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 1,15,0 0,54,4 cd 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 0,54,4 cd 0,5 1,1 5,0 4,4 m m Mediated game

1,1 3,3 3,6 2,8 Games with Incomplete Information 1,47,2 0,5 6,41,5 5,1 4,2 2,45,0 6,0 5,20,2 11 22 33 44

Games with Incomplete Information 0,53,6 7,21,4 2,85,1 1,56,4 5,02,4 4,23,3 0,25,2 1,16,0 11 22 33 44 Ex–post equilibrium - The strategies induce an equilibrium in every state

Implementing an Outcome Function by Mediation 2,20,0 3,05,2 3,0 0,02,2 AB a b a a bb No ex-post equilibrium in G G

Implementing an Outcome Function by Mediation 2,20,0 3,05,2 3,0 0,02,2 AB a b a a bb 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 2,20,0 3,05,2 3,0 0,02,2 AB a b a a bb 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 5,23,0 2,25,2 A 3,05,2 3,05,2 0,02,2 0,0 m b m-Ab m-B a a a b 2,20,0 5,2 2,2 0,02,2 0,02,2 3,05,2 3,0 m m-Ab m-B a B GMGM

Implementing an Outcome Function by Mediation (cont.) 5,23,0 2,25,2 A 3,05,2 3,05,2 0,02,2 0,0 m b m-Ab m-B a a a b 2,20,0 5,2 2,2 0,02,2 0,02,2 3,05,2 3,0 m m-Ab m-B a B GMGM 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 0,0 m b m-Ab m-B a a a b 2,20,0 5,2 2,2 0,02,2 0,02,2 3,05,2 3,0 m m-Ab m-B a B GMGM 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 v i - 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 p j ( b 1,…, b n ) – position j ’s payment under bid profile ( b 1,…, b n ) Quasi-linear utilities: utility for i if assigned to position j and pays q i per-click is  j ( v i - q i ) Outcome(b) = (allocation(b), position payment vector(b))

Some Position Auctions VCG p j ( b )=  l ¸ j +1 b ( 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 v1v2v1v2 v20v20 v1¸ v2v1¸ v2

Example self-price, single slot auction  1 = 1, n=2 c-mediator v1v2v1v2 v20v20 v1¸ v2v1¸ v2 vivi 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 v1v2v1v2 v20v20 v1¸ v2v1¸ v2 vivi 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 v1v2v1v2 v20v20 v1¸ v2v1¸ v2 vivi 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 The c-mediator is valid for c=1

Self-Price Position Auctions n=3, k=2 v 1 =5, v 2 =5, v 3 =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 v 1 =5, v 2 =5, v 3 =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 v 1 =5, v 2 =5, v 3 =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+ p 1 vcg ( v ) p 2 vcg ( v ) p 1 vcg ( v ) p k -1 vcg ( v ) p k vcg ( v ) p k vcg ( v )/2 Positions according to v If all players choose the mediator: M N ( v }= Mediator for the next-price auction

1+ p 1 vcg ( v ) p 2 vcg ( v ) p 1 vcg ( v ) p k -1 vcg ( v ) p k vcg ( v ) p k vcg ( v )/2 Positions according to v If some players play directly: M S ( v S )= v S 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 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 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 ). v i = ( v - i, Z ) - i has the “largest” value M N ( v )= b ( v ) M N \{ i } ( v )= b - i ( v i ) M S ( v s )= v S (other subsets S) *monotonicity is used for proving validity

Existence of Valid Mediators for Position Auctions (cont.) Corollaries 1. Suppose p j ( b )= w j b ( j +1), 0 · w j · 1. Valid mediators exist if and only if for every j, w j · w j 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) 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.

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