Transcript Slide Show - Microsoft Research

Jason Hartline
Vasilis Syrgkanis
Northwestern University
Cornell University
December 11, 2013
 PoA in auctions (as games of incomplete information):
 Single-Item First Price, All-Pay, Second Price Auctions
 Simultaneous Single Item Auctions
 Position Auctions: GSP, GFP
 Combinatorial auctions
2
 Reduce analysis of complex setting to simple setting.
 Conclusion for simple setting X, proved under restriction P,
extends to complex setting Y
 X: complete information PNE to Y: incomplete information BNE
 X: single auction to Y: composition of auctions
3
 Objective in X is good because each player doesn’t want to
deviate to strategy 𝑏𝑖′
 Extension from setting X to setting Y: if best response argument
satisfies condition P then conclusion extends to Y
4
5
Complete info PNE to BNE with correlated values
 Target setting. First Price Bayes-Nash
Equilibrium with asymmetric correlated
values
 Simple setting. Complete information Pure
Nash Equilibrium
 Thm. If proof of PNE PoA based on own-
value based deviation argument then PoA
of BNE also good
Complete info PNE
to BNE with
correlated values
References:
Roughgarden STOC’09
Lucier, Paes Leme EC’11
Roughgarden EC’12
Syrgkanis ‘12
Syrgkanis, Tardos STOC’13
6
𝑣1
𝑏1
𝑣𝑖
𝑏𝑖
𝑣𝑛
• Highest bidder wins:
𝑥𝑖 𝐛 = {𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑜𝑟 𝑡ℎ𝑎𝑡 𝑖 𝑤𝑖𝑛𝑠}
• Pays his bid: 𝑃𝑖 𝐛 = 𝑏𝑖 ⋅ 𝑥𝑖 𝐛
• Quasi-Linear preferences:
UTILITY = VALUE − PAYMENT
𝑢𝑖 𝐛 = (𝑣𝑖 − 𝑏𝑖 ) ⋅ 𝑥𝑖 𝐛
• Objective:
WELFARE = UTILITIES + PAYMENTS
𝑆𝑊 𝐛 =
𝑏𝑛
𝑢𝑖 𝐛 +
𝑖
=
𝑃𝑖 𝐛
𝑖
𝑢𝑖 𝐛 + 𝑏𝑖 ⋅ 𝑥𝑖 𝐛
𝑖
=
𝑣𝑖 ⋅ 𝑥𝑖 𝐛
𝑖
7
Target: BNE with correlated values
• 𝐯 = 𝑣1 , … , 𝑣𝑛 ∼ 𝐹: correlated distribution
𝑣1
𝑏1 𝑣1
𝐹∼
𝑣𝑖
𝑏𝑖 𝑣𝑖
• Conditional on value, maximizes utility:
𝐸 𝑢𝑖 𝐛 𝐯 | 𝑣𝑖 ≥ 𝐸 𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 𝐯−𝐢 | 𝑣𝑖
• Equilibrium Welfare:
𝐸 𝑆𝑊 𝐛 𝐯
𝑣𝑛
=𝐸
𝑣𝑖 ⋅ 𝑥𝑖 𝐛 𝐯
𝑖
𝑏𝑛 𝑣𝑛
• Optimal Welfare: highest value bidder
𝐸 𝑂𝑃𝑇 𝐯
𝑣𝑖 ⋅ 𝑥𝑖∗ 𝐯
=𝐸
𝑖
8
Target: BNE with correlated values
𝑣1
𝑏1 𝑣1
𝐹∼
𝑣𝑖
𝑏𝑖 𝑣𝑖
𝑃𝑜𝐴 =
𝐸 𝑂𝑃𝑇 𝐯
𝐸 𝑆𝑊 𝐛 𝐯
𝑣𝑛
𝑏𝑛 𝑣𝑛
9
Simpler: PNE and complete Information
• 𝑣 = (𝑣1 , … , 𝑣𝑛 ): common knowledge
𝑣1
≥
𝑏1
𝑣𝑖
• 𝑏𝑖 maximizes utility:
𝑢𝑖 𝑏 ≥ 𝑢𝑖 𝑏𝑖′ , 𝑏−𝑖
• Equilibrium Welfare:
𝑏𝑖
≥
𝑆𝑊 𝑏 =
𝑣𝑖 ⋅ 𝑥𝑖 𝐛
𝑖
𝑣𝑛
𝑏𝑛
• Optimal Welfare:
𝑣𝑖 ⋅ 𝑥𝑖∗ 𝐯
𝑂𝑃𝑇 𝑣 =
𝑖
10
Simpler: PNE and complete Information
𝑣1
≥
𝑏1
𝑣𝑖
≥
𝑂𝑃𝑇(𝐯)
𝑃𝑜𝐴 =
𝑆𝑊(𝐛)
𝑏𝑖
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
𝑖
𝑏𝑛
11
Simpler: PNE and complete Information
Theorem. 𝑃𝑜𝐴 = 1
𝑣1
Proof. Highest value player can deviate to 𝑝 𝐛
≥
𝑏1
𝑢1 𝑝 𝐛 + , 𝐛−𝐢 = 𝑣1 − 𝑝 𝐛
𝑢𝑖 0, 𝐛−𝐢 = 0
𝑣𝑖
≥
𝑏𝑖
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
𝑏𝑛
𝑖
+
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 = 𝑣1 − 𝑝 𝐛
𝑢𝑖 𝐛 ≥
𝑖
+
𝑖
By PNE condition
12
Simpler: PNE and complete Information
Theorem. 𝑃𝑜𝐴 = 1
𝑣1
Proof. Highest value player can deviate to 𝑝 𝐛
≥
𝑏1
𝑢1 𝑝 𝐛 + , 𝐛−𝐢 = 𝑣1 − 𝑝 𝐛
𝑢𝑖 0, 𝐛−𝐢 = 0
𝑣𝑖
≥
𝑏𝑖
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
𝑖
𝑏𝑛
+
+
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 = 𝑣1 − 𝑅𝐸𝑉(𝑏)
𝑈𝑇𝐼𝐿(𝑏) ≥
𝑖
𝑈𝑇𝐼𝐿 𝑏 + 𝑅𝐸𝑉 𝑏 ≥ 𝑣1
𝑆𝑊(𝑏) ≥ 𝑣1
13
 What if conclusions for PNE of complete information directly
extended to:
 incomplete information BNE
 simultaneous composition of single-item auctions
 Obviously: 𝑃𝑜𝐴 = 1 doesn’t carry over
 Possible, but we need to restrict the type of analysis
14
𝑣1
≥
• Recall. 𝑃𝑜𝐴 = 1 because highest value
player doesn’t want to deviate to 𝑝+
𝑣𝑖
𝑏𝑝1+′
• Challenge. Don’t know 𝑝 or 𝐯−𝐢 in
incomplete information
≥
𝑣𝑛
𝑝 = max 𝑏𝑖
𝑖
• Idea. Restrict deviation to not depend on
these parameters
15
Simpler: PNE and complete Information
Recall PoA=1 Proof
Theorem. 𝑃𝑜𝐴 = 1
𝑣1
≥
𝑏1
Proof. Highest value player can deviate to 𝑝 𝐛
𝑢1 𝑝 𝐛 + , 𝐛−𝐢 = 𝑣1 − 𝑝 𝐛
𝑢𝑖 0, 𝐛−𝐢 = 0
𝑣𝑖
𝑏𝑖
≥
Can we find 𝑏𝑖′ that
𝑝 𝐛 = max 𝑏𝑖 𝑣 ?
