The Price of Anarchy in a Network Pricing Game. g Sept 27, 2007 - PowerPoint PPT Presentation

The Price of Anarchy in a Network Pricing Game. g Sept 27, 2007 Allerton Conference John Musacchio Assistant Professor Technology and Information Management University of California Santa Cruz University of California, Santa Cruz johnm@soe.ucsc.edu Joint work with: Shuang Wu University of California, Santa Cruz

Overview � Model – Single source-destination pair g p – Competing providers – Non-atomic users – Traffic dependent latency – Elastic user demand � Model due to – Acemoglu and Ozdaglar [1] � Elastic user demand extension: El ti d d t i – Hayrapetyan, Tardos, Wexler [3] [1] D. Acemoglu and A. Ozdaglar, “Competition and Efficiency in Congested Markets,” Math. of OR , Feb. 2007. [3] A. Hayrapetyan, E. Tardos and T. Wexler, “A Network Pricing Game for Selfish Traffic,” Distributed Computing, March 2007.

Overview � Price of anarchy: Social Welfare with Optimal Prices Social Welfare Nash with Nash prices = 3 p 2 � Result due to Ozdaglar [2] Result due to Ozdaglar [2] � We prove the same result a different way – Ozdaglar proof: mathematical programming O dag a p oo a e a ca p og a g argument – Our proof: circuit analogy, linear algebra [2] A. Ozdaglar, ``Price Competition with Elastic Traffic,'‘ to appear in Networks

Some Other Related Work � Roughgarden 02, 03 – Selfish routing games – Taxes to induce optimal routing � Johari and Tsitsiklis 05 – Cournot rate allocation mechanisms – Different situation – Very similar structure John Musacchio – Allerton 07

Organization � Overview � Model Description � Model Description � Nash and Social Optimum Characterization � Circuit Analogy � Circuit Analogy � PoA Proof Overview John Musacchio – Allerton 07

Wardrop Equilibrium for given prices Path 1 Non-Atomic Users p 1 + l 1 ( f 1 ) l ( f ) Path 2 p 2 + l 2 ( f 2 ) Choose lowest “disutility”: p i + l ( f i ) Delay+Price p p 2 Path 2 Path 1 Delay+Price Delay y p 1 Delay Traffic Traffic 100% 100% 20% 80% John Musacchio – Allerton 07

Elastic Demand ? Demand or “Disutility” Curve Key assumption: Disutil Concave Decreasing User Surplus ity Total Flow (#of non-atomic users that connect) John Musacchio – Allerton 07

John Musacchio – Allerton 07 Path 2 Social Optimum Pricing Path 1 p 1 p 2 p

John Musacchio – Allerton 07 Path 2 Path 1 p 1 p 2 p Network Pricing

Wardrop Equilibrium for given ( p 1 , p 2 ,p 3 ) For now consider linear latency case: l i ( f i ) = a i f i + b i User Surplus User Surplus d - s s Disutility D p 2 p 1 Provider 1 Profit p 3 L ( f 1 ) a 1 b 1 f 1 f 2 f 3 Figure from [3] f f Total Flow (#of non-atomic users that connect) [3] A. Hayrapetyan, E. Tardos and T. Wexler, “A Network Pricing Game for Selfish Traffic,” Distributed Computing, March 2007.

Consequence of price change � Suppose: player 1 unilaterally reduces price. - New Wardrop equilibrium disutility: d - h . New Wardrop equilibrium disutility: d h d d d - h Disutility - s Provider 1 Profit p 1 p 1 p 3 p 3 y f 3 − h f 2 f 3 f 1 a 3 h f 2 − h f + h / s f a 2 Total Flow (#of non-atomic users that connect)

Nash Equilibrium Analysis Convenient definition: Convenient definition: New profit – old profit: =0 =0 Nash equilibrium condition: John Musacchio – Allerton 07

Social optimum pricing � Price so that users see the cost they impose on society. * + b i ) f i * � Latency cost on link i : ( a i f i * + b i � Marginal cost: 2 a i f i * + b i � Latency seen by user: a i f i * � Difference: a i f i * = a i f i * achieves social optimum * * � Conclusion: p i John Musacchio – Allerton 07

Circuit analogy Social Optimum: Social Optimum: Nash Equilibrium: Nash Eq ilibri m Voltage Voltag d d * d e e f f * Current f f Current d * d Provider P δ 1 δ 2 δ 3 Power p 1 a 1 a 2 a 3 a a 1 a a 2 a a 3 Profit = f 1 f a 1 a 2 a 3 a 1 a 2 a 3 + + + + + + + + + + b 1 b 2 b 3 b 1 b 2 b 3 - - - - - -

