Routing Games 1 Sandip Chakraborty Department of Computer Science and Engineering, INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR November 5, 2015 1 Source: Routing Games by Tim Roughgarden Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 0 / 11
Preface: Pigou s Example Suppose that there is one unit of traffic, from a very large population of network users. Each user chooses independently between two routes. Which route the user s will choose at the equilibrium? Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 1 / 11
Preface: Pigou s Example - A nonlinear view p is a very large number Suppose that there is one unit of traffic, from a very large population of network users. Each user chooses independently between two routes. Which route the user s will choose at the equilibrium? Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 2 / 11
Nonatomic Selfish Routing Selfish players choose routes to minimize the cost incurred A selfish routing game occurs in a multicommodity flow network A network is given by a directed graph G = (V, E): Vertex set V and edge set E, together with a set (s 1, t 1 ),..., (s k, t k ) of source-sink vertex pairs called commodities. P i is the commodity (s i, t i ), P = k P i. i=1 Routes chosen by a player for commodity P i flow f P amount of traffic of commodity i that chooses the path P to travel from s i to t i. r i total traffic for commodity i; the flow f is feasible if i {1...k}, P P i f P = r i Every edge is associated with a cost function c e : R + R +. Cost functions are non-negative, continuous and non-decreasing. Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 3 / 11
Nonatomic Selfish Routing Selfish players choose routes to minimize the cost incurred A selfish routing game occurs in a multicommodity flow network A network is given by a directed graph G = (V, E): Vertex set V and edge set E, together with a set (s 1, t 1 ),..., (s k, t k ) of source-sink vertex pairs called commodities. P i is the commodity (s i, t i ), P = k P i. i=1 Routes chosen by a player for commodity P i flow f P amount of traffic of commodity i that chooses the path P to travel from s i to t i. A nonatomic r i total traffic selfishforouting commodity game i; the is defined flow f is feasible by theiftriple i {1...k}, (G, r, c) f P = r i P P i Every edge is associated with a cost function c e : R + R +. Cost functions are non-negative, continuous and non-decreasing. Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 3 / 11
Equilibrium in Nonatomic Selfish Routing Game Define the cost of a path P with respect to a flow f as the sum of the costs of the constituent edges: where, c P (f ) = e P f e = P P:e P c e (f e ) denotes the amount of traffic using paths that contain the edge e. a notion of congestion cost! f p Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 4 / 11
Equilibrium in Nonatomic Selfish Routing Game Define the cost of a path P with respect to a flow f as the sum of the costs of the constituent edges: where, c P (f ) = e P f e = P P:e P c e (f e ) denotes the amount of traffic using paths that contain the edge e. a notion of congestion cost! Nonatomic Equilibrium Flow: Let f be a feasible flow for the nonatomic selfish routing game (G, r, c). The flow f is as equilibrium flow, if, for every commodity i {1, 2,..., k} and every pair P, P P i of s i t i paths with f p > 0, f p c P (f ) c P (f ) Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 4 / 11
An Example: Braess s Paradox Assume there is 1 unit of traffic. What will be the equilibrium flow? Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 5 / 11
An Example: Braess s Paradox Assume there is 1 unit of traffic. What will be the equilibrium flow? Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 6 / 11
Existence and Uniqueness of Equilibrium Flows Theorem Let (G, r, c) be a nonatomic selfish routing game. a) (G, r, c) admits at least one equilibrium flow. b) If f and f are equilibrium flows for (G, r, c), then c e (f e ) = c e (f e ) for every edge e. Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 7 / 11
Existence and Uniqueness of Equilibrium Flows Theorem Let (G, r, c) be a nonatomic selfish routing game. a) (G, r, c) admits at least one equilibrium flow. b) If f and f are equilibrium flows for (G, r, c), then c e (f e ) = c e (f e ) for every edge e. Potential function: a function defined on the outcomes of the game, such that the equilibria of the game are precisely the outcomes that optimize the potential function. Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 7 / 11
