Pure and Bayes-Nash Price of Anarchy for GSP

Renato Paes Leme Éva TardosCornell Cornell & MSR

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Selling one Ad Slot

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Pros

pect

ive

adve

rtise

rs

Selling one Ad Slot

$2

$5

$7

$3

Pays $5 per click

VickreyAuction

-Truthful- Efficient- Simple- …

Auction Model

b1

b2

b3

b4 b5

b6

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Auction Model

b1

b2

b3

b4 b5

b6

$$$$$$

$$

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$

$

VickreyAuction

VCGAuction

Generalized

SecondPrice

Auction

-Truthful- Efficient- Simple (?)- …

- Not Truthful- Not Efficient- Even Simpler- …

Our ResultsGeneralized Second Price Auction

(GSP), although not optimal, has good social welfare guarantees:

• 1.618 for Pure Price of Anarchy

• 8 for Bayes-Nash Price of Anarchy• GSP with uncertainty

(Simplified) Model• αj : click-rate of slot j• vi : value of player i• bi : bid (declared

value)

• Assumption: bi ≤ vi

Since playing bi > vi is dominated strategy.

α1

α2

α3

b1

b2

b3

v1

v2

v3

(Simplified) Model

v1

v2

v3

α1

α2

α3

b1

b2

b3

pays b1 per click

(Simplified) Model

vi αjbi

j = σ(i)i = π(j)

ui(b) = ασ(i) ( vi - bπ(σ(i) + 1))Utility of player i :

σ = π-1

Model

vi αjbi

j = σ(i)i = π(j)

σ = π-1

next highest bidui(b) = ασ(i) ( vi - bπ(σ(i) + 1))Utility of player i :

Model

vi αjbi

j = σ(i)i = π(j)

Nash equilibrium:

σ = π-1

ui(bi,b-i) ≥ ui(b’i,b-i)

Is truth-telling always Nash ?

Example Non-truthful

α1 = 1

α2 = 0.9

b1 = 2v1 = 2

v2 = 1 b1 = 1

b1 = 0.9

u1 = 1 (2-1)u1 = 0.9(2-0)

Measuring inefficiency

vi αjbi

j = σ(i)i = π(j)

σ = π-1

Social welfare =

∑i vi αi

∑i vi ασ(i)

Optimal allocation =

Measuring inefficiency

Price of Anarchy = max =OptSW(Nash)Nash

Main Theorem 1

Thm: Pure Price of Anarchy ≤ 1.618

If bi ≤ vi and (b1…bn) are bid in equilibrium, then for the allocation σ : ∑i vi ασ(i) ≥ 1.618-1 ∑i vi αi

Previously known [EOS, Varian]: Price of Stability = 1

GSP as a Bayesian Game

Modeling uncertainty:

GSP as a Bayesian Game

b

b ?

GSP as a Bayesian Game

b

b

b

b

b

b

b

b

b

b

b

b

b

Idea: Optimize against a distribution.

Bayes-Nash solution concept

Thm: Bayes-Nash PoA ≤ 8

• Bayes-Nash models the uncertainty of other players about valuations

• Values vi are independent random vars• Optimize against a distribution

Bayesian Model

V1

V2

V3

v1 ~

v2 ~

v3 ~

α1

α2

α3

b1(v1)

b2(v2)

b3(v3)

Model

Vivi ~ αjbi(vi)

j = σ(i)i = π(j)

E[ui(bi,b-i)|vi] ≥ E[ui(b’i,b-i)|vi] Bayes-Nash equilibrium:

Expectation over v-i

vi are random variablesμ(i) = slot that player i occupies in Opt (also a random variable)

Bayes-Nash PoA =

Bayes-Nash Equilibrium

E[∑i vi αμ(i)]E[∑i vi ασ(i)]

Previously known [G-S]: Price of Stability ≠ 1

Sketch of the proof

α2

α3

Opt

α1v1

v2

v3

αi

ασ(j)

vj

vπ(i)

therefore:ασ(j)αi vj

vπ(i) ≥ 12≥ 1

2 or

Simple and intuitive condition on matchings in equilibrium.

ασ(j)αi vj

vπ(i)+ ≥ 1

αi

ασ(j)

vj

vπ(i)

ασ(j)αi vj

vπ(i)+ ≥ 1

Need to show only for i < j and π(i) > π(j). It is a combination of 3 relations:ασ(j) ( vj – bπ(σ(j)+1) ) ≥ αi ( vj – bπ(i) ) [ Nash ]bπ(σ(j)+1) ≥ 0 bπ(i) ≤ vπ(i) [conservative]

Sketch of the proof

Sketch of the proof

ασ(j)αi vj

vπ(i)+ ≥ 1

2 SW = ∑i ασ(i) vi + αi vπ(i) = vπ(i)= ∑i αi vi ≥ ασ(i)

αi vi+

≥ ∑i αi vi = Opt

Proof idea:new structural condition

vi ασ(i) +αivπ(i) ≥ αivi

viE[ασ(i)|vi] + E[αμ(i) vπμ (i)|vi] ≥ ¼ viE[αμ(i)|vi]

Bayes Nash version:

Pure Nash version: αi

ασ(j)

vj

vπ(i)

Upcoming results[Lucier-Paes Leme-Tardos]

• Improved Bayes-Nash PoA to 3.164• Valid also for correlated distributions

• Future directions:• Tight Pure PoA (we think it is

1.259)