {w }

Individual rationality: µ(x) %

x

x, ∀x ∈ M ∪ W

Two-Sided Matching

Stability: µ is stable if:

1

µ is individually rational, and

2

@ {m, w } such that µ(m) 6= w and

w

m µ(m)

m

w µ(w )

Two-Sided Matching

Theorem (Gale, Shapley (1962))

For any two-sided market, the set of stable matchings is non-empty.

Proof.

Deferred acceptance algorithm (Gale-Shapley).

Two-Sided Matching

Example

P (m ₁ ) = w ₂ w ₁ w ₃ P(w ₁ ) = m ₁ m ₃ m ₂ P (m ₂ ) = w ₁ w ₃ w ₂ P(w ₂ ) = m ₃ m ₁ m ₂ P (m ₃ ) = w ₁ w ₂ w ₃ P(w ₃ ) = m ₁ m ₃ m ₂

µ ¹ =

w ₁ w ₂ w ₃ m 1 m 2 m 3

µ ² =

w ₂ w ₁ w ₃ m 1 m 2 m 3

Stable!

The Roommate Problem

P(a)=bcd P(b)=cad P(c)=abd P(d)=abc

µ ¹ = ab

| | cd µ ² = a — b

c — d µ ³ = ab

| | dc

None

is

stable!

Lattice Operator

µ ∨ _M µ ⁰ (m) =







µ(m), if µ(m)

m µ ⁰ (m) µ ⁰ (m), otherwise

µ ∨ _M µ ⁰ (w ) =







µ(w ), if µ(w )

w µ ⁰ (w ) µ ⁰ (w ), otherwise

and analogously for ∧ _M .

Lemma

Lemma (Conway)

Let µ, µ‘ be stable matchings. Then µ ∨ _M µ ⁰ and µ ∧ _M µ ⁰ are also stable matchings.

Consequence: M-Optimal and W-Optimal stable matchings.

Strategic Questions

Theorem

There is no mechanism such that:

1. It is a dominant strategy for all agents to state preferences truthfully.

2. A stable matching is chosen for every report.

Strategic Questions

Proof.

M = {m ₁ m ₂ } W = {w ₁ w ₂ } P (m ₁ ) = w ₁ w ₂ P (w ₁ ) = m ₂ m ₁ P (m ₂ ) = w ₂ w ₁ P (w ₂ ) = m ₁ m ₂

µ ^M = m ₁ m ₂

w 1 m 2 µ ^W = m ₁ m ₂

w 2 w 1

Suppose Γ(p) = µ ^M

Consider strategy Q (w ₂ ) = m ₁ then Γ(p _−w

2

, Q(w ₂ )) = µ ^W

Strategic Questions

Theorem

Let Γ = Γ ^M be the mechanism that assigns to each profile of reports the

M-optimal stable matching. Then truthful reporting is a dominant

strategy for all men.

Strategic Questions

Theorem

Consider a Nash equilibrium of Γ ^M where men use their dominant

strategies (and women respond optimally). Then the outcome is a stable

matching for the true preferences!

Strategic Questions

Proof.

P - True Preferences.

P ⁰ → stated preferences.

Let µ ⁰ = Γ(P ⁰ ). Assume {m,w} block µ ⁰ . Then, in Γ ^M (p ⁰ ) m must have

proposed to w, and she rejected him. Consider P ⁰ (w ) = m ⇒ then {m, w }

would be matched. Contradiction to P ⁰ being an equilibrium.

Many-to-one Matching

n firms: F _i = {F ₁ , · · · , F _n } m workers: W = {W ₁ , · · · , W m } matching: µ : F ∪ W → 2 ^{F ∪W}

1

|µ(w )| = 1, ∀w ∈ W

2

µ(w ) = w , if µ(w ) ∈ / F

3

µ(F ) ⊆ W [µ(F ) = ∅]

4

µ(w ) = F _i ⇔ w ∈ µ(F _i )

Many-to-one Matching

P (w _i ) = F _i , F _j , · · · , F _k , w _i , · · · P (F j ) = S 1 .S 2 . · · · , ∅, · · · where S i ⊆ W (sets of workers) Definition

Ch _F (S ) = S ⁰ ⇔ ∀S ⁰⁰ ⊆ S , it holds that S ⁰

F S ⁰⁰ .

Many-to-one Matching

Matching is blocked by {w , F } if:

1

µ(w ) 6= F

2

F

w µ(w )

3

w ∈ Ch _F (µ(F ) ∪ {w }) Individual rationality:

µ(w ) %

w

w

µ(F ) = Ch F (µ(F )) %

F

∅

Many-to-one Matching

Definition

Firms have substitutable preferences if: ∀S, w , w ⁰ ∈ S w ∈ Ch _F (S ) ⇒ w ∈ Ch _F (S \ w ⁰ ).

Theorem

If all firms have substitutable preferences, then the set of stable matchings is not empty.

Proof.

Run G-S algorithm with firms proposing.

The Assignment Game

M ∩ W = ∅

V (S ) = 0 if S ⊆ M , or S ⊆ W V (S ) = max

µ

P

m∈S v(m, µ ^s (m))

where µ ^S is a matching on coalition S.

The Assignment Game

µ ^∗ is optimal if V (M ∪ W ) = P

m∈M v(m, µ ^∗ (m)) w ₁ w ₂ w ₃

m 1 10 11 7

m 1 6 8 2

m ₃ 5 5 9

Optimal

matching

generically

unique.

