PRICE OF ANARCHY: QUANTIFYING THE INEFFICIENCY OF EQUILIBRIA Zongxu Mu “The Invisible Hand”

Equilibria and Efficiency ◦ Central to free market economics The Wealth of Nations (Smith, 1776) ◦ “… led by an invisible hand to promote an end which was no part of his intention” ◦ Self-interest agents  social-efficient outcomes

Source: http://en.wikipedia.org/wiki/File:AdamSmith.jpg

23-Jan-14 INEFFICIENCY OF EQUILIBRIA 2 Inefficiency of Equilibria

Inefficient equilibrium in markets: ◦ Of certain structures (e.g., monopoly) ◦ For certain kinds of goods (e.g., public goods) ◦ With externalities (e.g., pollution) ◦ … absence of order or government Government interventions can be beneficial -- Merriam Webster ◦ There is a price (efficiency lost) of “anarchy”

23-Jan-14 INEFFICIENCY OF EQUILIBRIA 3 Inefficiency of Equilibria

Nash equilibrium: DD Prisoner’s Dilemma ◦ Pareto-dominated 퐶 퐷 ◦ The only non-Pareto-optimal outcome! 퐶 −1, −1 −4, 0

Pareto-optimality: a qualitative 퐷 0, −4 −3, −3 observation A quantitative measure?

23-Jan-14 INEFFICIENCY OF EQUILIBRIA 4 Outline

• Pareto-optimality Inefficiency of Equilibria • Price of Anarchy • Pigou’s example Selfish Routing Games • Nonatomic games

Other Applications

Summary

23-Jan-14 OUTLINE 5 Inefficiency of Equilibria – A Short History

Source: http://en.wikipedia.org/wiki/File:John_Forbes_Nash,_Jr._by_Peter_Badge.jpg Source: http://cgi.di.uoa.gr/~elias/images/elias-bio.jpg

Inefficiency of Equilibrium Price of Anarchy Rapoport and Chammah Papadimitriou 1965 2001

1951 1999 Nash Equilibrium Origin of PoA: Coordination Ratio Nash Koutsoupias and Papadimitriou

Source: http://en.wikipedia.org/wiki/File:Anatol_Rapoport.jpg Source: http://www.cs.berkeley.edu/~christos/index_files/image002.png

23-Jan-14 INEFFICIENCY OF EQUILIBRIA 7 Inefficiency of Equilibria

Optimality in utilities? ◦ Utilities of different persons cannot be compared or summed up Cost or payoff may also have concrete interpretations ◦ Money, network delay, … Specific objective functions for “social cost”

◦ Utilitarian: 푓 표 = 푢푖

◦ Egalitarian: 푓 표 = max 푢푖 .

23-Jan-14 INEFFICIENCY OF EQUILIBRIA 8 Inefficiency of Equilibria

Objective Function  Quantify Prisoner’s Dilemma Price of Anarchy 퐶 퐷 ◦ Similar to approximation ratio 퐶 −1, −1 −4, 0 푓(푤표푟푠푡 푒푞푢푖푙푖푏푟푖푢푚) 푃표퐴 = 푓(표푝푡푖푚푎푙 표푢푡푐표푚푒) 퐷 0, −4 −3, −3

−3 +(−3) = = 3 −1 +(−1)

23-Jan-14 INEFFICIENCY OF EQUILIBRIA 9 Price of Anarchy – Properties and Interests

Can be unbounded Prisoner’s Dilemma ◦ 푑 → +∞ 퐶 퐷 −푑 +(−푑) ◦ 푃표퐴 = = 푑 → +∞ −1 +(−1) 퐶 −1, −1 −푑 − 1, 0 Can be bounded 퐷 0, −푑 − 1 −푑, −푑 Is central control needed? ◦ Mechanism design

23-Jan-14 INEFFICIENCY OF EQUILIBRIA 10 Selfish Routing Games

Pigou’s (1920) example ◦ 푠: source; 푡: sink ◦ 푐(푥): unit cost of an edge 푐 푥 = 1 ◦ 1 unit of traffic in total 풔 풕 ◦ What is the Nash equilibrium? 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 12 Selfish Routing Games

Pigou’s (1920) example ◦ Nash equilibrium: ◦ All traffic on the lower edge 0 traffic, 푐 푥 = 1 ◦ Total cost: 1 × 푐 1 = 1 풔 풕

1 traffic, 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 13 Selfish Routing Games

Pigou’s (1920) example ◦ Optimal solution: ◦ Half traffic on each edge 0.5 traffic, 푐 푥 = 1 ◦ Total cost: 0.5 × 1 + 0.5 × 0.5 = 0.75 풔 풕 1 4 ◦ Price of anarchy = = 0.75 3 0.5 traffic, 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 14 Selfish Routing Games

Modified Pigou’s example ◦ A small change in cost function 푐 푥 = 1

풔 풕

푐 푥 = 푥푝

23-Jan-14 SELFISH ROUTING GAMES 15 Selfish Routing Games

Modified Pigou’s example ◦ Nash equilibrium: ◦ All traffic on the lower edge 0 traffic, 푐 푥 = 1 ◦ Total cost: 1 × 11 = 1 풔 풕

1 traffic, 푐 푥 = 푥푝

23-Jan-14 SELFISH ROUTING GAMES 16 Selfish Routing Games

Modified Pigou’s example ◦ When is the cost optimized? ◦ 휖 ∈ 0,1 : traffic on upper edge 휖 traffic, 푐 푥 = 1 ◦ Cost = 휖 + 1 − 휖 푝+1 1 풔 풕 − ◦ Minimized when 휖 = 1 − 푝 + 1 푝 푝 ◦ As 푝 → ∞, optimal cost → 0 (1 − 휖) traffic, 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 17 Selfish Routing Games

Modified Pigou’s example

푓(푤표푟푠푡 푒푞푢푖푙푖푏푟푖푢푚) PoA = 휖 traffic, 푐 푥 = 1 푓(표푝푡푖푚푎푙 표푢푡푐표푚푒) As 푝 → ∞ 풔 풕 ◦ 푓(표푝푡푖푚푎푙 표푢푡푐표푚푒) → 0 푝 ◦ PoA → ∞ (1 − 휖) traffic, 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 18 Is That a Game?..