𝑖
𝑣depend only on
𝑖
𝑛
𝑏𝑛
+
+
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 = 𝑣1 − 𝑅𝐸𝑉(𝑏)
𝑈(𝑏) ≥
𝑖
𝑈 𝑏 + 𝑅𝐸𝑉 𝑏 ≥ 𝑣1
𝑆𝑊(𝑏) ≥ 𝑣1
16
(price and other values oblivious)
New Theorem. 𝑃𝑜𝐴 ≤ 𝟐
𝑣1
≥
𝑏1
Proof. Each player can deviate to
𝑥(𝑏𝑖 , 𝐛−𝐢 )
𝑥(𝑏𝑖 , 𝐛−𝐢 )
𝑣𝑖
≥
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
=
𝑣𝑖
2
1
1
𝑏𝑖
𝑏𝑖′
𝑣𝑖
2
𝑖
OR
𝑣𝑖
𝑝 𝑏 ≥
2
𝑏𝑛
𝑝 𝐛
𝑣𝑖
2
𝑣𝑖
𝑏𝑖
𝑣𝑖
𝑝 𝐛
2
𝑣17
𝑖
𝑏𝑖
(price and other values oblivious)
New Theorem. 𝑃𝑜𝐴 ≤ 𝟐
𝑣1
≥
𝑏1
Proof. Each player can deviate to
𝑥(𝑏𝑖 , 𝐛−𝐢 )
𝑣𝑖
𝑏𝑖′
=
𝑥(𝑏𝑖 , 𝐛−𝐢 )
𝑣𝑖
2
1
≥
𝑏𝑖
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
𝑖
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢
𝑝 𝑏
≥
𝑏𝑛
𝑝 𝐛
𝑣𝑖
2
𝑣𝑖
1
⋅
2
𝑏𝑖
𝑣𝑖
𝑝 𝐛
≥
𝑣𝑖18
𝑏𝑖
(price and other values oblivious)
New Theorem. 𝑃𝑜𝐴 ≤ 𝟐
𝑣1
≥
𝑏1
Proof. Each player can deviate to
𝑣𝑖
𝑏𝑖′
=
𝑣𝑖
2
≥
𝑣𝑖
𝑣𝑖
𝑢𝑖
, 𝐛−𝐢 + 𝑝 𝐛 ≥
2
2
𝑏𝑖
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
𝑖
𝑏𝑛
19
(price and other values oblivious)
New Theorem. 𝑃𝑜𝐴 ≤ 𝟐
𝑣1
≥
𝑏1
Proof. Each player can deviate to
𝑣𝑖
≥
𝑏𝑖
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
𝑖
𝑏𝑛
𝑏𝑖′
=
𝑣𝑖
2
𝑣𝑖
𝑣𝑖 ∗
∗
𝑢𝑖
, 𝐛−𝐢 + 𝑝 𝐛 ⋅ 𝑥𝑖 𝐯 ≥ ⋅ 𝑥𝑖 (𝐯)
2
2
𝑣𝑖
1
𝑈𝑇𝐼𝐿(𝐛) ≥
𝑢𝑖
, 𝐛−𝐢 + 𝑝 𝐛 ≥ 𝑂𝑃𝑇(𝐯)
2
2
𝑖
1
𝑈𝑇𝐼𝐿 𝐛 + 𝑅𝐸𝑉 𝐛 ≥ 𝑂𝑃𝑇(𝐯)
2
1
𝑆𝑊(𝐛) ≥ 𝑂𝑃𝑇(𝐯)
2
20
(price and other Key
values
oblivious)
Deviation Property
Smoothness Property
New Theorem. 𝑃𝑜𝐴 ≤ 2
𝑣1
≥
𝑏1
𝑣𝑖
Exists
≥
𝑏𝑖
𝑝 𝐛 = max 𝑏𝑖
𝑣𝑛
𝑖
𝑏𝑛
Proof. Each player can deviate to
𝑏𝑖′
𝑏𝑖′
=
depending only on own value
𝑣𝑖
2
𝑣𝑖
𝑣𝑖 ∗
∗
𝑢𝑖
, 𝐛−𝐢 + 𝑝 𝐛 ⋅ 𝑥𝑖 𝐯 ≥ ⋅ 𝑥𝑖 (𝐯)
2
2
𝑣′ 𝑖 ′
11 1
𝑈𝑇𝐼𝐿(𝐛) ≥
𝑢𝑖𝑖𝑢𝑖𝑏𝑖𝑏, 𝑖𝐛,,𝐛−𝐢
𝑝𝑝 𝐛𝐛 𝐛≥
++
𝑂𝑃𝑇(𝐯)
𝐛−𝐢
+𝑅𝐸𝑉
≥≥ 𝑂𝑃𝑇(𝐯)
𝑂𝑃𝑇(𝐯)
−𝐢
2
22 2
𝑖𝑖 𝑖
1
𝑈𝑇𝐼𝐿 𝐛 + 𝑅𝐸𝑉 𝐛 ≥ 𝑂𝑃𝑇(𝐯)
2
1
𝑆𝑊(𝑏) ≥ 𝑂𝑃𝑇(𝐯)
2
21
𝜆, 𝜇 −Smoothness via own-value deviations
Exists 𝑏𝑖′ depending only on own value
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
For any bid vector 𝐛
𝑖
22
𝜆, 𝜇 −Smoothness via own-value deviations
Exists 𝑏𝑖′ depending only on own value
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
For any bid vector 𝐛
𝑖
Note. Smoothness is property of auction not equilibrium
