Pure and Bayes-Nash Price
of Anarchy for GSP
Renato Paes Leme
Cornell
Éva Tardos
Cornell & MSR
Keyword Auctions
organic search results
sponsored
search links
Keyword Auctions
Keyword Auctions
Prospective advertisers
Selling one Ad Slot
$2
$5
$7
$3
Selling one Ad Slot
$2
Pays $5 per click
$5
$7
$3
Vickrey
Auction
-Truthful
- Efficient
- Simple
-…
Auction Model
b1
$$$
$$$
b3
b2
$$
b4
b5
b6
$$
$
$
Auction Model
Vickrey
Auction
b2
- Notb Truthful
4
- Not Efficient
- Even Simpler
-…
b1
$$$
VCG
bAuction
$$$
3
-Truthful
$$
- Efficient
Generalized
- Simple (?)
b5
-Second
…$$
Price
Auction
$
b6
$
Our Results
Generalized 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)
v1
v2
• Assumption: bi ≤ vi
Since playing bi > vi is v3
dominated strategy.
b1
α1
α2
b2
b3
α3
(Simplified) Model
v1
b1
v2
b2
v3
pays b1
per click
α1
b3
α2
α3
(Simplified) Model
vi
bi
σ = π-1
i = π(j)
αj
j = σ(i)
Utility of player i :
ui(b) = ασ(i) ( vi - bπ(σ(i) + 1))
Model
vi
bi
σ = π-1
i = π(j)
αj
j = σ(i)
Utility of player i :
ui(b) = ασ(i) ( vi - bπ(σ(i) + 1))
next highest bid
Model
vi
bi
σ = π-1
i = π(j)
αj
j = σ(i)
Nash equilibrium:
ui(bi,b-i) ≥ ui(b’i,b-i)
Is truth-telling always Nash ?
Example Non-truthful
u1 = 10.9(2-0)
(2-1)
v1 = 2
bb1 1==0.9
2
α1 = 1
v2 = 1
b1 = 1
α2 = 0.9
Measuring inefficiency
vi
bi
σ = π-1
i = π(j)
Social welfare = ∑i vi ασ(i)
Optimal allocation = ∑i vi αi
αj
j = σ(i)
Measuring inefficiency
Price of Anarchy = max
Opt
Nash SW(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
Modeling uncertainty:
GSP as a Bayesian Game
GSP as a Bayesian Game
b
b?
GSP as a Bayesian Game
Idea: Optimize against a distribution.
b
b
b
b
b
b
b
b
b
b
b
b
b
Bayes-Nash solution concept
• Bayes-Nash models the uncertainty of
other players about valuations
• Values vi are independent random vars
• Optimize against a distribution
Thm: Bayes-Nash PoA ≤ 8
Bayesian Model
v1 ~ V 1
b1(v1)
α1
v2 ~ V 2
v3 ~ V 3
b2(v2)
b3(v3)
α2
α3
Model
vi ~ V i
bi(vi)
αj
i = π(j)
j = σ(i)
Bayes-Nash equilibrium:
E[ui(bi,b-i)|vi] ≥ E[ui(b’i,b-i)|vi]
Expectation over v-i
Bayes-Nash Equilibrium
vi are random variables
μ(i) = slot that player i occupies in
Opt (also a random variable)
Bayes-Nash PoA =
E[∑i vi αμ(i)]
E[∑i vi ασ(i)]
Previously known [G-S]: Price of Stability ≠ 1
Sketch of the proof
ασ(j) vπ(i)
≥
1
+
vj
αi
v1
α1
therefore:
ασ(j) 1
vπ(i)
1
≥
≥
or
αi
vj
2
2
v2j
αi2
Simple and intuitive
condition on matchings
in equilibrium.
ασ(j)
3
vvπ(i)
3
Opt
Sketch of the proof
ασ(j) vπ(i)
≥
1
+
v
αi
j
vj
vπ(i)
αi
ασ(j)
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
ασ(j) vπ(i)
≥
1
+
vj
αi
2 SW = ∑i ασ(i) vi + αi vπ(i) =
= ∑i αi vi ασ(i) +vπ(i)
vi
αi
≥ ∑i αi vi = Opt
≥
Proof idea:
new structural condition
Pure Nash version:
vi ασ(i) +αivπ(i) ≥ αivi
vj
vπ(i)
αi
ασ(j)
Bayes Nash version:
viE[ασ(i)|vi] + E[αμ(i) vπμ (i)|vi] ≥ ¼ viE[αμ(i)|vi]
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)