Mediators in Position Auctions

Mediators in Position Auctions

Itai Ashlagi

Dov 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 5,0 1,1 0,5 4,4

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 5,0 1,1 0,5 4,4

c d

0,5 1,1

m

Mediated

1,1 3,3 2,8 3,6

Games with Incomplete Information

s1 s₂

t1

t2

1,4 7,2

0,5 6,4

1,5

5,1

4,2

2,4 5,0

6,0 5,2 0,2

₁ ₂

₃ ₄

Games with Incomplete Information

s1 s₂

t1

t2

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

₁ ₂

₃ ₄

Ex–post equilibrium -

Implementing an Outcome Function by Mediation

2,2 0,0

3,0 5,2

5,2 3,0

0,0 2,2

A B

a b

a b a b

No ex-post

equilibrium in G

G

Implementing an Outcome Function by Mediation

2,2 0,0

3,0 5,2

5,2 3,0

0,0 2,2

A B

a b

a b a 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} =a

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

Mediator M

Implementing an Outcome Function by Mediation

2,2 0,0

3,0 5,2

5,2 3,0

0,0 2,2

A B

a b

a b a 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} =a

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

Mediator M

5,2 3,0

2,2 5,2

A

3,0 5,2

3,0 5,2

0,0

2,2 0,0 2,2

m

b

m-A m-B a b

a a

b

2,2 0,0

5,2 2,2

0,0 2,2

0,0 2,2

3,0

5,2 3,0 5,2

m

m-A m-B a b

B

G^M

Implementing an Outcome Function by Mediation (cont.)

5,2 3,0

2,2 5,2

A

3,0 5,2

3,0 5,2

0,0

2,2 0,0 2,2

m

b

m-A m-B a b

a a

b

2,2 0,0

5,2 2,2

0,0 2,2

0,0 2,2

3,0

5,2 3,0 5,2

m

m-A m-B a b

B

G^M

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.)

5,2 3,0

2,2 5,2

A

3,0 5,2

3,0 5,2

0,0

2,2 0,0 2,2

m

b

m-A m-B a b

a a

b

2,2 0,0

5,2 2,2

0,0 2,2

0,0 2,2

3,0

5,2 3,0 5,2

m

m-A m-B a b

B

G^M

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 – j^th highest bid to j^th highest position

Tie breaks - fixed order priority rule (1,2,…,n) Payment scheme

pj(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 isj(vi-qi)

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

Some Position Auctions

• VCG p_j(b)=_l¸j+1b_(l)(_k-1-_k)/_j

• Self-price p_j(b)=b_(j)

• Next –price p_j(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?

Example

self-price, single slot auction

₁=1, n=2

c-mediator v₁

v₂

v₂ 0 v₁¸ v₂

Example

self-price, single slot auction

₁=1, n=2

c-mediator v₁

v₂

v₂ 0 v₁¸ v₂

c-mediator

v_i cv

i

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.

Example

self-price, single slot auction

₁=1, n=2

c-mediator v₁

v₂

v₂ 0 v₁¸ v₂

c-mediator

v_i cv

i

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.

Example

self-price, single slot auction

₁=1, n=2

c-mediator v₁

v₂

v₂ 0 v₁¸ v₂

c-mediator

v_i cv

i

For every c¸1 ^vcg can be

implemented in the single- slot self-price auction.

Valid Mediators – players who trust the mediator never loose money

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.

VCG

player 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.

VCG

player 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 mediator where participation is mandatory.

1+p₁^vcg(v)

p₂^vcg(v) p₁^vcg(v)

p_k-1^vcg(v)

p_k^vcg(v) p_k^vcg(v)/2 p_k^vcg(v)/2 Position s

accordi ng to v If all players

choose the mediator:

M_N(v}=

Mediator for the next-

price auction

1+p₁^vcg(v)

p₂^vcg(v) p₁^vcg(v)

p_k-1^vcg(v)

p_k^vcg(v) p_k^vcg(v)/2 p_k^vcg(v)/2 Position s

accordi ng to v

If some players play directly:

If all players choose the mediator:

M_N(v}=

Mediator for the next-

price auction

Proof:

1. p_j-1^vcg(v)¸ p_j^vcg(v) for every j¸ 2

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

Proof:

1. p_j-1^vcg(v)¸ p_j^vcg(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. p_j-1^vcg(v)¸ p_j^vcg(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. p_j^vcg(v) · v_(j+1) for every j

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

Proof:

1. p_j-1^vcg(v)¸ p_j^vcg(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. p_j^vcg(v) · v_(j+1) for every j

h - i’s position without deviation h’ – i’s position after deviation VCG utility in

h position ¸ VCG utility in h’ position

Proof:

1. p_j-1^vcg(v)¸ p_j^vcg(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. p_j^vcg(v) · v_(j+1) for every j

h - i’s position without deviation h’ – i’s position after deviation VCG utility in

h position ¸ VCG utility in

h’ position next-price utility in ¸ h’ position

Proof:

1. p_j-1^vcg(v)¸ p_j^vcg(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. p_j^vcg(v) · v_(j+1) for every j

h - i’s position without deviation h’ – i’s position after deviation VCG utility in

h position ¸ VCG utility in

h’ position next-price utility in ¸ h’ position

Existence of Valid

Mediators for Position Auctions

Theorem:

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

M_N(v)=b(v)

M_N\{i}(v)=b_-i(vⁱ)

M_S(v_s)=v_S (other subsets S)

Existence of Valid Mediators for Position Auctions (cont.)

Corollaries

1. Suppose p_j(b)=w_jb_(j+1) , 0· w_j· 1.

Valid mediators exist if and only if for every j, w_j· w_j+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 ®_i_j

Related Work

Mediators in Incomplete Information Games Collusive 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 Auctions

Internet Advertising and the Generalized

Second Price Auction (Edelman, Ostrovsky and Schwarz 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