23
𝜆, 𝜇 −Smoothness via own-value deviations
Exists 𝑏𝑖′ depending only on own value
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
For any bid vector 𝐛
𝑖
Applies to any auction: Not First-Price Auction specific
24
Proof. If 𝐛 PNE then
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
𝑈𝑇𝐼𝐿 𝐛 + 𝜇 ⋅ 𝑅𝐸𝑉(𝐛) ≥
𝑖
Note. UTIL 𝐛 = 𝑆𝑊 𝐛 − 𝑅𝐸𝑉 𝐛
Note. SW 𝐛 ≥ 𝑅𝐸𝑉 𝐛
𝑈𝑇𝐼𝐿 𝐛 + 𝜇 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝜆 ⋅ 𝑂𝑃𝑇 𝐯
𝑆𝑊 𝐛 + (𝜇 − 1) ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝜆 ⋅ 𝑂𝑃𝑇 𝐯
𝑆𝑊 𝐛 + (𝜇 − 1) ⋅ 𝑆𝑊 𝐛 ≥ 𝜆 ⋅ 𝑂𝑃𝑇 𝐯
𝜇 ⋅ 𝑆𝑊 𝐛 ≥ 𝜆 ⋅ 𝑂𝑃𝑇 𝐯
25
First Extension Theorem. If PNE PoA proved by
showing 𝜆, 𝜇 −smoothness property via own-value
deviations, then PoA bound extends to BNE with
correlated values
Note. Not specific to First-Price Auction
26
Proof. If 𝒃(⋅) BNE then
𝐸𝑣 [
𝐸 𝑢𝑖 𝐛 𝐯
≥ 𝐸 𝑢𝑖
𝑣𝑖
, 𝐛−𝐢 𝐯−𝐢
2
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
𝑈𝑇𝐼𝐿 𝑏 + 𝜇 ⋅ 𝑅𝐸𝑉(𝑏) ≥
𝑖
]
Just redo PNE proof in expectation over values.
27
𝑣1
• Is half value best own-value deviation?
𝑣𝑖
• Bid 𝑏𝑖′ ∼ 𝐻 𝑣𝑖 with support 0, 1 − 𝑣𝑖 and
𝑒
1
′
ℎ 𝑏𝑖 =
𝑣𝑖 − 𝑏𝑖′
1
𝑏1′ ∼ 𝐻 𝑣1
𝑏𝑖′ ∼ 𝐻 𝑣𝑖
𝑝(𝐛) = max 𝑏𝑖
𝑖
𝑣𝑛
𝑏𝑛′ ∼ 𝐻 𝑣𝑛
28
•
1
1
Bid 𝑏𝑖′ ∼ 𝐻 𝑣𝑖 with support 0, 1 − 𝑒 𝑣𝑖 and ℎ 𝑏𝑖′ = 𝑣 −𝑏′
𝑖
𝑖
𝑥(𝑏𝑖 , 𝐛−𝐢 )
𝑣1
𝑏1′ ∼ 𝐻 𝑣1
𝑣𝑖
𝑢𝑖 (𝑏𝑖′ )
𝑏𝑖′ ∼ 𝐻 𝑣𝑖
𝑝(𝐛) = max 𝑏𝑖
𝑖
𝑣𝑛
𝑏𝑛′ ∼ 𝐻 𝑣𝑛
𝑝 𝐛
𝑏𝑖′
w.p.
1
𝑣𝑖
1−
𝑣𝑖
𝑒
𝑏𝑖
1
𝑣𝑖 −𝑏𝑖′
29
•
1
1
Bid 𝑏𝑖′ ∼ 𝐻 𝑣𝑖 with support 0, 1 − 𝑒 𝑣𝑖 and ℎ 𝑏𝑖′ = 𝑣 −𝑏′
𝑖
𝑖
𝑥(𝑏𝑖 , 𝐛−𝐢 )
𝑣1
𝑏1′ ∼ 𝐻 𝑣1
𝑣𝑖
𝑏𝑖′ ∼ 𝐻 𝑣𝑖
𝑝(𝐛) = max 𝑏𝑖
𝑖
𝑣𝑛
𝑏𝑛′ ∼ 𝐻 𝑣𝑛
𝑝 𝐛
𝑏𝑖′
w.p.
1
𝑣𝑖
1−
𝑣𝑖
𝑒
𝑏𝑖
1
𝑣𝑖 −𝑏𝑖′
30
•
1
1
Bid 𝑏𝑖′ ∼ 𝐻 𝑣𝑖 with support 0, 1 − 𝑒 𝑣𝑖 and ℎ 𝑏𝑖′ = 𝑣 −𝑏′
𝑖
𝑣1
𝑖
𝑥(𝑏𝑖 , 𝐛−𝐢 )
𝑏1′ ∼ 𝐻 𝑣1
𝐸 𝑢𝑖 𝑏𝑖′
𝑝 𝑏
𝑣𝑖
𝑏𝑖′ ∼ 𝐻 𝑣𝑖
𝑝 𝐛
𝑝(𝐛) = max 𝑏𝑖
𝑖
𝑣𝑛
𝑏𝑛′ ∼ 𝐻 𝑣𝑛
𝐸 𝑢𝑖 𝑏𝑖′
•
1
𝑏𝑖
1
𝑣𝑖
1−
𝑣𝑖
𝑒
+𝑝 𝑏 > 1−
1
𝑣
𝑒 𝑖
𝑒
So in fact: 1 − 𝑒 , 1 -smooth. 𝑃𝑜𝐴 ≤ 𝑒−1 ≈ 1.58
31
RECAP
 First Extension Thm. If proof of PNE PoA
based on 𝜆, 𝜇 −smoothness via ownvalue based deviations then PoA of BNE
with correlated values also 𝜇/𝜆
Complete info PNE
to BNE with
correlated values
QUESTIONS?