The Assignment Game

Interpretation:

M-sellers, each has indivisible object; reserve price: c _m .

W-buyers, each wants at most one object; value for object m: r _wm .

v (m, w ) = max{0, r wm − c m }

The Assignment Game

Stable payoff vectors

1

feasible: P

i∈M∪W x _i ≤ V (M ∪ W )

2

ind. rational: x _i ≥ 0, ∀i ∈ M ∪ W

3

no blocking: x _m + x _w ≥ v (m, w ), ∀m, w

The Assignment Game

Example

w 1 w 2

m ₁ 5 3

m ₂ 7 8

(x _m

, x _w

, x _m

, x _w

) (2.5, 2.5, 4, 4)

(1, 4, 3, 5)

Stable

The Assignment Game

Theorem (Shapley-Shubik)

The set of stable payoff vectors is not empty.

Proof.

min X

m∈M

x _m + X

w∈W

x _w

!

s.t.

1

x _i ≥ 0 ∀i ∈ M ∪ W

2

x _m + x _w ≥ v(m, w ), ∀m, w

The Assignment Game

Proof. (cont.)

Program has solution. Let R be the minimum. Need to show feasibility R ≤ V (M ∪ W ).

Dual program

max X

i,j

p ij v(i, j ) s.t.

1

P

i p ij ≤ 1, ∀j

2

P

j p ij ≤ 1, ∀i

3

p _ij ≥ 0, ∀i, j

The Assignment Game

Proof. (cont.)

Dual has solution with p _ij ∈ {0, 1}, p _ij = 1 ⇒ i ∈ M , j ∈ W or vice-versa.

The max achieved equals R (duality theorem). Obvious now that

R ≤ V (M ∪ W )

The Assignment Game

Lemma

Let x , y be stable payoff vectors. Define:

u _m = max(x _m , y _m ), ∀m ∈ M u w = min(x w , y w ), ∀w ∈ W

Then u is also a stable payoff vector (and analogously for women).

Proof.

Exercise!

The Assignment Game

Consequence:

The set of stable payoff vectors is a lattice with a maximal and minimal element.

Proof.

Above Lemma + compactness

The Assignment Game

Example (homogeneous objects)

W M

25 11 23 16 20 19 17 21 10 22 6 24 5

competitive prices 19 ≤ P ≤ 20

19 20

w 1 6 5

w ₂ 46 3

w ₃ 1 0

m 1 8 9

m ₂ 3 4

m ₃ 0 1

The Assignment Game

Example

(m ₁ , m ₂ , w ₁ , w ₂ ) P ₁ , P ₂ (1, 0, 2, 3) −→ (1, 0) (3, 2, 0, 1) −→ (3, 2)

w 1 w 2

m 1 3 4

m ₂ 1 3

The Assignment Game

Example (cont.) Vickrey Prices P ₁ = 1; P ₂ = 0

(each buyer pays externality)

w 1 w 2

m ₁ 3 4

m ₂ 1 3

⇒ Outcome equivalent to minimal stable payoff vector (buyer optimal).

The Simultaneous Ascending Clock Auction

M - objects; W - buyers

c m

r wm

(c

and r

are integers)

Assume M contains dummy object m ₀ : c _m

= 0

r _wm

= 0, ∀w

Dummy can be assigned to several buyers

The Simultaneous Ascending Clock Auction

Let P be a vector of prices, one for each object in M D w (P ) = {m ∈ M|r _wm − P m ≥ max

m

∈M (r _wm

− P _m

)}

→ Demand of w at P

The Simultaneous Ascending Clock Auction

Definition

P is quasi-competitive if there exists a matching µ such that:

1

µ(w ) = m ⇒ m ∈ D _w (P )

2

µ(w ) = w ⇒ m ₀ ∈ D _w (P )

Competitive equilibrium (P , µ):

1

quasi-competitive +

2

P m = c m if m ∈ / µ(W )

The Simultaneous Ascending Clock Auction

Lemma

1

Every competitive equilibrium yields a stable payoff vector.

2

Every stable payoff vector can be obtained via a competitive equilibrium.

Proof.

Exercise.

Hall’s Theorem

Let B,C be disjoint sets ∀b ∈ B, D b ⊆ C Can we find a matching µ : B → C such that

1

∀b, µ(b) ∈ D _b

2

µ(b) 6= µ(b ⁰ ), ∀b, b ⁰

Hall’s Theorem

Necessary condition:

∀B ⁰ ⊆ B,

| S

b∈B

D _b | ≥ |B ⁰ |

Theorem

The above condition is also sufficient!

Auction

P _m (i ) = c _m , ∀m

Bidders announce D w (P (1)) Find µ such that

µ(w ) ∈ D w (P (1))

If possible → stop otherwise

∃W ⁰ such that

|W ⁰ | > | ∪ _w∈W D w (P (1))|

⇒ There exists an overdemanded set of objects

Choose M ⁰ ⊆ M to be a minimal over-demanded set.

Auction

P m (2) =

( P _m (1) + 1 , m ∈ M ⁰ P m (1) , otherwise Continue...

Algorithm must stop since at high prices only m ₀ ∈ D _w (P (K )), ∀w

Auction

Theorem

The ascending auction stops at the minimum quasi-competitive price

vector. In the corresponding direct revelation mechanism, truthfully

revealing valuations is a dominant strategy for each buyer.