Existence and Uniqueness of Equilibrium Flows Theorem Let (G, r, c) be a nonatomic selfish routing game. a) (G, r, c) admits at least one equilibrium flow. b) If f and f are equilibrium flows for (G, r, c), then c e (f e ) = c e (f e ) for every edge e. Potential function: a function defined on the outcomes of the game, such that the equilibria of the game are precisely the outcomes that optimize the potential function. What can be a potential function for the nonatomic selfish routing game (G, r, c)? Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 7 / 11
Characterization of Optimal Flows x.c e (x) : the contribution to the social cost function by the traffic on the edge e. The cost incurred by a player choosing the path P in the flow f is c P (f ) f P denotes amount of traffic choosing path P. cost of a flow f can be defined as, C(f ) = P P c P (f )f P Now, c P (f ) = c e (f e ) Therefore, e P C(f ) = e E c e (f e ).f e Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 8 / 11
Characterization of Optimal Flows x.c e (x) : the contribution to the social cost function by the traffic on the edge e. The cost incurred by a player choosing the path P in the flow f is c P (f ) f P denotes amount of traffic choosing path P. cost of a flow f can be defined as, C(f ) = P P c P (f )f P Now, c P (f ) = c e (f e ) Therefore, e P C(f ) = e E c e (f e ).f e We assume x.c e (x) is continuously differentiable and convex. Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 8 / 11
Characterization of Optimal Flows Let c e (x) = d dx (x.c e(x)) = c e (x) + x.c e(x). We call it the marginal cost function for the edge e. Let cp (f ) = ce (f ) e P Let (G, r, c) be a nonatomic selfish routing game such that, for every edge e, the function x.c e (x) is convex and continuously differentiable. Let c e denote the marginal cost function of the edge e. Then f is an optimal flow for (G, r, c) if and only if, for every commodity i {1, 2,..., k} and every pair P, P P i of s i t i paths with f p > 0, c P (f ) C P (f ) Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 9 / 11
Equivalence of Equilibrium and Optimal Flows Let (G, r, c) be a nonatomic selfish routing game such that, for every edge e, the function x.c e (x) is convex and continuously differentiable. Let c e denote the marginal cost function of the edge e. Then f is an optimal flow for (G, r, c) is and only if it is an equilibrium flow for (G, r, c ) Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 10 / 11
Equivalence of Equilibrium and Optimal Flows Let (G, r, c) be a nonatomic selfish routing game such that, for every edge e, the function x.c e (x) is convex and continuously differentiable. Let c e denote the marginal cost function of the edge e. Then f is an optimal flow for (G, r, c) is and only if it is an equilibrium flow for (G, r, c ) We seek a function h e (x) for each edge e such that h e(x) = c e (x). Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 10 / 11
Equivalence of Equilibrium and Optimal Flows Let (G, r, c) be a nonatomic selfish routing game such that, for every edge e, the function x.c e (x) is convex and continuously differentiable. Let c e denote the marginal cost function of the edge e. Then f is an optimal flow for (G, r, c) is and only if it is an equilibrium flow for (G, r, c ) We seek a function h e (x) for each edge e such that h e(x) = c e (x). Therefore, we can set x h e (x) = c e (y)dy 0 Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 10 / 11
Equivalence of Equilibrium and Optimal Flows Let (G, r, c) be a nonatomic selfish routing game such that, for every edge e, the function x.c e (x) is convex and continuously differentiable. Let c e denote the marginal cost function of the edge e. Then f is an optimal flow for (G, r, c) is and only if it is an equilibrium flow for (G, r, c ) We seek a function h e (x) for each edge e such that h e(x) = c e (x). Therefore, we can set h e (x) = x c e (y)dy So, we define the potential function as, 0 φ(f ) = e E f e 0 c e (x)dx Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 10 / 11
Proof of the Theorem Since edge cost functions are continuous, the potential function is a continuous function on this set. Every point where φ attains a minimum is an equilibrium flow for (G, r, c). Suppose f and f are equilibrium flows for (G, r, c), then both will be the minimum for the potential functions. Therefore the cost function would be same for both. Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 11 / 11
Thank You Sandip Chakraborty (IIT Kharagpur) CS 60019 November 5, 2015 11 / 11