32
33
Single auction to simultaneous auctions
PNE complete information
 Target setting. Simultaneous single-item
first price auctions with unit-demand
bidders (complete information PNE).
 Simple setting. Single-item first price
auction (complete information PNE).
 Thm. If proof of PNE PoA of single-item
based on proving (𝜆, 𝜇)-smoothness via
own-value deviation then PNE PoA of
simultaneous auctions also 𝜇/𝜆.
Single auction to
simultaneous
auctions
PNE complete
information
References:
Roughgarden STOC’09
Roughgarden EC’12
Syrgkanis ‘12
Syrgkanis, Tardos STOC’13
34
1
𝑣𝑖1
𝑣𝑖2
𝑖
Unit-Demand Valuation
𝑗
𝑣𝑖 𝑆 = max 𝑣𝑖
𝑗∈𝑆
𝑛
𝑣𝑖3
1
𝑏𝑖1
𝑏𝑖2
𝑖
Unit-Demand Valuation
𝑗
𝑣𝑖 𝑆 = max 𝑣𝑖
𝑗∈𝑆
𝑛
𝑏𝑖3
Can we derive global efficiency guarantees from local
1
, 1 −smoothness of each first price auction?
2
𝑗𝑖∗
APPROACH: Prove smoothness of the global
mechanism
GOAL: Construct global deviation
𝑝1
𝑗
𝑣𝑖
2
0
IDEA: Pick your item in the optimal allocation
and perform the smoothness deviation for your
local value
𝑗
𝑣𝑖 ,
i.e.
𝑗
𝑣𝑖
2
0
Smoothness locally:
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝑝𝑗𝑖∗ 𝐛 ≥
Summing over players:
𝑖
𝑖
Implying
1
,1
2
′
𝑢
𝑏
𝑢𝑖𝑖 𝑏𝑖𝑖′,, 𝐛
𝐛−𝐢
−𝐢
𝑗𝑖∗
𝑣𝑖
2
11
+
+ 𝑅𝐸𝑉(𝐛)
𝑅𝐸𝑉 𝐛 ≥
≥ 2 𝑂𝑃𝑇(𝐯)
⋅ 𝑂𝑃𝑇(𝐯)
2
−smoothness property globally.
Second Extension Theorem. If proof of PNE PoA of single-item
auction based on proving (𝜆, 𝜇)-smoothness smoothness via ownvalue deviation then PNE PoA of simultaneous auctions also ≤
𝜇/𝜆.
39
 BNE PoA of simultaneous single-item auctions with correlated
unit-demand values ≤ 1/2?
 Not really: deviation not oblivious to opponent valuations
 Item in the optimal matching depends on values of opponents
40
 What we showed:
Exists 𝑏𝑖′ depending only on valuation profile 𝐯
(not 𝐛−𝐢 )
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
For any bid vector 𝐛
𝑖
41
RECAP
Second Extension Theorem. If proof of
PNE PoA of single-item auction based on
proving (𝜆, 𝜇)-smoothness then PNE PoA of
simultaneous auctions also ≤ 𝜇/𝜆.
Single auction to
simultaneous
auctions
Next we will extend above to BNE
PNE complete
information
QUESTIONS?
42
43
Complete info PNE to BNE with independent values
 Target setting. First Price Bayes-Nash
Equilibrium with asymmetric
independent values
 Simple setting. Complete information Pure
Nash Equilibrium
 Thm. If proof of PNE PoA based on (𝜆, 𝜇)-
smoothness via valuation profile
dependent deviation then PoA of BNE with
independent values also 𝜇/𝜆
Complete info PNE
to BNE with
independent values
References:
Christodoulou et al. ICALP’08
Roughgarden EC’12
Syrgkanis ‘12
Syrgkanis, Tardos STOC’13
44
𝜆, 𝜇 −Smoothness via valuation profile deviations
Exists 𝑏𝑖′ depending only on valuation profile 𝐯
(not 𝐛−𝐢 )
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
For any bid vector 𝐛
𝑖
45
Recall First Extension Theorem.
If PNE PoA proved by showing 𝜆, 𝜇 −smoothness
property via own-value deviations, then PoA bound
extends to BNE with correlated values
 Relax First Extension Theorem to allow for dependence
on opponents values
 To counterbalance: assume independent values
46
𝐹1 ∼
𝐹𝑖 ∼
𝐹𝑛 ∼
𝑣1
𝑣𝑖
𝑣𝑛
𝑗𝑖∗ 𝑣𝑖 , 𝐰−𝐢
𝑏𝑖′
𝑣𝑖 , 𝐰−𝐢
1 𝑗𝑖∗
= ⋅ 𝑣𝑖
2
𝑣𝑖 ,𝐰−𝐢
• Need to construct feasible BNE
deviations
• Each player random samples the others
values and deviates as if that was the
true values of his opponents
• Above works out, due to independence of
value distributions
47
𝐸
𝑣𝑖
𝑢𝑖
𝑏𝑖′
𝑣𝑖 , 𝐰−𝐢 , 𝐛−𝐢 𝐯−𝐢
=𝐸
Utility of deviation of player 𝑖
In expectation over his own
value too.
𝐹𝑖 ∼
𝑣𝑖
𝑤−𝑖 ∼ 𝐹−𝑖
𝑏𝑖′ 𝑣𝑖 , 𝐰−𝐢
1 𝑗𝑖∗
= ⋅ 𝑣𝑖
2
𝑤𝑖
𝑢𝑖
𝑏𝑖′ 𝐰 , 𝐛−𝐢 𝐯−𝐢
Utility of deviation from a random sample of
player 𝑖 who knows the values of all other
players.
But where players play non equilibrium
strategies.
𝑣𝑖 ,𝐰−𝐢
𝑏1 𝑣1
𝑏𝑗 𝑣𝑗
𝐯−𝐢 ∼ 𝐹𝑖
𝑏𝑛 𝑣𝑛
48
𝐸
𝑣𝑖
𝑢𝑖
𝑏𝑖′
𝑣𝑖 , 𝐰−𝐢 , 𝐛−𝐢 𝐯−𝐢
Utility of deviation of player 𝑖
In expectation over his own
value too.
𝐹𝑖 ∼
𝑤𝑖
𝑏𝑖′
1 𝑗𝑖∗
𝐰 = ⋅ 𝑣𝑖
2
𝐰
=𝐸
𝑤𝑖
𝑢𝑖
𝑏𝑖′ 𝐰 , 𝐛−𝐢 𝐯−𝐢
Utility of deviation from a random sample of
player 𝑖 who knows the values of all other
players.
