The Impact of Social Ignorance on Weighted Congestion Games Vasilis - - PowerPoint PPT Presentation

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

The Impact of Social Ignorance on Weighted Congestion Games

Vasilis Gkatzelis

C.I.M.S. New York University

WINE 2009 Joint work with Dimitris Fotakis1, Alexis Kaporis2,3 and Paul Spirakis3

• 1. E.C.E. Dept., National Technical University of Athens
• 2. I.C.S.E. Dept., University of the Aegean
• 3. Research Academic Computer Technology Institute, Patras

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Outline

1 Introduction

Motivation and Previous Work Contribution

2 Model and Preliminaries

Weighted Linear Congestion Games Weighted Graphical Congestion Games

3 Potential Function and Cost Approximation

Potential Function Cost Approximation

4 Inefficiency of Pure Nash Equilibria

The Price of Anarchy The Price of Stability

5 Convergence of ǫ-Nash Dynamics

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Congestion Games

Rosenthal (1973) “A class of games possessing pure strategy Nash equilibria”

A set V = {1, 2, ..., n} of selfish players A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A non-decreasing delay function de : N → R≥0 for e ∈ R Linear congestion games, i.e. de(x) = aex + be s = (s1, s2, ..., sn) ∈ Σ =Xi∈V Σi where si ∈ Σi for i ∈ V A congestion se = |{i : e ∈ si}| for e ∈ R A cost ci(s) =

e∈si de(se) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, ci(s−i, s′ i) ≥ ci(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Potential Games

Monderer and Shapley (1996) “Potential Games”

Exact potential function Φ(s) : Σ → R such that Φ(s) − Φ(s−i, s′

i ) = ci(s) − ci(s−i, s′ i ) ∀i ∈ V , s′ i ∈ Σi

Nash dynamics converges to PNE (the directed state graph with payoff improving individual defections is a DAG)

For congestion games Φ(s) =

e∈R

se

k=1 de(k) is exact

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Potential Games

Monderer and Shapley (1996) “Potential Games”

Exact potential function Φ(s) : Σ → R such that Φ(s) − Φ(s−i, s′

i ) = ci(s) − ci(s−i, s′ i ) ∀i ∈ V , s′ i ∈ Σi

Nash dynamics converges to PNE (the directed state graph with payoff improving individual defections is a DAG)

For congestion games Φ(s) =

e∈R

se

k=1 de(k) is exact

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Potential Games

Monderer and Shapley (1996) “Potential Games”

Exact potential function Φ(s) : Σ → R such that Φ(s) − Φ(s−i, s′

i ) = ci(s) − ci(s−i, s′ i ) ∀i ∈ V , s′ i ∈ Σi

Nash dynamics converges to PNE (the directed state graph with payoff improving individual defections is a DAG)

For congestion games Φ(s) =

e∈R

se

k=1 de(k) is exact

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Inefficiency of Pure Nash Equilibria

Social cost function: C(s) =

i∈V ci(s)

Let configuration sopt∈ arg mins∈Σ C(s)

Σ

PNE

OPT

The (pure) Price of Anarchy (PoA): maxs∈PNE

C(s) C(sopt)

The (pure) Price of Stability (PoS): mins∈PNE

C(s) sopt

Awerbuch et al. and Christodoulou et al. independently show: Unweighted linear congestion games have PoA=2.5 Awerbuch et al. also show: Weighted linear congestion games have PoA≈ 2.62 Christodoulou et al. and Caragiannis et al. show: Unweighted linear congestion games have PoS≈ 1.58

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Inefficiency of Pure Nash Equilibria

Social cost function: C(s) =

i∈V ci(s)

Let configuration sopt∈ arg mins∈Σ C(s)

Σ

PNE

OPT

The (pure) Price of Anarchy (PoA): maxs∈PNE

C(s) C(sopt)

The (pure) Price of Stability (PoS): mins∈PNE

C(s) sopt

Awerbuch et al. and Christodoulou et al. independently show: Unweighted linear congestion games have PoA=2.5 Awerbuch et al. also show: Weighted linear congestion games have PoA≈ 2.62 Christodoulou et al. and Caragiannis et al. show: Unweighted linear congestion games have PoS≈ 1.58

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Inefficiency of Pure Nash Equilibria

Social cost function: C(s) =

i∈V ci(s)

Let configuration sopt∈ arg mins∈Σ C(s)

Σ

PNE

OPT

The (pure) Price of Anarchy (PoA): maxs∈PNE

C(s) C(sopt)

The (pure) Price of Stability (PoS): mins∈PNE

C(s) sopt

Awerbuch et al. and Christodoulou et al. independently show: Unweighted linear congestion games have PoA=2.5 Awerbuch et al. also show: Weighted linear congestion games have PoA≈ 2.62 Christodoulou et al. and Caragiannis et al. show: Unweighted linear congestion games have PoS≈ 1.58