Nash vs. Social Opt. – Original Game Nash Equilibrium: Nash Equilibrium: Social Optimum: Social Optimum: - sd d * John Musacchio – Allerton 07

Nash vs. Social Opt. – Modification 1 Nash Equilibrium: N h E ilib i S Social Optimum: i l O ti - sd d * Flow & Social Welfare Fl & S i l W lf Flow & Social Welfare Fl & S i l W lf Not Reduced Unchanged Price of Anarchy Not Reduced John Musacchio – Allerton 07

Nash vs. Social Opt. – Modification 2 V V Social Optimum: Social Optimum: Nash Equilibrium: Nash Equilibrium: - sd d * • Flow Unchanged Fl U h d • Flow Unchanged • Social Welfare reduced • Social Welfare reduced by light green area by light green area y g g Price of Anarchy Not Reduced

Circuit analogy Social Optimum: Social Optimum: Nash Equilibrium: Nash Equilibrium: V V s s δ 1 δ 2 δ 3 Power = Power = Provider Provider a a 1 a a 2 a 3 a a 1 a 2 a 3 Surplus Surplus a 1 a a a 2 a a 3 a 1 a 2 a 3 + + b 1 + b 2 b 3 b 1 + + + b 2 b 3 - - - - - -

Matrix – Vector Notation ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ f ∗ f 1 1 1 ... 1 ⎢ ⎥ ⎢ ⎥ f ∗ ⎢ ⎥ F ∗ = f 2 1 1 ... F = ⎣ . ⎣ ⎦ ⎦ M = ⎣ . ⎣ ⎦ ⎦ 2 ⎣ . ⎣ ⎦ ⎦ . ... . . . . . f ∗ . f n . . n ⎡ ⎡ δ 1 ⎤ ⎤ ⎡ a 1 ⎡ ⎤ ⎤ 0 ... 0 ... ⎢ ⎥ ⎢ ⎥ ⎣ 0 0 a 2 ... ⎣ 0 0 δ 2 δ ... A = ∆ = ⎦ ⎦ . . . . ... ... . . . . . . . . Providers used in Nash equilibrium can become Providers used in Nash equilibrium can become “undercut” in social optimum w l o g w.l.o.g. Providers: 1 Providers: 1 ,…, m not undercut m not undercut m + 1 ,…, n undercut · ¯ · ¸ ¸ · ¯ · ¸ ¸ · · ¯ ¸ ¸ F ∆ , 0 A, 0 F = F = A = A = ∆ = ∆ = F 0 , A 0 , ∆ John Musacchio – Allerton 07

Relations between flow vectors V V V V Soc. Opt: Nash: s s δ 3 δ 1 δ 2 a a 1 a a 2 a 3 a a 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 a 3 - b 2 + b - b 3 b b 1 b - b 2 b - b 3 b b 1 b + + + + + - -

Social Welfare V V V V Soc. Opt: Nash: s s δ 3 δ 1 δ 2 1 a a 1 a a 2 a 3 a a 3 a 1 a 2 a 1 a 2 a 3 a 1 a 2 a 3 b 1 b - b 2 b - b 3 b - b 2 + - b 3 b 1 + + + + + - -

John Musacchio – Allerton 07 Social Welfare Comparison Metric

Useful Algebraic Identities Define: Then: Proof: • relation of δ i ’s and A ; matrix inversion lemma ; i John Musacchio – Allerton 07

John Musacchio – Allerton 07 Use Identities

Change Coordinates (i) 1 1 (i) Parabola in Z always ≥ 0. (i) P b l i Z l 0 (ii) Thi (ii) This might be negative i ht b ti 2 β || Q || 2 1 Z (ii) Positive � done. Case 1: ¯ a j C Case 2: Use existence of a “small” to show that total “undercut 2 U i t f “ ll” t h th t t t l “ d t flow” Z is small. Tedious algebra � |(i)| > |(ii)|

Worst Case Nash: Social Optimum: 1 1 D Disutility 2 1 2 Flow Flow Price = 1 Price = 0 Flow 1 Flow = 1 Flow = 2 Flow = 2 Social Welfare =1 Social Welfare = 3/2 John Musacchio – Allerton 07

• Provided a pure strategy equilibrium exists... • Linearize at equilibrium Convex Latency

Conclusions � Analysis of network pricing game can be reduced to analysis of a circuit. d d t l i f i it � Potential for using method for extended model. d l � Analysis of circuit a bit more tedious than desired desired. John Musacchio – Allerton 07