But where players play non equilibrium
strategies.
𝑏1 𝑣1
𝑏𝑗 𝑣𝑗
𝐰−𝐢 ∼ 𝐹𝑖
𝑏𝑛 𝑣𝑛
49
𝑣
𝐸 𝑢𝑖 𝑖 𝑏𝑖′ 𝑣𝑖 , 𝐰−𝐢 , 𝐛−𝐢 𝐯−𝐢
𝑖
𝑤𝑖
𝑢𝑖
𝑏𝑖′ 𝐰 , 𝐛−𝐢 𝐯−𝐢
𝑖
Sum of deviating utilities
𝐹𝑖 ∼
𝑤𝑖
=𝐸
𝑏𝑖′
1 𝑗𝑖∗
𝐰 = ⋅ 𝑣𝑖
2
𝐰
Sum of complete information
setting deviating utilities
𝑏1 𝑣1
𝑏𝑗 𝑣𝑗
𝐰−𝐢 ∼ 𝐹𝑖
𝑏𝑛 𝑣𝑛
50
Recall. Exists 𝑏𝑖′ depending
only on valuation profile 𝐯
(not 𝐛𝑣𝑖−𝐢 )′
𝐸 𝑢𝑖 𝑏𝑖 𝑣𝑖 , 𝐰−𝐢 , 𝐛−𝐢 𝐯−𝐢
𝑖
𝑤𝑖
=𝐸
𝑢𝑖
𝑏𝑖′ 𝐰 , 𝐛−𝐢 𝐯−𝐢
𝑖
For any bid vector 𝐛
Utility of deviation of player 𝑖 ≥ 𝐸 𝜆 ⋅ 𝑂𝑃𝑇 𝐰 − 𝜇 ⋅ 𝑅𝐸𝑉 𝐛 𝐯
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
𝑖
𝐹𝑖 ∼
𝑤𝑖
𝑢𝑖 𝑏𝑖′ 𝐰 , 𝐛−𝐢 𝐯−𝐢
𝑏𝑖′
1 𝑗𝑖∗
𝐰 = ⋅ 𝑣𝑖
2
1 𝑗𝑖∗
≥ ⋅ 𝑣𝑖
2
𝐰
− 𝑝𝑗𝑖∗
𝐰
By smoothness on the left
𝑏1 𝑣1
𝐰
𝑏𝑗 𝑣𝑗
𝐛 𝐯
𝐰−𝐢 ∼ 𝐹𝑖
𝑏𝑛 𝑣𝑛
51
𝑣
𝐸 𝑢𝑖 𝑖 𝑏𝑖′ 𝑣𝑖 , 𝐰−𝐢 , 𝐛−𝐢 𝐯−𝐢
𝑤𝑖
=𝐸
𝑖
𝑢𝑖
𝑏𝑖′ 𝐰 , 𝐛−𝐢 𝐯−𝐢
𝑖
≥ 𝐸 𝜆 ⋅ 𝑂𝑃𝑇 𝐰 − 𝜇 ⋅ 𝑅𝐸𝑉 𝐛 𝐯
Found 𝑏𝑖′ that depend only on 𝑣𝑖 such that:
𝐸 𝑢𝑖 𝑏𝑖′ 𝑣𝑖 , 𝐛−𝐢 𝐯−𝐢
+ 𝝁 ⋅ 𝐸 𝑅𝐸𝑉 𝐛 𝐯
≥ 𝝀 ⋅ 𝐸 𝑂𝑃𝑇 𝐯
𝑖
Rest is easy
52
Third Extension Theorem. If PNE PoA proved by
showing 𝜆, 𝜇 −smoothness property via valuation
profile dependent deviations, then PoA bound extends to
BNE with independent values
53
RECAP
 Thm. If proof of PNE PoA based on (𝜆, 𝜇)smoothness via valuation profile
dependent deviation then PoA of BNE with
independent values also 𝜇/𝜆
 Corollary. If PNE PoA of single-item
auction proved via (𝜆, 𝜇)-smoothness via
valuation profile dependent deviation,
then BNE of simultaneous auctions with
unit-demand and independent also 𝜇/𝜆
Complete info PNE
to BNE with
independent values
Corollary. BNE PoA of simultaneous first
price auctions with submodular bidders ≤
𝑒
𝑒−1
QUESTIONS?
54
RECAP
 Thm. If proof of PNE PoA based on (𝜆, 𝜇)smoothness via valuation profile
dependent deviation then PoA of BNE with
independent values also 𝜇/𝜆
 Corollary. If PNE PoA of single-item
auction proved via (𝜆, 𝜇)-smoothness via
valuation profile dependent deviation,
then BNE of simultaneous auctions with
submodular and independent also 𝜇/𝜆
Complete info PNE
to BNE with
independent values
 Corollary. BNE PoA of simultaneous first
price auctions with submodular bidders
𝑒
≤
𝑒−1
QUESTIONS?
