Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary The Price of Anarchy of Selfish Routing Arthur van Goethem & Sk´ uli Arnlaugsson June 8, 2011 Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 1/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Overview Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 2/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Introduction Price of Anarchy The Price of Anarchy of a non-atomic 1 selfish routing game is the ratio between the cost of an equilibrium flow and that of an optimal flow. p ( G , r , c ) = C ( f ) C ( f ∗ ) ◮ f denotes the equilibrium flow, and f ∗ denotes the optimal flow ◮ What is the potential function? 1 Very large number of players, each controlling a negligible fraction of the overall traffic Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 3/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Introduction Price of Anarchy The Price of Anarchy of a non-atomic 1 selfish routing game is the ratio between the cost of an equilibrium flow and that of an optimal flow. p ( G , r , c ) = C ( f ) C ( f ∗ ) ◮ f denotes the equilibrium flow, and f ∗ denotes the optimal flow ◮ What is the potential function? 1 Very large number of players, each controlling a negligible fraction of the overall traffic Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 3/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Proving an upper bound on the PoA for non-atomic selfish routing ◮ PoA depends solely on the (non-)linearity cost function ◮ Assume � x xc e ( x ) ≤ γ c e ( y ) dy 0 ∀ e ∈ E and ∀ x ≥ 0 I.e. the cost function for a single edge is smaller that the potential function for this edge. Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 4/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Pigou bound ◮ The potential function gives a good, but not optimal bound on the PoA. Can we do better? ◮ To show this, we introduce the Pigou bound Definition 18.18 (Pigou bound) Let C be a nonempty set of cost functions.The Pigou bound α ( C ) for C is r ∗ c ( r ) α ( C ) = sup sup x ∗ c ( x ) + ( r − x ) ∗ c ( r ) c ∈C x , r ≥ 0 with the understanding that 0 / 0 = 1. ◮ The Pigou bound is a natural lower bound on the PoA based on “Pigou-like examples”. Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 5/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Pigou bound ◮ The potential function gives a good, but not optimal bound on the PoA. Can we do better? ◮ To show this, we introduce the Pigou bound Definition 18.18 (Pigou bound) Let C be a nonempty set of cost functions.The Pigou bound α ( C ) for C is r ∗ c ( r ) α ( C ) = sup sup x ∗ c ( x ) + ( r − x ) ∗ c ( r ) c ∈C x , r ≥ 0 with the understanding that 0 / 0 = 1. ◮ The Pigou bound is a natural lower bound on the PoA based on “Pigou-like examples”. Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 5/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Pigou bound ◮ The potential function gives a good, but not optimal bound on the PoA. Can we do better? ◮ To show this, we introduce the Pigou bound Definition 18.18 (Pigou bound) Let C be a nonempty set of cost functions.The Pigou bound α ( C ) for C is r ∗ c ( r ) α ( C ) = sup sup x ∗ c ( x ) + ( r − x ) ∗ c ( r ) c ∈C x , r ≥ 0 with the understanding that 0 / 0 = 1. ◮ The Pigou bound is a natural lower bound on the PoA based on “Pigou-like examples”. Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 5/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Pigou bound, cont’d Proposition 18.19 Let C be a set of cost functions that contains all of the constant cost functions. Then the price of anarchy in non-atomic instances with cost functions on C can be arbitrarily close to α ( C ) . Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 6/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Variational Inequality ◮ It is also shown that Pigou bound is an upper bound on the PoA in general multi-commodity flow networks ◮ To show this we need the Variational inequality characterization Proposition 18.20 (Variational inequality characterization) Let f be a feasible flow for the non-atomic instance (G, r, c). The flow is an equilibrium flow if and only if � � c e ( f e ) f ∗ c e ( f e ) f e ≤ e e ∈ E e ∈ E for every flow f ∗ feasible for (G, r, c). Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 7/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Variational Inequality ◮ It is also shown that Pigou bound is an upper bound on the PoA in general multi-commodity flow networks ◮ To show this we need the Variational inequality characterization Proposition 18.20 (Variational inequality characterization) Let f be a feasible flow for the non-atomic instance (G, r, c). The flow is an equilibrium flow if and only if � � c e ( f e ) f ∗ c e ( f e ) f e ≤ e e ∈ E e ∈ E for every flow f ∗ feasible for (G, r, c). Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 7/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Variational Inequality ◮ It is also shown that Pigou bound is an upper bound on the PoA in general multi-commodity flow networks ◮ To show this we need the Variational inequality characterization Proposition 18.20 (Variational inequality characterization) Let f be a feasible flow for the non-atomic instance (G, r, c). The flow is an equilibrium flow if and only if � � c e ( f e ) f ∗ c e ( f e ) f e ≤ e e ∈ E e ∈ E for every flow f ∗ feasible for (G, r, c). Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 7/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Tightness of the Pigou bound ◮ We can now show that the Pigou bound is tight (bounding from above and below) Theorem 18.21 (Tightness of the Pigou bound) Let C be a set of cost functions and α ( C ) be the Pigou bound for C . If (G, r, c) is a non-atomic instance with cost functions in C , then the price of anarchy of (G, r, c) is at most α ( C ) . Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 8/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Tightness of the Pigou bound ◮ We can now show that the Pigou bound is tight (bounding from above and below) Theorem 18.21 (Tightness of the Pigou bound) Let C be a set of cost functions and α ( C ) be the Pigou bound for C . If (G, r, c) is a non-atomic instance with cost functions in C , then the price of anarchy of (G, r, c) is at most α ( C ) . Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 8/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Largest possible PoA ◮ Determining the largest possible PoA in Pigou-like examples is a tractable problem in many cases ◮ When C is a set of affine cost functions it is precisely 4 3 ◮ When C is a set of polynomials with degree at most p and non-negative coefficients: (1 − p ( p + 1) − ( p +1) / p ) − 1 ≈ p ln p Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 9/21
Introduction Potential Function Pigou bound Braess’s Paradox Atomic Selfish Routing Reducing the Price Summary Largest possible PoA ◮ Determining the largest possible PoA in Pigou-like examples is a tractable problem in many cases ◮ When C is a set of affine cost functions it is precisely 4 3 ◮ When C is a set of polynomials with degree at most p and non-negative coefficients: (1 − p ( p + 1) − ( p +1) / p ) − 1 ≈ p ln p Arthur van Goethem & Sk´ uli Arnlaugsson, The Price of Anarchy of Selfish Routing 9/21
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