This document discusses the intrinsic robustness of the price of anarchy concept in game theory. It defines key concepts like pure Nash equilibrium, price of anarchy, and smooth games. Smooth games are games where the cost of any outcome is close to the cost of the optimal outcome. The document proves several games are smooth, including congestion games with affine costs, valid utility games, and simultaneous second-price auctions. It introduces the concept of robust price of anarchy as being defined based on the smoothness parameters of a game. The document also presents an extension theorem showing price of anarchy bounds extend beyond pure Nash equilibrium to other equilibrium concepts and outcome sequences.

1. Intrinsic Robustness of
the Price of Anarchy
Tim Roughgarden
July 3,2013
1

2. Basic Knowledge
• PNE
• Optimal Solution
Improvement upon given dictatorial control over everyone’s actions
• Price of Anarchy
largest cost of an equilibrium
cost of an optimal outcome
2

3. Introduction
• Why need more robust bounds?
• Hard to coordinate on one of multiple Equilibrium
• PNE is computationally intractable
• PNE does not exist
Need a more robust bounds to some wider range of outcome
3

4. Basic Knowledge
• MNE
Ex: “Rock-Paper-Scissors”
[Always exist/hard to compute]
• CorEq
[Easy to compute/hard to learn]
• No Regret [CCE]
[Easy to compute /learn]
4
PNE
MNE
CorEq
No Regret [CCE]

5. Smooth Game
• Definition [on Cost-minimization game]
5
𝜆, 𝜇 − 𝑠𝑚𝑜𝑜𝑡ℎ 𝑔𝑎𝑚𝑒 [ F𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑡𝑤𝑜 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑠 , 𝑠∗]
C(s)= 𝑖=1
𝑘
𝐶𝑖(𝑠)

6. Smooth Game
• Definition of Robust POA [on Cost-minimization game]
inf{
λ
1−𝜇
: 𝜆, 𝜇 𝑠. 𝑡. 𝐺 𝑖𝑠 𝜆, 𝜇 − 𝑠𝑚𝑜𝑜𝑡ℎ } (𝜇 𝑎𝑙𝑤𝑎𝑦𝑠 ≤ 1)
• Relaxation of Smoothness
𝐶 𝑠 ≤ 𝑖=1
𝑘
𝐶𝑖 𝑠 ;
6

7. Smooth Game
• Definition [on Payoff-Maximization Game]
• Definition of Robust POA
sup {
λ
1+𝜇
: 𝜆, 𝜇 𝑠. 𝑡. 𝐺 𝑖𝑠 𝜆, 𝜇 − 𝑠𝑚𝑜𝑜𝑡ℎ }
7
𝜆, 𝜇 − 𝑠𝑚𝑜𝑜𝑡ℎ 𝑔𝑎𝑚𝑒 [ F𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑡𝑤𝑜 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑠 , 𝑠∗
]

8. Example & Non-Example
Example
• Congestion Game With Affine Cost Function
• Valid Utility Game
• Simultaneous Second-Price Auctions
Non-Example
• Network Formation Game
• Symmetric Congestion Games with Singleton Strategies
8

9. Example
Congestion Game With Affine Cost Function
𝐸: 𝑎 𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑒𝑠𝑜𝑢𝑟𝑠𝑒 , 𝑎 𝑠𝑒𝑡 𝑜𝑓 𝑘 𝑝𝑙𝑎𝑦𝑒𝑟𝑠
𝑐 𝑒 𝑥 = 𝑎 𝑒 𝑥 + 𝑏 𝑒 , 𝑎 𝑒 > 0, 𝑏 𝑒 > 0
𝑥 𝑒 = |{𝑖: 𝑒 ∈ 𝑠𝑖}|
9

10. Example
Congestion Game With Affine Cost Function
We claim that Congestion Game With Affine Cost Function 𝑖𝑠
5
3
,
1
3
− 𝑠𝑚𝑜𝑜𝑡ℎ , 𝑟𝑜𝑏𝑢𝑠𝑡 𝑃𝑂𝐴 𝑖𝑠 𝑎𝑡 𝑚𝑜𝑠𝑡
5
2
10
(𝑓𝑟𝑜𝑚 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑠𝑡𝑢𝑑𝑖𝑒𝑠)