55
56
 Focusing on complete info PNE,
might be restrictive in some settings
Arguing about
distributions
 Working with the distributions
directly can potentially yield better
bounds
References:
Feldman et al. STOC’13
57
 Price of the item follows a distribution D
𝑣1 ∼ 𝐹1
 What if a player deviates to bidding a
𝑏1 (𝑣1 )
random sample from price distribution
𝑣𝑖 ∼ 𝐹𝑖
 The probability that he wins is ½ by
𝑏𝑖 (𝑣𝑖 )
𝑝∼𝐷
𝑣𝑛 ∼ 𝐹𝑛
𝑏𝑛 (𝑣𝑛 )
symmetry of the two distributions
 He pays at most 𝐸[𝑝]
𝐸
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢
𝐯−𝐢
𝑣𝑖
≥ − 𝐸[𝑝]
2
58
𝑣1 ∼ 𝐹1
 Same spirit: exists deviations that depend on
price distribution such that
𝑏1 (𝑣1 )
𝐸
𝑣𝑖 ∼ 𝐹𝑖
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢
𝑖
𝑏𝑖 (𝑣𝑖 )
𝑝∼𝐷
𝑣𝑛 ∼ 𝐹𝑛
𝐯−𝐢
+ 𝐸 𝑅𝐸𝑉 𝐛 𝐯
𝐸 𝑂𝑃𝑇 𝐯
≥
2
 BNE PoA≤ 2
𝑏𝑛 (𝑣𝑛 )
59
 Correlated deviating strategies across multiple auctions
 Decomposition of deviation analysis to separate deviations imposes
independent randomness
1
′
𝑏𝑖1
∼ 𝐷1
𝑖
𝑛
′
𝑏𝑖2
∼ 𝐷2
60
 Correlation can achieve higher deviating utility
𝑆
1
𝑃1
′
𝑏𝑖1
𝑖
𝐵𝑖′ ∼ 𝐷
∼𝐷
′
𝑏𝑖2
𝑃2
Sub-additive valuations
𝑣𝑖 𝑆 + 𝑣𝑖 𝑇 ≥ 𝑣𝑖 (𝑆 ∪ 𝑇)
𝑛
61
• Draw bid from price distribution
 Correlation can achieve higher deviating utility
𝑆
1
𝑝1
𝑏1′
𝑖
• X(𝑏, 𝑝): set of won items with
bid vector b and price vector p
𝒃′ ∼ 𝐷
∼𝐷
𝑏2′
𝑝2
Sub-additive valuations
𝑣𝑖 𝑆 + 𝑣𝑖 𝑇 ≥ 𝑣𝑖 (𝑆 ∪ 𝑇)
• By symmetry:
𝐸 𝑣 𝑋 𝑏′ , 𝑝
= 𝐸 𝑣 𝑋 𝑝, 𝑏 ′
𝑛
• Value collected: 𝐸 𝑣 𝑋
• Either I win or price wins:
𝑋 𝑏, 𝑝 + 𝑋 𝑝, 𝑏 = 𝑆
𝑏′ , 𝑝
=
1
𝐸
2
𝑣 𝑋
𝑏′ , 𝑝
+𝑣 𝑋
𝑝, 𝑏 ′
≥
1
𝐸
2
𝑣 𝑆
62
 Drawing deviation from price
distribution!
Arguing about
distributions
 Buys correlation across auctions
 Better bounds beyond submodular
63
64
Vickrey Auction - Truthful, efficient, simple
(second price)
$2
$99
$5
$0
Pays
$0
$7
$0
$3
$0
Pays
$5
$4
$0
but has many bad Nash equilibria
Assume bid ≤ value (no overbidding)
Theorem. All Nash equilibria efficient. highest
value wins
65
 Same approach but replace Payments with “Winning Bids” and use
no-overbidding
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝐵𝐼𝐷𝑆 𝐛 ≥ 𝝀 ⋅ 𝑂𝑃𝑇(𝐯)
For any bid vector 𝐛
𝑖
 No overbidding assumption:
𝐵𝐼𝐷𝑆 ≤ 𝑊𝐸𝐿𝐹𝐴𝑅𝐸
Then 𝑃𝑜𝐴 ≤
1+𝜇
𝜆
66
 Deviate to bidding your value: 𝑏𝑖′ 𝑣𝑖 = 𝑣𝑖
 𝐵(𝐛): winning bid
 Either winning bid B(𝐛) ≥ 𝑣𝑖 or 𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 = 𝑣𝑖 − 𝐵𝑖 𝐛
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝐵𝑖 𝐛 ≥ 𝑣𝑖 ⇒ 𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝐵𝑖 𝐛 ⋅ 𝑥𝑖∗ 𝐯 ≥ 𝑣𝑖 ⋅ 𝑥𝑖∗ 𝐯
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝐵𝐼𝐷𝑆 𝐛 ≥ 𝑂𝑃𝑇(𝐯)
𝑖
67
 Vickrey auction (1,1)-smooth using bids
 𝑃𝑜𝐴 ≤ 2: under no-overbidding
 Vickrey is efficient?
 𝑃𝑜𝐴 ≤ 2: extends to simultaneous Vickrey auctions even under
BNE with independent values
68
69
Advertisers
Slots
 Allocate slots by bid
𝑣1 ∼ 𝐹1
1
𝑏1
𝑎1
 Charge bid per-click
𝑎2
