Algorithms and Economics of Networks

Algorithms and Economics of Networks Abraham Flaxman and Vahab Mirrokni, Microsoft Research

Outline • Network Congestion Games • Congestion Games • Rosenthal’s Theorem: Congestion games are potential Games: • PoA for Congestion Games. • Market Sharing Games. • Network Design Games.

Network Congestion Games • A directed graph G=(V,E) with n users, • Each edge e in E(G) has a delay function fe, • Strategy of user i is to choose a path Aj from a source si to a destination ti, • The delay of a path is the sum of delays of edges on the path, • Each user wants to minimize his own delay by choosing the best path.

Example: Network Congestion Game s1 t1 t3 t2 s2 s3

Example: Network Congestion Game Agent 2 path 2 s1 t1 t3 t2 s2 Agent 2 path 1 s3

Congestion Games • n players, a set of facilities E, • Strategy of player i is to choose a subset of facilities (from a given family of subsets Ti), • Facility i have a cost (delay) function fe which depends on the number of players playing this facility, • Each player minimizes its total cost,

Example: Congestion Games Picture from Kapelushnik Lior f1 f2 f3 f4 F5 F6

Example: Congestion Games f1 f2 f3 f4 F5 F6

Congestion Games: Pure NE • Rosenthal’s Theorem (1979): Any congestion game is an exact potential game. • Proof is based on the following Potential Function

Classes of Congestion Games • Every network congestion game is a congestion game • Symmetric and Asymmetric Players • Network Design Games. • Maximizing Congestion Games: Each player wants to maximize his payoff (instead of minimizing his delay)  Market Sharing Games. • Generalizations: • Weighted Congestion Games • Player-specific Congestion Games

Congestion Games: Social Cost • Two social Cost functions: • Consider a pure Strategy A = (A1, A2, …, An). Defintion 1: Defintion 2:

Congestion Games: PoA • PoA for two social Cost functions: Defintion 1: Defintion 2: We Prove bounds for

Congestion Games: PoA • PoA for congestion game with linear delay functions is at most 5/2. • Proof: • Lemma 1: for a pair of nonnegative integers a,b: • Proof: …

Congestion Games: Lower Bound S1 t1 S2 t2 S3 t3

Congestion Games: PoA for mixed NE • Theorem: PoA for mixed Nash equilibria in congestion games with linear latency function is 2.61. • Theorem: PoA for mixed Nash equilibria of weighted congestion games with linear latency function is 2.61. • Theorem: PoA for polynomial delay functions of constant degree is a constant.

Other Variants • Atomic Congestion Games: Many infinitesimal users. The load of each user is very small. • Theorem: PoA for non-atomic congestion games with linear latency functions is 4/3. • Splittable Network Congestion Games

Market Sharing Games • Congestion Game • Facilities are Markets. • Cost function  Profit Function. • Players share the profit of markets (equally). • Each player has some packing constraint for the set of markets he can satisfy. • PoA: 1/2.

Network Design Games • Players want to construct a network. • They share the cost of buying links in the network. • Known Results: Price of Stability, Convergence…

Next Lecture • Coordination Mechanism Design

Algorithms and Economics of Networks