11. Example
Valid Utility Game
𝐸: 𝑎 𝑔𝑟𝑜𝑢𝑛𝑑 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑒𝑠𝑜𝑢𝑟𝑠𝑒 , 𝑎 𝑠𝑒𝑡 𝑜𝑓 𝑘 𝑝𝑙𝑎𝑦𝑒𝑟𝑠
𝑉: 𝑛𝑜𝑛𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑜𝑛 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐸(𝑣𝑎𝑙𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑙𝑎𝑦𝑒𝑟)
Π𝑖: 𝑝𝑎𝑦𝑜𝑓𝑓 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑝𝑙𝑎𝑦𝑒𝑟
𝑈 𝑠 ⊆ 𝐸 ∶ 𝑡ℎ𝑒 𝑢𝑛𝑖𝑜𝑛 𝑜𝑓 𝑝𝑙𝑎𝑦𝑒𝑟𝑠′ 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑖𝑒𝑠 𝑖𝑛 𝑠
𝑊 𝑠 = 𝑉 𝑈 𝑠
Definition of “Valid”:
𝑖 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖 , 𝑠 , Π𝑖 𝑠 ≥ 𝑊 𝑠 − 𝑊 𝜙, 𝑠−𝑖
𝑖𝑖 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑠 , 𝑖=1
𝑘
Π𝑖 𝑠 ≤ 𝑊 𝑠
11

12. Example
Valid Utility Game
𝑍𝑖 ⊆ 𝐸: 𝑡ℎ𝑒 𝑢𝑛𝑖𝑜𝑛 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑦𝑒𝑟𝑠′
𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑖𝑒𝑠 𝑖𝑛 𝑠 ,
𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑖𝑒𝑠 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑 𝑏𝑦 𝑝𝑙𝑎𝑦𝑒𝑟𝑠 1,2, . . . , 𝑖 𝑖𝑛 𝑠∗
We claim that Congestion Game With Affine Cost Function 𝑖𝑠 1,1 − 𝑠𝑚𝑜𝑜𝑡ℎ , 𝑟𝑜𝑏𝑢𝑠𝑡 𝑃𝑂𝐴 𝑖𝑠 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡
1
2
12
(𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑎𝑟𝑖𝑡𝑦 𝑜𝑓 𝑉)

13. Example
Simultaneous Second-price Auctions
𝐴 𝑠𝑒𝑡 𝑜𝑓 𝑔𝑜𝑜𝑑𝑠 1,2, … , 𝑚 𝑓𝑜𝑟 𝑠𝑎𝑙𝑒
𝑣𝑖 𝑇 : 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑔𝑜𝑜𝑑𝑠 𝑎𝑛𝑑 𝑖𝑠 𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑜𝑟
𝑏𝑖𝑗: 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑔𝑜𝑜𝑑 𝑗 , 𝑝𝑙𝑎𝑦𝑒𝑟 𝑖′ 𝑠 𝑏𝑖𝑑 𝑜𝑛 𝑖𝑡
𝑣𝑖 𝑇 ≥
𝑗∈𝑇
𝑏𝑖𝑗
Each good is allocated independent, at a price equal to the second highest price
13

14. Example
Simultaneous Second-price Auctions
𝑋𝑖 𝑏 ⊆ 1,2, … , 𝑚 : 𝑡ℎ𝑒 𝑔𝑜𝑜𝑑𝑠 𝑡ℎ𝑎𝑡 𝑖 𝑤𝑖𝑛𝑠 , 𝑑𝑒𝑓𝑖𝑛𝑒 𝑜𝑛 𝑎 𝑏𝑖𝑑 𝑝𝑟𝑜𝑓𝑖𝑙𝑒
𝑝𝑖 𝑏 =
𝑗∈𝑋 𝑖(𝑏)
𝑏 2 𝑗 : 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑜𝑓 𝑏𝑖𝑑𝑑𝑒𝑟 𝑖
Π𝑖 𝑏 = 𝑣𝑖 𝑋𝑖 𝑏 − 𝑝𝑖 𝑏 : 𝑝𝑎𝑦𝑜𝑓𝑓 𝑜𝑓 𝑖
𝑊 𝑏 =
𝑖=1
𝑘
𝑣𝑖 𝑋𝑖 𝑏 : 𝑠𝑜𝑐𝑖𝑎𝑙 𝑤𝑒𝑙𝑓𝑎𝑟𝑒 , 𝑖𝑛𝑐𝑙𝑢𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑒𝑙𝑙𝑒𝑟
14
This game satisfies the following relaxation of (1,1)-smoothness

15. Example
Simultaneous Second-price Auctions [the relaxed smoothness condition]
𝑇𝑖: 𝑔𝑜𝑜𝑑𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑒𝑑 𝑡𝑜 𝑏𝑖𝑑𝑑𝑒𝑟 𝑖
15

16. Tight Class of Game
Definition
A set 𝒢 𝑜𝑓 𝑐𝑜𝑠𝑡 − 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑚𝑒 𝑖𝑠 𝑡𝑖𝑔ℎ𝑡 𝑖𝑓
𝐴 𝒢 : 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑣𝑎𝑙𝑢𝑒𝑠 λ, 𝜇 𝑠. 𝑡. 𝒢 𝑖𝑠 𝜆, 𝜇 − 𝑠𝑚𝑜𝑜𝑡ℎ
𝒢 ⊆ 𝒢: the games with at least one PNE
𝜌 𝑝𝑢𝑟𝑒 𝐺 : 𝑡ℎ𝑒 𝑃𝑂𝐴 𝑜𝑓 𝒢
16

