Mediators in Position Auctions

Itai AshlagiDov Monderer Moshe

Tennenholtz Technion

Talk Outline

• Mediators in games with complete information.

• Mediators and mediated equilibrium in games with incomplete information.

• Apply the theory to position auctions.

Mediators- Complete Information

Monderer & Tennenholtz 06• A mediator is defined to be a

reliable entity, which can ask the agents for the right to play on their behalf, and is guaranteed to behave in a pre-specified way based on messages received from the agents.

• However, a mediator can not enforce behavior; agents can play in the game directly without the mediator's help.

Mediators – Complete Information

c

d 1,15,0

0,54,4

c d

Mediator:If both use the mediator services – (c,c)If a single player chooses the mediator, the mediator plays d on behalf of this player.

c

d 1,15,0

0,54,4c d

0,5

1,1

1,15,0 4,4

m

m

Mediated game

1,1 3,3

3,62,8

Games with Incomplete Information

1s 2s

1t

2t

1,47,2

0,5

6,41,5

5,1

4,2

2,45,0

6,0

5,20,2

12

3 4

Games with Incomplete Information

1s 2s

1t

2t

0,5 3,6

7,2 1,4

2,8 5,1

1,5 6,4

5,0 2,4

4,2 3,3

0,2 5,2

1,1 6,0

12

3 4

Ex–post equilibrium - The strategies induce an equilibrium in every state

Implementing an Outcome Function by Mediation

2,20,0

3,05,2

5,23,0

0,02,2

A B

a

b

a ab b

No ex-post equilibrium in G

G

Implementing an Outcome Function by Mediation

2,20,0

3,05,2

5,23,0

0,02,2

A B

a

b

a ab b

No ex-post equilibirum in G

G

M{1,2}(m,m-A)=(a,a) M{1,2}(m,m-B)=(b,b)M{1} =aM{2}(m-A)=b, M{2}(m-B)=a

Mediator M

Implementing an Outcome Function by Mediation

2,20,0

3,05,2

5,23,0

0,02,2

A B

a

b

a ab b

No ex-post equilibirum in G

G

M{1,2}(m,m-A)=(a,a) M{1,2}(m,m-B)=(b,b)M{1} =aM{2}(m-A)=b, M{2}(m-B)=a

Mediator M

5,23,0

2,25,2

A

3,05,2

3,05,2

0,02,2 2,20,0

m

b

m-A bm-B a

a a

b

2,20,0

5,22,2

0,02,2

0,02,2

3,05,2 5,23,0

mm-A bm-B a

B

GM

Implementing an Outcome Function by Mediation (cont.)

5,23,0

2,25,2

A

3,05,2

3,05,2

0,02,2 2,20,0

m

b

m-A bm-B a

a a

b

2,20,0

5,22,2

0,02,2

0,02,2

3,05,2 5,23,0

mm-A bm-B a

B

GM

M{1,2}(m,m-A)=(a,a) M{1,2}(m,m-B)=(b,b)M{1} =a M{2}(m-A)=b, M{2}(m-B)=a

Implementing an Outcome Function by Mediation (cont.)

5,23,0

2,25,2

A

3,05,2

3,05,2

0,02,2 2,20,0

m

b

m-A bm-B a

a a

b

2,20,0

5,22,2

0,02,2

0,02,2

3,05,2 5,23,0

mm-A bm-B a

B

GM

The mediator implements the following outcome function:A)=(a,a) (B)=(b,b)

M{1,2}(m,m-A)=(a,a) M{1,2}(m,m-B)=(b,b)M{1} =a M{2}(m-A)=b, M{2}(m-B)=a

Mediators & Mechanism Design

Mechanism design – find a game to implement

Mediators – find a mediator to implement for a given game.

Position Auctions - Model• k – #positions, n - #players n>k

• vi - player i’s valuation per-click

• j- position j’s click-through rate 1>2>>

Allocation rule – jth highest bid to jth highest position Tie breaks - fixed order priority rule (1,2,…,n)

Payment schemepj(b1,…,bn) – position j’s payment under bid profile (b1,…,bn)

Quasi-linear utilities: utility for i if assigned to position j and pays qi per-click isj(vi-qi)

Outcome(b) = (allocation(b), position payment vector(b))

Some Position Auctions• VCG pj(b)=l¸j+1b(l)(k-1-k)/j

• Self-price pj(b)=b(j)

• Next –price pj(b)=b(j+1)

There is no (ex-post) equilibrium in the self-price and next-price position auctions.

In which position auctions can the VCG outcome function be implemented? Why should we do it?

Exampleself-price, single slot auction

1=1, n=2

c-mediatorv1

v2

v2

0v1¸ v2

Exampleself-price, single slot auction

1=1, n=2

c-mediatorv1

v2

v2

0v1¸ v2

c-mediatorvi cvi

For every c¸1 vcg can be implemented in the single-slot self-price auction.

c>1 can lead to negative utilities for players who trust the mediator.

Exampleself-price, single slot auction

1=1, n=2

c-mediatorv1

v2

v2

0v1¸ v2

c-mediatorvi cvi

For every c¸1 vcg can be implemented in the single-slot self-price auction.

c>1 can lead to negative utilities for players who trust the mediator.

Exampleself-price, single slot auction

1=1, n=2

c-mediatorv1

v2

v2

0v1¸ v2

c-mediatorvi cvi

For every c¸1 vcg can be implemented in the single-slot self-price auction.

