Network Formation Games Ofir Geri Price of Anarchy

Outline • General cost-sharing connection games • Fair cost-sharing connection games • Capacitated symmetric

General Cost-Sharing Connection Games E. Anshelevich, A. Dasgupta, É. Tardos, and T. Wexler, “Near-Optimal

Basic Properties of Nash Equilibria • The bought edges form a graph that is

A Nash Equilibrium May Require Cost-Sharing •

Single Source Games: Price of Stability •

Single Source Games: Price of Stability (Figure taken from Anshelevich et al. , “Near-Optimal

Single Source Games: Price of Stability • (Figure taken from Anshelevich et al. ,

General Connection Games • (Figure taken from Anshelevich et al. , “Near-Optimal Network Design

3 -Approximate Nash Equilibrium • (Figure taken from Anshelevich et al. , “Near-Optimal Network

Fair Cost-Sharing Connection Games E. Anshelevich, A. Dasgupta, J. Kleinberg, É. Tardos, T. Wexler,

Price of Anarchy and Price of Stability •

Capacitated Cost-Sharing Connection Games M. Feldman and T. Ron, “Capacitated Network Design Games”

The Price of Anarchy • (Figure taken from Feldman and Ron, “Capacitated Network Design

Graph-Theoretic Preliminaries s 1 … s 2 t 1 … … s 1 s

The Price of Stability • (Figure taken from Feldman and Ron, “Capacitated Network Design

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Network Formation Games Ofir Geri Price of Anarchy Seminar Supervised by Prof. Michal Feldman Tel-Aviv University 2/4/2014

Introduction •

Outline • General cost-sharing connection games • Fair cost-sharing connection games • Capacitated symmetric cost-sharing connection games

General Cost-Sharing Connection Games E. Anshelevich, A. Dasgupta, É. Tardos, and T. Wexler, “Near-Optimal Network Design with Selfish Agents”

The Connection Game •

Basic Properties of Nash Equilibria • The bought edges form a graph that is a forest • A player only contributes to the edges they use • Each edge is either fully paid for or not paid at all

A Game Without Nash Equilibrium

A Nash Equilibrium May Require Cost-Sharing •

The Price of Anarchy •

Single Source Games •

Single Source Games: Price of Stability •

Single Source Games: Price of Stability •

Single Source Games: Price of Stability (Figure taken from Anshelevich et al. , “Near-Optimal Network Design with Selfish Agents”)

Single Source Games: Price of Stability •

Single Source Games: Price of Stability • (Figure taken from Anshelevich et al. , “Near-Optimal Network Design with Selfish Agents”)

Single Source Games: Price of Stability •

Approximate Nash Equilibria •

Single Source Games •

General Connection Games • (Figure taken from Anshelevich et al. , “Near-Optimal Network Design with Selfish Agents”)

General Connection Games •

Stable Payments •

Connection Set: Definition •

Connection Sets •

3 -Approximate Nash Equilibrium •

3 -Approximate Nash Equilibrium • (Figure taken from Anshelevich et al. , “Near-Optimal Network Design with Selfish Agents”)

3 -Approximate Nash Equilibrium •

Fair Cost-Sharing Connection Games E. Anshelevich, A. Dasgupta, J. Kleinberg, É. Tardos, T. Wexler, and T. Roughgarden, “The Price of Stability for Network Design with Fair Cost Allocation”

The Fair Connection Game •

Price of Anarchy and Price of Stability •

Concave Cost Functions •