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Inefficiency of Pure Nash Equilibria

Social cost function: C(s) =

i∈V ci(s)

Let configuration sopt∈ arg mins∈Σ C(s)

Σ

PNE

OPT

The (pure) Price of Anarchy (PoA): maxs∈PNE

C(s) C(sopt)

The (pure) Price of Stability (PoS): mins∈PNE

C(s) sopt

Awerbuch et al. and Christodoulou et al. independently show: Unweighted linear congestion games have PoA=2.5 Awerbuch et al. also show: Weighted linear congestion games have PoA≈ 2.62 Christodoulou et al. and Caragiannis et al. show: Unweighted linear congestion games have PoS≈ 1.58

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Inefficiency of Pure Nash Equilibria

Social cost function: C(s) =

i∈V ci(s)

Let configuration sopt∈ arg mins∈Σ C(s)

Σ

PNE

OPT

The (pure) Price of Anarchy (PoA): maxs∈PNE

C(s) C(sopt)

The (pure) Price of Stability (PoS): mins∈PNE

C(s) sopt

Awerbuch et al. and Christodoulou et al. independently show: Unweighted linear congestion games have PoA=2.5 Awerbuch et al. also show: Weighted linear congestion games have PoA≈ 2.62 Christodoulou et al. and Caragiannis et al. show: Unweighted linear congestion games have PoS≈ 1.58

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Inefficiency of Pure Nash Equilibria

Social cost function: C(s) =

i∈V ci(s)

Let configuration sopt∈ arg mins∈Σ C(s)

Σ

PNE

OPT

The (pure) Price of Anarchy (PoA): maxs∈PNE

C(s) C(sopt)

The (pure) Price of Stability (PoS): mins∈PNE

C(s) sopt

Awerbuch et al. and Christodoulou et al. independently show: Unweighted linear congestion games have PoA=2.5 Awerbuch et al. also show: Weighted linear congestion games have PoA≈ 2.62 Christodoulou et al. and Caragiannis et al. show: Unweighted linear congestion games have PoS≈ 1.58

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Convergence Time of Dynamics

(-) Ackermann et al. show: It is PLS-complete to compute Nash equilibria for congestion games with linear delay functions (∃ exponentially long paths) (+) Chien and Sinclair show: ǫ-Nash dynamics for symmetric congestion games converges to ǫ-Nash equilibria in a number of steps polynomial in |V | and ǫ−1 (given a “γ -bounded jump” condition for delays) (-) Voecking et al. show: The above approach cannot be extended to asymmetric congestion games (PLS-complete to approximate) (+) Awerbuch et al. show: ǫ-Nash dynamics for congestion games converges to almost

• ptimal solutions w.r.t. social cost in a number of steps

polynomial in |V | and ǫ−1 (“γ -bounded jump” condition)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Convergence Time of Dynamics

(-) Ackermann et al. show: It is PLS-complete to compute Nash equilibria for congestion games with linear delay functions (∃ exponentially long paths) (+) Chien and Sinclair show: ǫ-Nash dynamics for symmetric congestion games converges to ǫ-Nash equilibria in a number of steps polynomial in |V | and ǫ−1 (given a “γ -bounded jump” condition for delays) (-) Voecking et al. show: The above approach cannot be extended to asymmetric congestion games (PLS-complete to approximate) (+) Awerbuch et al. show: ǫ-Nash dynamics for congestion games converges to almost

• ptimal solutions w.r.t. social cost in a number of steps

polynomial in |V | and ǫ−1 (“γ -bounded jump” condition)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Convergence Time of Dynamics

(-) Ackermann et al. show: It is PLS-complete to compute Nash equilibria for congestion games with linear delay functions (∃ exponentially long paths) (+) Chien and Sinclair show: ǫ-Nash dynamics for symmetric congestion games converges to ǫ-Nash equilibria in a number of steps polynomial in |V | and ǫ−1 (given a “γ -bounded jump” condition for delays) (-) Voecking et al. show: The above approach cannot be extended to asymmetric congestion games (PLS-complete to approximate) (+) Awerbuch et al. show: ǫ-Nash dynamics for congestion games converges to almost

• ptimal solutions w.r.t. social cost in a number of steps

polynomial in |V | and ǫ−1 (“γ -bounded jump” condition)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Convergence Time of Dynamics

(-) Ackermann et al. show: It is PLS-complete to compute Nash equilibria for congestion games with linear delay functions (∃ exponentially long paths) (+) Chien and Sinclair show: ǫ-Nash dynamics for symmetric congestion games converges to ǫ-Nash equilibria in a number of steps polynomial in |V | and ǫ−1 (given a “γ -bounded jump” condition for delays) (-) Voecking et al. show: The above approach cannot be extended to asymmetric congestion games (PLS-complete to approximate) (+) Awerbuch et al. show: ǫ-Nash dynamics for congestion games converges to almost