Players? Some agents wanting their traffic get across Actions? Each agent can choose a path Payoffs? The utility is the negative of network delay

Familiar?? Congestion games!!

23-Jan-14 SELFISH ROUTING GAMES 19 Selfish Routing Games

Atomic routing games Nonatomic routing games ◦ Some players ◦ Some players ◦ Each controls a non-negligible ◦ Each controls a negligible fraction of traffic fraction of traffic

Oligopoly Perfect competition

23-Jan-14 SELFISH ROUTING GAMES 20 Selfish Routing Games

Marginal Social Cost ◦ Increase in total cost due to additional traffic ◦ Cost of 푥 traffic: 푥 ⋅ 푐 푥 ′ ◦ Marginal cost function: 푐∗ 푥 = 푥 ⋅ 푐 푥 = 푐 푥 + 푥 ⋅ 푐′ 푥

Potential Function ◦ Use of integration in nonatomic games

23-Jan-14 SELFISH ROUTING GAMES 21 Selfish Routing Games

General Equilibrium Properties Nonatomic games Atomic games ◦ At least one equilibrium flow ◦ Equilibrium flow exists ◦ Uniqueness of equilibrium ◦ If all players control the same amount of traffic ◦ With affine cost functions

23-Jan-14 SELFISH ROUTING GAMES 22 Nonatomic Routing Games

Braess’s Paradox in nonatomic routing games 풗 푐 푥 = 푥 푐 푥 = 1 ◦ 1 unit of total traffic 풔 풕

푐 푥 = 1 풘 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 23 Nonatomic Routing Games

Braess’s Paradox in nonatomic routing games 풗 푐 푥 = 푥 푐 푥 = 1 ◦ Equilibrium: ◦ 푠 → 푣 → 푡: 0.5 traffic 풔 풕 ◦ 푠 → 푤 → 푡: 0.5 traffic ◦ Cost = 1.5 푐 푥 = 1 풘 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 24 Nonatomic Routing Games

Braess’s Paradox in nonatomic routing games 풗 푐 푥 = 푥 푐 푥 = 1

풔 푐 푥 = 0 풕

푐 푥 = 1 풘 푐 푥 = 푥

23-Jan-14 SELFISH ROUTING GAMES 25 Nonatomic Routing Games

Braess’s Paradox in nonatomic routing games 풗 푐 푥 = 푥 푐 푥 = 1 ◦ Equilibrium: ◦ 푠 → 푣 → 푤 → 푡: 1 풔 푐 푥 = 0 풕 ◦ Cost = 2 2 4 ◦ PoA = = 푐 푥 = 1 푐 푥 = 푥 1.5 3 풘

23-Jan-14 SELFISH ROUTING GAMES 26 Nonatomic Routing Games

Price of anarchy ◦ Maximized in Pigou-like examples ◦ Dependent on “nonlinearity” of cost functions ◦ Pigou bound: tight upper bound Polynomial degree ≤ 푝 Non-negative coefficients ◦ Independent of ◦ Network size or structure 푝+1 −1 − 푝 ◦ Number of different source-sink pairs 1 − 푝 ⋅ 푝 + 1 푝 ≈ ln 푝 4 푝 = 1 ⇒ 3

23-Jan-14 SELFISH ROUTING GAMES 27 Applications

Other games: Reduce PoA: ◦ Facility location ◦ Marginal cost pricing ◦ Pure Nash equilibrium exists ◦ Pigouvian taxes ◦ Price of anarchy is small ◦ Capacity augmentation ◦ Load balancing ◦ Makespan scheduling ◦ Resource allocation ◦ PoA as a design metric

23-Jan-14 OTHER APPLICATIONS 29 Summary

Price of anarchy quantifies the inefficiency of equilibrium ◦ Ratio of “social cost” of worst equilibrium over optimum Selfish routing is intensively studied ◦ Equilibrium flow always exists in nonatomic routing games ◦ Pigou’s example shows that PoA can be bounded or unbounded ◦ PoA depends on cost functions but not on other network properties PoA presents in the study of other domains

23-Jan-14 SUMMARY 30 References

D. Braess. On a paradox of traffic planning. Transport. Sci., 39(4):446-450, 2005. E. Koutsoupias and C. H. Papadimitriou. Worst-case equilibria. In Proc. 16th Symp. Theoretical Aspects of Computer Science, LNCS 1563:404-413, 1999. N Nisan, T Roughgarden, E. Tardos, V. V. Vazirani (eds.). Algorithmic Game Theory. Cambridge University Press, 2007. A. C. Pigou. The Economics of Welfare. Macmillan, 1920. C. H. Papadimitriou. Algorithms, games, and the Internet. In Proc. 33rd Symp. Theory of Computing, pp 749-753, 2001. T. Roughgarden. The price of anarchy is independent of the network topology. J. Comput. System Sci., 67(2):341-364, 2003. T. Roughgarden. Selfish Routing and the Price of Anarchy. MIT Press, 2005. T. Roughgarden and E. Tardos. How bad is selfish routing? J. ACM, 49(2):236-259, 2002.

23-Jan-14 REFERENCES 31