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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.

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Mediators in Position Auctions

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  1. Mediators in Position Auctions Itai Ashlagi Dov Monderer Moshe Tennenholtz Technion

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

  3. 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.

  4. Mediators – Complete Information c d 4,4 0,5 c 5,0 1,1 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 4,4 0,5 c Mediated game 5,0 1,1 1,1 d 5,0 1,1 4,4 m

  5. Games with Incomplete Information 2,8 5,1 0,5 3,6 1,5 6,4 7,2 1,4 2 1 5,0 2,4 0,2 5,2 4,2 3,3 1,1 6,0 4 3

  6. Games with Incomplete Information 2 1 4 3 Ex–post equilibrium - The strategies induce an equilibrium in every state

  7. b b a a 5,2 3,0 2,2 0,0 a 0,0 2,2 3,0 5,2 b A B Implementing an Outcome Function by Mediation No ex-post equilibrium inG G

  8. b b a a 5,2 3,0 2,2 0,0 a 0,0 2,2 3,0 5,2 b A 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 Mediator M Implementing an Outcome Function by Mediation No ex-post equilibirum inG G

  9. b b a a 5,2 3,0 2,2 0,0 a 0,0 2,2 3,0 5,2 b A 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 Mediator M m-A a b m-A a b m-B m-B m 2,2 5,2 2,2 0,0 5,2 2,2 5,2 3,0 m 0,0 2,2 2,2 0,0 3,0 5,2 5,2 3,0 GM a a 5,2 3,0 3,0 5,2 2,2 0,0 0,0 2,2 b b A B Implementing an Outcome Function by Mediation No ex-post equilibirum inG G

  10. m-A a b m-A a b m-B m-B m 2,2 5,2 2,2 0,0 5,2 2,2 5,2 3,0 m 0,0 2,2 2,2 0,0 3,0 5,2 5,2 3,0 GM a a 5,2 3,0 3,0 5,2 2,2 0,0 0,0 2,2 b b A B Implementing an Outcome Function by Mediation (cont.) 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

  11. m-A a b m-A a b m-B m-B m 2,2 5,2 2,2 0,0 5,2 2,2 5,2 3,0 m 0,0 2,2 2,2 0,0 3,0 5,2 5,2 3,0 GM a a 5,2 3,0 3,0 5,2 2,2 0,0 0,0 2,2 b b A B Implementing an Outcome Function by Mediation (cont.) 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 The mediator implements the following outcome function: (A)=(a,a) (B)=(b,b)

  12. Mediators & Mechanism Design Mechanism design – find a game to implement Mediators – find a mediator to implement  for a given game.

  13. 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>>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 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))

  14. 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?

  15. Exampleself-price, single slot auction 1=1, n=2 v1 v2 v2 0 c-mediator v1¸v2

  16. Exampleself-price, single slot auction 1=1, n=2 For every c¸1vcg can be implemented in the single-slot self-price auction. v1 v2 v2 0 c-mediator v1¸v2 c-mediator vi cvi

  17. Exampleself-price, single slot auction 1=1, n=2 For every c¸1vcg can be implemented in the single-slot self-price auction. v1 v2 v2 0 c-mediator v1¸v2 c-mediator vi cvi c>1 can lead to negative utilities for players who trust the mediator.

  18. Exampleself-price, single slot auction 1=1, n=2 For every c¸1vcg can be implemented in the single-slot self-price auction. v1 v2 v2 0 c-mediator v1¸v2 c-mediator vi cvi c>1 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 c=1

  19. Self-Price Position Auctions The VCG outcome function can not be implemented in the self-price position auction unless k=1. n=3, k=2 v1=5, v2=5, v3=10

  20. Self-Price Position Auctions The VCG outcome function can not be implemented in the self-price position auction unless k=1. n=3, k=2 v1=5, v2=5, v3=10 VCG player 3, pays 5 player 1, pays 5 player 2, pays 0

  21. Self-Price Position Auctions The VCG outcome function can not be implemented in the self-price position auction unless k=1. n=3, k=2 v1=5, v2=5, v3=10 VCG 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!

  22. 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.

  23. 1+p1vcg(v) p1vcg(v) Positions according to v p2vcg(v) pk-1vcg(v) pkvcg(v) pkvcg(v)/2 pkvcg(v)/2 Mediator for the next-price auction If all players choose the mediator: MN(v}=

  24. 1+p1vcg(v) p1vcg(v) Positions according to v p2vcg(v) pk-1vcg(v) pkvcg(v) pkvcg(v)/2 pkvcg(v)/2 Mediator for the next-price auction If all players choose the mediator: MN(v}= If some players play directly: MS(vS)=vS

  25. Proof: 1. pj-1vcg(v)¸pjvcg(v) for every j¸ 2 where equality holds if and only if v(j)=…=v(k+1)

  26. Proof: • 1. pj-1vcg(v)¸pjvcg(v) for every j¸ 2 • where equality holds if and only if v(j)=…=v(k+1) • Reporting untruthfully to the mediator • is non-beneficial.

  27. Proof: • 1. pj-1vcg(v)¸pjvcg(v) for every j¸ 2 • where equality holds if and only if v(j)=…=v(k+1) • Reporting untruthfully to the mediator • is non-beneficial. • 3. pjvcg(v) ·v(j+1) for every j • h - i’s position without deviation • h’ – i’s position after deviation

  28. Proof: • 1. pj-1vcg(v)¸pjvcg(v) for every j¸ 2 • where equality holds if and only if v(j)=…=v(k+1) • Reporting untruthfully to the mediator • is non-beneficial. • 3. pjvcg(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 ¸

  29. Proof: • 1. pj-1vcg(v)¸pjvcg(v) for every j¸ 2 • where equality holds if and only if v(j)=…=v(k+1) • Reporting untruthfully to the mediator • is non-beneficial. • 3. pjvcg(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 ¸ ¸

  30. Proof: • 1. pj-1vcg(v)¸pjvcg(v) for every j¸ 2 • where equality holds if and only if v(j)=…=v(k+1) • Reporting untruthfully to the mediator • is non-beneficial. • 3. pjvcg(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 ¸ ¸

  31. 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.

  32. 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

  33. Existence of Valid Mediators for Position Auctions (cont.) Corollaries 1. 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 ®ij

  34. 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)

  35. 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.

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

  37. Thank You

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