Mediators in Position Auctions

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Mediators in Position Auctions

• Itai Ashlagi
• Dov Monderer Moshe Tennenholtz
• Technion

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Talk Outline

• Mediators in games with complete information.
• Mediators and mediated equilibrium in games with
  incomplete information.
• Apply the theory to position auctions.

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Mediators- Complete InformationMonderer
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.

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Mediators Complete Information
c
d
0,5
4,4
c
1,1
5,0
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
m
0,5
0,5
4,4
c
Mediated game
1,1
5,0
1,1
d
1,1
5,0
4,4
m
5
Games with Incomplete Information
2,8
5,1
3,6
0,5
6,4
1,5
1,4
7,2
?2
?1
2,4
5,0
5,2
0,2
3,3
4,2
1,1
6,0
?4
?3
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Games with Incomplete Information
?2
?1
?4
?3
Expost equilibrium - The strategies induce an
equilibrium in every state
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Implementing an Outcome Function by Mediation
No ex-post equilibrium in G
G
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Implementing an Outcome Function by Mediation
No ex-post equilibirum in G
G
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Implementing an Outcome Function by Mediation
No ex-post equilibirum in G
G
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Implementing an Outcome Function by Mediation
(cont.)
M1,2(m,m-A)(a,a) M1,2(m,m-B)(b,b) M1
a M2(m-A)b, M2(m-B)a
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Implementing an Outcome Function by Mediation
(cont.)
M1,2(m,m-A)(a,a) M1,2(m,m-B)(b,b) M1
a M2(m-A)b, M2(m-B)a
The mediator implements the following outcome
function ?(A)(a,a) ?(B)(b,b)
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Mediators Mechanism Design

• Mechanism design find a game to implement ?
• Mediators find a mediator to implement ? for a
  given game.

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Position Auctions - Model

• k positions, n - players ngtk
• vi - player is valuation per-click
• ?j- position js click-through rate
  ?1gt?2gt?gt?k
• Allocation rule jth highest bid to jth highest
  position
• Tie breaks - fixed order priority rule
  (1,2,,n)
• Payment scheme
• pj(b1,,bn) position js 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))

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Some Position Auctions

• VCG pj(b)?lj1b(l)(?k-1-?k)/?j
• Self-price pj(b)b(j)
• Next price pj(b)b(j1)
• 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?

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Exampleself-price, single slot auction
?11, n2
v1 v2
v2 0
c-mediator
v1 v2
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Exampleself-price, single slot auction
?11, n2
For every c1 ?vcg can be implemented in the
single-slot self-price auction.
v1 v2
v2 0
c-mediator
v1 v2
c-mediator
vi
cvi
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Exampleself-price, single slot auction
?11, n2
For every c1 ?vcg can be implemented in the
single-slot self-price auction.
v1 v2
v2 0
c-mediator
v1 v2
c-mediator
vi
cvi
cgt1 can lead to negative utilities for players
who trust the mediator.
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Exampleself-price, single slot auction
?11, n2
For every c1 ?vcg can be implemented in the
single-slot self-price auction.
v1 v2
v2 0
c-mediator
v1 v2
c-mediator
vi
cvi
cgt1 can lead to negative utilities for players
who trust the mediator.
Valid Mediators players who trust the mediator
never loose money The c-mediator is valid for
c1
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Self-Price Position Auctions
The VCG outcome function can not be implemented
in the self-price position auction unless k1.

• n3, k2
• v15, v25, v310

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Self-Price Position Auctions
The VCG outcome function can not be implemented
in the self-price position auction unless k1.
VCG

• n3, k2
• v15, v25, v310

player 3, pays 5
player 1, pays 5
player 2, pays 0
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Self-Price Position Auctions
The VCG outcome function can not be implemented
in the self-price position auction unless k1.
VCG

• n3, k2
• v15, v25, v310

player 3, pays 5
player 1, pays 5
player 2, pays 0
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!
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Next-price Position Auctions

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

Edelman, Ostrovsky and Schwarz provided a
mechanism that can be viewed as a simplified
form of a mediator where participation is
mandatory.
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Mediator for the next-price auction
If all players choose the mediator MN(v
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Mediator for the next-price auction
If all players choose the mediator MN(v
If some players play directly MS(vS)vS
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Proof
1. pj-1vcg(v) pjvcg(v) for every j 2 where
equality holds if and only if v(j)v(k1)
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Proof

• 1. pj-1vcg(v) pjvcg(v) for every j 2
• where equality holds if and only if v(j)v(k1)
• Reporting untruthfully to the mediator
• is non-beneficial.

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Proof

• 1. pj-1vcg(v) pjvcg(v) for every j 2
• where equality holds if and only if v(j)v(k1)
• Reporting untruthfully to the mediator
• is non-beneficial.
• 3. pjvcg(v) v(j1) for every j
• h - is position without deviation
• h is position after deviation

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Proof

• 1. pj-1vcg(v) pjvcg(v) for every j 2
• where equality holds if and only if v(j)v(k1)
• Reporting untruthfully to the mediator
• is non-beneficial.
• 3. pjvcg(v) v(j1) for every j
• h - is position without deviation
• h is position after deviation

VCG utility in h position
VCG utility in h position

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Proof

• 1. pj-1vcg(v) pjvcg(v) for every j 2
• where equality holds if and only if v(j)v(k1)
• Reporting untruthfully to the mediator
• is non-beneficial.
• 3. pjvcg(v) v(j1) for every j
• h - is position without deviation
• h is position after deviation

VCG utility in h position
VCG utility in h position
next-price utility in h position


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Proof

• 1. pj-1vcg(v) pjvcg(v) for every j 2
• where equality holds if and only if v(j)v(k1)
• Reporting untruthfully to the mediator
• is non-beneficial.
• 3. pjvcg(v) v(j1) for every j
• h - is position without deviation
• h is position after deviation
• 4. Mediator is valid

VCG utility in h position
VCG utility in h position
next-price utility in h position


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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
  positions 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 dont pay more than their
  bid.

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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

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Existence of Valid Mediators for Position
Auctions (cont.)

• Corollaries
• 1. Suppose pj(b)wjb(j1) , 0 wj 1.
• Valid mediators exist if and only if for every
  j, wj wj1
• 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

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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)
• Position Auctions (Varian 2005)

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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.

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Future Work

• Stronger implementations in position auctions
  (2-strong, k-strong).
• Mediator in other applications.
• Mediators and Learning.

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Thank You