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 t1t3 t2

s2

s3

xx 26 xsin

62 x

xln2x4

3

xx 28

x2

Example: Network Congestion Game

s1 t1t3 t2

s2

s3

Agent 2 path 1

Agent 2 path 2

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

6,5,4,22 ffff

f1 f2 f3 f4 F5 F6

4,3,2,3,11 fffff

4,13 ff

Example: Congestion Games

6,5,4,22 ffff

f1 f2 f3 f4 F5 F6

4,3,2,3,11 fffff

4,13 ff

1

Example: Congestion Games

6,5,4,22 ffff

f1 f2 f3 f4 F5 F6

4,3,2,3,11 fffff

4,13 ff

2

Example: Congestion Games

6,5,4,22 ffff

f1 f2 f3 f4 F5 F6

4,3,2,3,11 fffff

4,13 ff

3

Congestion Games: Pure NE

Rosenthal’s Theorem (1979): Any congestion game is an exact potential game.

Proof is based on the following Potential Function

Ee

An

te

e

tf1

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:

AcAMax iNimax

Ni

i AcASumDefintion 2:

Congestion Games: PoA PoA for two social Cost functions: Defintion 1:

Defintion 2:

Ni

i AcASum

We Prove bounds for

opt

AMaxNEaisAmax

opt

ASumNEaisAmax

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: …

22

3

5

3

11 baab

Congestion Games: Lower Bound

0)( xfexxfe )(

opt

S1

S2

S3

t1

t2

t3

NE

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…

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