• ptimal solutions w.r.t. social cost in a number of steps

polynomial in |V | and ǫ−1 (“γ -bounded jump” condition)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Motivation: Most of the work is based on the full information setting In many typical applications players have partial information How does the social context affect inefficiency and convergence? Gairing, Monien and Tiemann: Bayesian approach and mostly parallel-link games Koutsoupias, Panagopoulou and Spirakis: Directed social graph (player knows neighbors’ weights and

• prob. distr. for others’) for two identical parallel links

Karakostas, Kim, Viglas and Xia: Fraction of players is totally ignorant of the presence of others Ashlagi, Krysta and Tennenholtz: Social graph not modelling information but individual costs

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Social Ignorance in Congestion Games

Bil`

• et al. introduce “Graphical Congestion Games”:

Players are vertices of a social graph G = (V , E) Each player has:

full information about the players in his social neighborhood no information whatsoever about the remaining players

They show that such games with linear delay functions and unweighted players playing based on their presumed costs:

admit a potential function have PoS≤|V | have PoA= Θ(|V | (degmax(G) + 1)) where degmax(G) is the maximum degree of the graph

In follow-up paper they show that one can dramatically decrease the PoA by enforcing a carefully selected social graph

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Motivation and Previous Work Contribution

Contribution

Is there a more natural parameter of the social graph G that characterizes the inefficiency of PNE and the dynamics? We extend the graphical congestion games’ model for weighted players to investigate the impact of different weights We use the independence number α(G) and show that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Linear Congestion Games

A set V = {1, 2, ..., n} of selfish players A positive integer weight wi for i ∈ V A set R of resources A strategy space Σi ⊆ 2R − {∅} for i ∈ V A delay function de(x) = aex + be for e ∈ R A configuration s = (s1, s2, ..., sn) ∈ Σ A congestion se =

i:e∈siwi for e ∈ R

A cost ci(s) =wi

• e∈si de(se) for i ∈ V

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Graphical Congestion Games

An undirected social graph G = (V , E) A neighborhood Γi = {j ∈ V : {i, j} ∈ E} An actual congestion se=

j ∈ V :e∈sj wj for e ∈ R

A presumed congestion si

e= wi + j ∈ Γi:e∈sj wj for e ∈ R

An actual cost ci(s) =

e∈si wi(ae se+be) for i ∈ V

A presumed cost pi(s) =

e∈si wi(ae si e +be) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, pi(s−i, s′ i) ≥pi(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Graphical Congestion Games

An undirected social graph G = (V , E) A neighborhood Γi = {j ∈ V : {i, j} ∈ E} An actual congestion se=

j ∈ V :e∈sj wj for e ∈ R

A presumed congestion si

e= wi + j ∈ Γi:e∈sj wj for e ∈ R

An actual cost ci(s) =

e∈si wi(ae se+be) for i ∈ V

A presumed cost pi(s) =

e∈si wi(ae si e +be) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, pi(s−i, s′ i) ≥pi(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Graphical Congestion Games

An undirected social graph G = (V , E) A neighborhood Γi = {j ∈ V : {i, j} ∈ E} An actual congestion se=

j ∈ V :e∈sj wj for e ∈ R

A presumed congestion si

e= wi + j ∈ Γi:e∈sj wj for e ∈ R

An actual cost ci(s) =

e∈si wi(ae se+be) for i ∈ V

A presumed cost pi(s) =

e∈si wi(ae si e +be) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, pi(s−i, s′ i) ≥pi(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Graphical Congestion Games

An undirected social graph G = (V , E) A neighborhood Γi = {j ∈ V : {i, j} ∈ E} An actual congestion se=

j ∈ V :e∈sj wj for e ∈ R

A presumed congestion si

e= wi + j ∈ Γi:e∈sj wj for e ∈ R

An actual cost ci(s) =

e∈si wi(ae se+be) for i ∈ V

A presumed cost pi(s) =

e∈si wi(ae si e +be) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, pi(s−i, s′ i) ≥pi(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Graphical Congestion Games

An undirected social graph G = (V , E) A neighborhood Γi = {j ∈ V : {i, j} ∈ E} An actual congestion se=

j ∈ V :e∈sj wj for e ∈ R

A presumed congestion si

e= wi + j ∈ Γi:e∈sj wj for e ∈ R

An actual cost ci(s) =

e∈si wi(ae se+be) for i ∈ V

A presumed cost pi(s) =

e∈si wi(ae si e +be) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, pi(s−i, s′ i) ≥pi(s)}