 Utility:
𝑣𝑖 ∼ 𝐹𝑖
𝑖
𝑏𝑖
CTRs
𝑢𝑖 𝑏 = 𝑎𝜎
𝑖
𝑣𝑖 − 𝑏𝑖
𝑎3
𝑣𝑛 ∼ 𝐹𝑛
𝑛
𝑏𝑛
𝑎4
70
Items
Unit-Demand Bidders
 Allocated items greedily to
highest remaining bid
1
𝑗 𝑏
 If allocated item 𝑗 𝑏 , charge 𝑏𝑖
𝑏𝑖1
𝑖
Unit-Demand
𝑗
𝑣𝑖 𝑆 = max 𝑣𝑖
𝑗∈𝑆
𝑏𝑖2
 Utility:
𝑗(𝑏)
𝑢𝑖 𝑏 = 𝑣𝑖
𝑗 𝑏
− 𝑏𝑖
𝑏𝑖3
𝑛
71
Single-Minded Bidders
Items
𝑆1
 Each bidder submits 𝑏𝑖 and 𝑇𝑖
1
𝑆2
2
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
 Run some algorithm (optimal or
greedy 𝑂 𝑚 -approx.) over reported
single-minded values
 Charge bid 𝑏𝑖 if allocated
3
𝑆3
72
GFP
 Allocate slots by bid
 Charge bid per-click
 Utility:
𝑢𝑖 𝑏 = 𝑎𝜎
𝑖
𝑣𝑖 − 𝑏𝑖
Matching
Markets-Greedy
Allocation
Single-Minded
Combinatorial
Auctions
 Allocated items greedily to
 Each bidder submits 𝑏𝑖
highest remaining bid
 If allocated item 𝑗 𝑏 ,
charge
 Utility:
𝑗 𝑏
𝑏𝑖
𝑗(𝑏)
𝑢𝑖 𝑏 = 𝑣𝑖
𝑗 𝑏
− 𝑏𝑖
and 𝑇𝑖
 Run some algorithm
(optimal or greedy
𝑂 𝑚 -approx.) over
reported single-minded
values
 Charge bid 𝑏𝑖 if
allocated
73
74
Advertisers
Slots
 Allocate slots by bid
𝑣1 ∼ 𝐹1
1
𝑏1
𝑎1
 Charge bid per-click
𝑎2
 Utility:
𝑣𝑖 ∼ 𝐹𝑖
𝑖
𝑏𝑖
CTRs
𝑢𝑖 𝑏 = 𝑎𝜎
𝑖
𝑣𝑖 − 𝑏𝑖
𝑎3
𝑣𝑛 ∼ 𝐹𝑛
𝑛
𝑏𝑛
75
Advertisers
Slots
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅
𝑣1
𝑖
≥
1
𝑜𝑝𝑡 1
𝑖
 𝑏𝑖′ =
𝑎1
𝑣2
𝑣𝑖
2
 Either bid of player at slot opt(𝑖) ≥
𝑜𝑝𝑡 2
𝑎2
≥
2
 Or utility ≥
𝑜𝑝𝑡 3
3
𝑎𝑜𝑝𝑡 𝑖 𝑣𝑖
2
𝑣𝑖
𝑢𝑖
,𝑏
+ 𝑎𝑜𝑝𝑡
2 −𝑖
CTRs
𝑎3
𝑣3
𝑎𝑜𝑝𝑡 𝑖 𝑣𝑖
𝑖
𝑖
⋅ 𝑏𝜋
𝑜𝑝𝑡 𝑖
𝑣𝑖
2
𝑎𝑜𝑝𝑡 𝑖 𝑣𝑖
≥
2
𝑎𝑜𝑝𝑡 𝑖 𝑣𝑖
𝑣𝑖
𝑢𝑖
,𝑏
+
𝑎𝑜𝑝𝑡 𝑖 ⋅ 𝑏𝜋 𝑜𝑝𝑡 𝑖 ≥
2 −𝑖
2
𝑖
𝑖
𝑣𝑖
1
𝑢𝑖
, 𝑏−𝑖 + 𝑅𝐸𝑉 𝑏 ≥ ⋅ 𝑂𝑃𝑇(𝑣)
2
2
𝑖
76
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Slots
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅
𝑣1
𝑖
≥
1
𝑜𝑝𝑡 1
𝑖
𝑎1
𝑢𝑖
𝑣2
𝑖
𝑜𝑝𝑡 2
≥
CTRs
𝑎3
𝑣3
𝑜𝑝𝑡 3
3
𝑣𝑖
1
, 𝑏−𝑖 + 𝑅𝐸𝑉 𝑏 ≥ ⋅ 𝑂𝑃𝑇(𝑣)
2
2
Thm. 𝑃𝑜𝐴 ≤ 2
𝑎2
2
𝑎𝑜𝑝𝑡 𝑖 𝑣𝑖
Proof.
𝑣𝑖
𝑢𝑖 𝑏 ≥
𝑢𝑖
, 𝑏−𝑖
2
𝑖
𝑖
1
𝑈𝑇𝐼𝐿 𝑏 + 𝑅𝐸𝑉 𝑏 ≥ ⋅ 𝑂𝑃𝑇(𝑣)
2
1
𝑆𝑊 𝑣 ≥ ⋅ 𝑂𝑃𝑇(𝑣)
2
77
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Slots
𝑢𝑖 𝑏𝑖′ , 𝐛−𝐢 + 𝝁 ⋅ 𝑅𝐸𝑉 𝐛 ≥ 𝝀 ⋅
𝑣1
𝑖
≥
1
𝑜𝑝𝑡 1
𝑖
𝑎1
𝑢𝑖
𝑣2
𝑖
𝑜𝑝𝑡 2
≥
CTRs
𝑎3
𝑣3
𝑜𝑝𝑡 3
3
𝑣𝑖
1
, 𝑏−𝑖 + 𝑅𝐸𝑉 𝑏 ≥ ⋅ 𝑂𝑃𝑇(𝑣)
2
2
Thm. Bayes-Nash 𝑃𝑜𝐴 ≤ 2
𝑎2
2
𝑎𝑜𝑝𝑡 𝑖 𝑣𝑖
Proof.
𝑣𝑖
𝐸 𝑢𝑖 𝑏 𝐯 ≥
𝐸 𝑢𝑖
,𝑏 𝑣
2 −𝑖 −𝑖
𝑖
𝑖
1
𝐸 𝑈𝑇𝐼𝐿 𝑏 𝐯 + 𝐸 𝑅𝐸𝑉 𝑏 𝐯 ≥ ⋅ 𝐸 𝑂𝑃𝑇 𝐯
2
1
𝐸 𝑆𝑊 b 𝐯 ≥ ⋅ 𝐸 𝑂𝑃𝑇 𝐯
2
78
Items
Unit-Demand Bidders
 Allocated items greedily to
highest remaining bid
1
𝑗 𝑏
 If allocated item 𝑗 𝑏 , charge 𝑏𝑖
𝑏𝑖1
𝑖
Unit-Demand
𝑗
𝑣𝑖 𝑆 = max 𝑣𝑖
𝑗∈𝑆
𝑏𝑖2
 Utility:
𝑗(𝑏)
𝑢𝑖 𝑏 = 𝑣𝑖
𝑗 𝑏
− 𝑏𝑖
𝑏𝑖3
𝑛
79
Items
Unit-Demand Bidders
𝑗
𝑣𝑖
𝑗
𝑏𝑖 =
2
 Only for 𝑗 =item in optimal matching
1
𝑏𝑖1
𝑖
Unit-Demand
𝑗
𝑣𝑖 𝑆 = max 𝑣𝑖
𝑗∈𝑆
𝑛
 Deviation
𝑏𝑖2
 If 𝑝𝑗 𝑏 is price of item 𝑗
𝑗
𝑢𝑖 𝑏𝑖′ , 𝑏−𝑖
𝑏𝑖3
 Thus
𝑣𝑖
≥
− 𝑝𝑗 (𝑏)
2
1
,1
2
-smooth via valuation profile
dependent deviations
80
Items
Unit-Demand Bidders
 In fact
𝑗
𝑗
𝑏𝑖 ∼ 𝐻 𝑣𝑖
 Only for 𝑗 =item in optimal matching
1
𝑢𝑖 𝑏𝑖′ , 𝑏−𝑖
𝑏𝑖1
𝑖
Unit-Demand
𝑗
𝑣𝑖 𝑆 = max 𝑣𝑖
𝑗∈𝑆
𝑏𝑖2
𝑏𝑖3
1 𝑗
≥ 1−
𝑣 − 𝑝𝑗 (𝑏)
𝑒 𝑖
1
𝑒
 Thus 1 − , 1 -smooth
 Greedy on true values: 2-approx.