Valid Mediators – players who trust the mediator never loose moneyThe c-mediator is valid for c=1

Self-Price Position Auctions

n=3, k=2

v1=5, v2=5, v3=10

The VCG outcome function can not be implemented in the self-price position auction unless k=1.

Self-Price Position Auctions

n=3, k=2

v1=5, v2=5, v3=10

The VCG outcome function can not be implemented in the self-price position auction unless k=1.

VCGplayer 3, pays 5

player 1, pays 5

player 2, pays 0

Self-Price Position Auctions

n=3, k=2

v1=5, v2=5, v3=10

The mediator must submit 5 on behalf of both players 1 and 3. But then player 3 will not be assigned to the first position!

The VCG outcome function can not be implemented in the self-price position auction unless k=1.

VCGplayer 3, pays 5

player 1, pays 5

player 2, pays 0

Theorem: There exists a valid mediator that implements vcg in the next-price position auction

Next-price Position Auctions

Edelman, Ostrovsky and Schwarz provided a mechanism that can be viewed as a “simplified” form of a mediatorwhere participation is mandatory.

1+p1vcg(v)

p2vcg(v)

p1vcg(v)

pk-1vcg(v)

pkvcg(v) pk

vcg(v)/2 pkvcg(v)/2

Positions according to v

If all players choose the mediator:

MN(v}=

Mediator for the next-price auction

1+p1vcg(v)

p2vcg(v)

p1vcg(v)

pk-1vcg(v)

pkvcg(v) pk

vcg(v)/2 pkvcg(v)/2

Positions according to v

If some players play directly: MS(vS)=vS

If all players choose the mediator:

MN(v}=

Mediator for the next-price auction

Proof:1. pj-1

vcg(v)¸ pjvcg(v) for every j¸ 2

where equality holds if and only if v(j)=…=v(k+1)

Proof:1. pj-1

vcg(v)¸ pjvcg(v) for every j¸ 2

where equality holds if and only if v(j)=…=v(k+1)

2.Reporting untruthfully to the mediator is non-beneficial.

Proof:1. pj-1

vcg(v)¸ pjvcg(v) for every j¸ 2

where equality holds if and only if v(j)=…=v(k+1)

2.Reporting untruthfully to the mediator is non-beneficial.

3. pjvcg(v) · v(j+1) for every j

h - i’s position without deviationh’ – i’s position after deviation

Proof:1. pj-1

vcg(v)¸ pjvcg(v) for every j¸ 2

where equality holds if and only if v(j)=…=v(k+1)

2.Reporting untruthfully to the mediator is non-beneficial.

3. pjvcg(v) · v(j+1) for every j

h - i’s position without deviationh’ – i’s position after deviationVCG utility in h position ¸

VCG utility in h’ position

Proof:1. pj-1

vcg(v)¸ pjvcg(v) for every j¸ 2

where equality holds if and only if v(j)=…=v(k+1)

2.Reporting untruthfully to the mediator is non-beneficial.

3. pjvcg(v) · v(j+1) for every j

h - i’s position without deviationh’ – i’s position after deviationVCG utility in h position ¸

VCG utility in h’ position ¸

next-price utility in h’ position

Proof:1. pj-1

vcg(v)¸ pjvcg(v) for every j¸ 2

where equality holds if and only if v(j)=…=v(k+1)

2.Reporting untruthfully to the mediator is non-beneficial.

3. pjvcg(v) · v(j+1) for every j

h - i’s position without deviationh’ – i’s position after deviation

4. Mediator is valid

VCG utility in h position ¸

VCG utility in h’ position ¸

next-price utility in h’ position

Existence of Valid Mediators for Position

AuctionsTheorem: Let G be a position auction. If the following

conditions hold then there exists a valid mediator that implements vcg in G:C1: position payment depends only on lower

position’s bids.C2: VCG cover – any VCG outcome can be

obtained by some bid profile.C3: G is monotone

Each one of these conditions are necessary.

*assumption – players don’t pay more than their bid.

The Mediator

b(v) – a “good” profile for v (obtains the desired outcome for v).

vi = (v-i, Z) - i has the “largest” value

MN(v)=b(v)

MN\{i}(v)=b-i(vi)

MS(vs)=vS (other subsets S)

*monotonicity is used for proving validity

Existence of Valid Mediators for Position Auctions (cont.)

Corollaries1. Suppose pj(b)=wjb(j+1) , 0· wj· 1.

Valid mediators exist if and only if for every j, wj· wj+1

2. Valid mediators exist in k-price position auctions

Quality effect Valid mediators exist in the existing (Google, Yahoo) position auctions, where the click-through rate for player i in position j is ®ij

Related WorkMediators in Incomplete Information GamesCollusive Bidder Behavior at Single-Object

Second-Price and English Auctions (Graham and Masrshall 1987)

Bidding Rings (McAfee and McMillan 1992)Bidding Rings Revisited (Bhat, Leyton-

Brown, Shoham and Tennenholtz 2005)Position AuctionsInternet Advertising and the Generalized

Second Price Auction (Edelman, Ostrovsky and Schwarz 2005)

Position Auctions (Varian 2005)

Conclusions• Introduced the study of mediators in

games with incomplete information.• Applied mediators to the context of

position auctions.• Characterization of the position

auctions in which the VCG outcome function can be implemented.

Future Work• Stronger implementations in

position auctions (2-strong, k-strong).

• Mediator in other applications.• Mediators and Learning.

Thank You