17. Extension Theorem
Static Version
𝐹𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑐𝑜𝑠𝑡 − 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑚𝑒 𝐺 𝑤𝑖𝑡ℎ 𝑅𝑜𝑏𝑢𝑠𝑡 𝑃𝑂𝐴 𝜌 𝐺 ,
𝑒𝑣𝑒𝑟𝑦 𝐶𝐶𝐸 𝑜𝑓 𝐺, 𝑎𝑛𝑑 𝑒𝑣𝑒𝑟𝑦 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑠∗
𝑜𝑓 𝐺,
17

18. Extension Theorem
Repeat Play and No-Regret Sequences
18
𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑎 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑠1 , 𝑠2, … , 𝑠 𝑇 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑜𝑓 𝑎 (𝜆, µ) − 𝑠𝑚𝑜𝑜𝑡ℎ 𝑔𝑎𝑚𝑒
(ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑡𝑖𝑐𝑎𝑙 𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑝𝑙𝑎𝑦𝑒𝑟 𝑖 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑡 )
(𝑏𝑦 𝑡ℎ𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑚𝑜𝑜𝑡ℎ𝑛𝑒𝑠𝑠)

19. Extension Theorem
Repeat Play and No-Regret Sequences
19
𝑖𝑓 𝑤𝑒 𝑐𝑜𝑛𝑐𝑒𝑟𝑛 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑖𝑛 𝑒𝑣𝑒𝑟𝑦 𝑝𝑙𝑎𝑦𝑒𝑟𝑠 𝑒𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒𝑠 "𝑣𝑎𝑛𝑖𝑠ℎ 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑒𝑔𝑟𝑒𝑡 " ∶
(𝑜 1 𝑖𝑠 𝑠𝑜𝑚𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 𝑔𝑜𝑒𝑠 𝑡𝑜 0 𝑎𝑠 𝑇 → 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦)
(𝑛𝑜 𝑟𝑒𝑔𝑟𝑒𝑡 𝑚𝑎𝑘𝑒𝑠 𝑡ℎ𝑒 𝑙𝑎𝑠𝑡 𝑡𝑒𝑟𝑚 𝑔𝑜𝑒𝑠 𝑡𝑜 0 𝑎𝑠 𝑇 → 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦)

20. Extension Theorem
Repeat Version
20
𝑡ℎ𝑎𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠 24 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑝𝑙𝑎𝑦𝑒𝑟, 𝑎𝑛𝑑 𝑒𝑣𝑒𝑟𝑦 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑠∗
𝑜𝑓𝐺, 𝐹𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑐𝑜𝑠𝑡
− 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑚𝑒 𝐺 𝑤𝑖𝑡ℎ 𝑟𝑜𝑏𝑢𝑠𝑡 𝑃𝑂𝐴 𝜌 𝐺 , 𝑒𝑣𝑒𝑟𝑦 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑠1
, … , 𝑠 𝑇
(𝑎𝑠 𝑇 → 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦) (𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑎𝑛𝑎𝑟𝑐ℎ𝑦)

21. Extension Theorem
Repeat Version (Mixed-Strategy )
21
(𝑎𝑠 𝑇 → 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦)

22. Approximate Equilibria
• Approximate Equilibria
𝜖 − 𝑁𝑎𝑠ℎ 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 ∶ Ci s ≤ 1 + 𝜖 Ci(si
′
, s−i) (cost-minimization games )
𝜖 − 𝑁𝑎𝑠ℎ 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 ∶ (𝑝𝑎𝑦𝑜𝑓𝑓 𝑚𝑎𝑧𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑚𝑒𝑠)
22

23. Bicriteria Bound
• Smooth Closed Sets of Cost-Minimization Games
23
𝒢: 𝑎 𝑠𝑒𝑡 𝑜𝑓 𝑐𝑜𝑠𝑡𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑚𝑒𝑠 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑢𝑛𝑑𝑒𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 𝑑𝑒𝑙𝑒𝑡𝑖𝑜𝑛𝑠 𝑎𝑛𝑑 𝑑𝑢𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠
𝐺 ∈ 𝒢 ∶ 𝐺 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝜆, 𝜇 − 𝑠𝑚𝑜𝑜𝑡ℎ
𝐺: 𝐺𝑎𝑚𝑒 𝑡ℎ𝑎𝑡 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝐺 𝑏𝑦 𝑑𝑢𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑛𝑔 𝑒𝑎𝑐ℎ 𝑝𝑙𝑎𝑦𝑒𝑟 𝑖 𝑛𝑖 𝑡𝑖𝑚𝑒𝑠

24. Other Topic in this paper
• Congestion games Are Tight [To General Case]
• Shortest Best-Response Sequencing (Best-Response Dynamics)
24

25. Subsequent Work
• Guarantees with Irrational Players
• Relaxing the Smoothness Condition
• The POA in Games of Incomplete Information
• Limits of Smoothness
25

26. Thanks for your
attention!
26