 Greedy on reported values: 1.58-approx.!
𝑛
81
 Greedy on true values: 2-approx.
Unit-Demand Bidders
1−𝜖
0
1
Items
 At equilibrium:
 Player 2 never goes for first item
 Too expensive
 So allocation is efficient
1−𝜖
82
Single-Minded Bidders
Items
 Each bidder submits 𝑏𝑖 and 𝑇𝑖
𝑆1
1
 Run some algorithm over
𝑆2
2
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
3
reported single-minded values
 Charge bid 𝑏𝑖 if allocated
𝑆3
83
Single-Minded Bidders
Items
𝑆1
 Each bidder submits 𝑏𝑖 and 𝑇𝑖
1
𝑆2
2
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
 Run optimal algorithm over
reported single-minded values
 Charge bid 𝑏𝑖 if allocated
3
𝑆3
84
𝑚 Items
𝑣 =1−𝜖
𝑣1 = 1
𝑣 =1−𝜖
• 𝑆𝑊 = 1 but 𝑂𝑃𝑇 = 𝑚
…
…
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
• Other players bid 0
…
…
𝑣2 = 1
At equilibrium:
• 1 and 2 bid 𝑏 = 1, T = 𝑚
𝑣 =1−𝜖
𝑆1 = 𝑆2
85
Single-Minded Bidders
Items
𝑆1
 Each bidder submits 𝑏𝑖 and 𝑇𝑖
1
𝑆2
2
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
3
𝑆3
𝒎 −Approximation
Algorithm over reported
single-minded values
 Run
 Charge bid 𝑏𝑖 if allocated
86
Single-Minded Bidders
Items
𝒎 −Approximation Algorithm
𝑆1
1
𝑏𝑖
 Reweight bids as: 𝒃𝒊 =
|𝑇𝑖 |
𝑆2
2
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
 Allocate in decreasing order of 𝒃𝒊
 Charge bid 𝑏𝑖 if allocated
 Idea: A player can block at most
3
𝑆3
𝑚 other players of same value
from being allocated
87
𝑚 Items
𝑣 =1−𝜖
𝑣1 = 1
…
…
𝑣2 = 1
𝑣 =1−𝜖
…
…
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
Large players cannot block
all small players
𝑣 =1−𝜖
𝑆1 = 𝑆2
88
Single-Minded Bidders
Items
𝑣𝑖
 Deviation 𝑏𝑖′ : bid
for 𝑆𝑖
2
𝑆1
1
 Let 𝜏𝑖 (b): Threshold bid for being
allocated 𝑆𝑖 (including bid of player)
𝑆2
2
 By similar analysis:
Single-minded:
𝑣𝑖 for whole set 𝑆𝑖
3
𝑢𝑖 𝑏𝑖′ , 𝑏−𝑖
𝑆3
 Need to show:
𝑣𝑖
+ 𝜏𝑖 (𝑏) ≥
2
𝑖 𝜏𝑖
𝑏 ≤ 𝑐 ⋅ 𝑅𝐸𝑉
89
 Fact: Algorithm is
𝑚 −approximation
 Think of hypothetical situation where each bidder is duplicated
 Duplicate bidder bids: 𝑏𝑖 = 𝜏𝑖 𝑏 − 𝜖 for set 𝑆𝑖
 By definition of 𝜏𝑖 (𝑏): algorithm doesn’t allocate to them
 Allocating to duplicate bidders yields welfare
𝜏𝑖 (𝑏)
𝑖
 Since algorithm is
𝑚 −approximation: 𝑅𝐸𝑉 =
𝑖 𝑏𝑖 𝑋𝑖 (𝑏) ≥
1
𝑚
𝑖 𝜏𝑖 (𝑏)
90
 Approximate mechanism:
 Welfare at equilibrium 𝑂
1
,
2
𝑚 −smooth
𝑚 -approximate NOT 𝑂 𝑚 −approximate
91
 Smoothness
Roughgarden STOC’09, Lucier, Paes Leme EC’11, Roughgarden EC’12, Syrgkanis ‘12,
Syrgkanis, Tardos STOC’13
 Simultaneous First-Second Price Single-Item Auctions
Bikhchandani GEB’96, Christodoulou, Kovacs, Schapira ICALP’08, Bhawalkar, Roughgarden
SODA’11, Hassidim, Kaplan, Mansour, Nisan EC’11, Feldman, Fu, Gravin, Lucier STOC’13
 Auctions based on Greedy Allocation Algorithms
Lucier, Borodin SODA’10
 AdAuctions (GSP, GFP)
Paes-Leme Tardos FOCS’10, Lucier, Paes-Leme + CKKK EC’11
 Sequential First/Second Price Auctions
Paes Leme, Syrgkanis, Tardos SODA’12, Syrgkanis, Tardos EC’12
 Multi-Unit Auctions
Bart de Keijzer et al. ESA’13
All above can be thought as smoothness proofs and some are compositions of auctions
Price of Anarchy in Auctions and Mechanisms
 Dutting, Henzinger, Stanberger. Valuation Compressions in VCG-Based Combinatorial Auctions
 Jose R. Correa, Andreas S. Schulz and Nicolas E. Stier-Moses. The Price of Anarchy of the
Proportional Allocation Mechanism Revisited
 Jason Hartline, Darrell Hoy and Sam Taggart. Interim Smoothness for Auction Welfare and
Revenue. (poster)
 Michal Feldman, Vasilis Syrgkanis and Brendan Lucier. Limits of Efficiency in Sequential Auctions
 Brendan Lucier, Yaron Singer, Vasilis Syrgkanis and Eva Tardos. Equilibrium in Combinatorial
Public Projects
Price of Anarchy in Games
 Xinran He and David Kempe. Price of Anarchy for the N-player Competitive Cascade Game with
Submodular Activation Functions
 Mona Rahn and Guido Schäfer. Bounding the Inefficiency of Altruism Through Social Contribution
Games
 Yoram Bachrach, Vasilis Syrgkanis and Milan Vojnovic. Incentives and Efficiency in Uncertain
Collaborative Environments