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Graphical Congestion Games

An undirected social graph G = (V , E) A neighborhood Γi = {j ∈ V : {i, j} ∈ E} An actual congestion se=

j ∈ V :e∈sj wj for e ∈ R

A presumed congestion si

e= wi + j ∈ Γi:e∈sj wj for e ∈ R

An actual cost ci(s) =

e∈si wi(ae se+be) for i ∈ V

A presumed cost pi(s) =

e∈si wi(ae si e +be) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, pi(s−i, s′ i) ≥pi(s)}

Σ

PNE

OPT

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Weighted Graphical Congestion Games

An undirected social graph G = (V , E) A neighborhood Γi = {j ∈ V : {i, j} ∈ E} An actual congestion se=

j ∈ V :e∈sj wj for e ∈ R

A presumed congestion si

e= wi + j ∈ Γi:e∈sj wj for e ∈ R

An actual cost ci(s) =

e∈si wi(ae se+be) for i ∈ V

A presumed cost pi(s) =

e∈si wi(ae si e +be) for i ∈ V

PNE= {s ∈ Σ : ∀i ∈ V , ∀s′

i ∈ Σi, pi(s−i, s′ i) ≥pi(s)}

Σ

PNE

OPT

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

Social graph G = (V , E), resource set R and a configuration s.

2 3 1 2 4 3 2 1 2 1 4 2 3

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

Let e be some resource with de(x) = aex + be.

2 3 1 2 4 3 2 1 2 1 4 2 3 e

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

The set Ve(s) of players using it induces subgraph Ge.

2 3 1 2 4 3 2 1 2 1 4 2 3 e

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

The actual congestion se of resource e in s is 20.

2 3 1 2 4 3 2 1 2 1 4 2 3 e

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

This player’s presumed congestion for resource e was 14.

2 3 1 2 4 3 2 1 2 1 4 2 3 e

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

This player’s presumed congestion for resource e was 3.

2 3 1 2 4 3 2 1 2 1 4 2 3 e

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

Ce(s)= 20(ae20 + be) and Pe(s)= ae164 + 20be.

2 3 1 2 4 3 2 1 2 1 4 2 3 e

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Weighted Linear Congestion Games Weighted Graphical Congestion Games

Social Graph Example

We will show that Ce(s)≤ α(Ge)Pe(s) for all e ∈ R and s ∈ Σ.

2 3 1 2 4 3 2 1 2 1 4 2 3 e

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Potential Function

Theorem Every graphical linear congestion game with weighted players admits a potential function and thus a pure Nash equilibrium. Φ(s) = P(s) + U(s) 2 where U(s) = n

i=1

• e∈si wi(aewi + be) is a potential function.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Potential Function

Theorem Every graphical linear congestion game with weighted players admits a potential function and thus a pure Nash equilibrium. Φ(s) = P(s) + U(s) 2 where U(s) = n

i=1

• e∈si wi(aewi + be) is a potential function.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation

Total actual cost of s: C(s) =

i∈V ci(s)

Total presumed cost of s: P(s) =

i∈V pi(s)

Lemma For any configuration s, C(s) ≤ α(G)P(s) where α(G) is the independence number of the social graph G Proof sketch: For any resource e ∈ R let Ve(s) = {i ∈ V : e ∈ si} and Ge = (Ve(s), Ee(s)) be the graph induced by these players. It suffices to show Ce(s)≤ α(Ge)Pe(s) We first prove that this is true for the case of unweighted players. We then reduce the case of weighted players to the unweighted.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation

Total actual cost of s: C(s) =

i∈V ci(s)

Total presumed cost of s: P(s) =

i∈V pi(s)

Lemma For any configuration s, C(s) ≤ α(G)P(s) where α(G) is the independence number of the social graph G Proof sketch: For any resource e ∈ R let Ve(s) = {i ∈ V : e ∈ si} and Ge = (Ve(s), Ee(s)) be the graph induced by these players. It suffices to show Ce(s)≤ α(Ge)Pe(s) We first prove that this is true for the case of unweighted players. We then reduce the case of weighted players to the unweighted.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation

Total actual cost of s: C(s) =

i∈V ci(s)

Total presumed cost of s: P(s) =

i∈V pi(s)

Lemma For any configuration s, C(s) ≤ α(G)P(s) where α(G) is the independence number of the social graph G Proof sketch: For any resource e ∈ R let Ve(s) = {i ∈ V : e ∈ si} and Ge = (Ve(s), Ee(s)) be the graph induced by these players. It suffices to show Ce(s)≤ α(Ge)Pe(s) We first prove that this is true for the case of unweighted players. We then reduce the case of weighted players to the unweighted.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation

Total actual cost of s: C(s) =

i∈V ci(s)

Total presumed cost of s: P(s) =

i∈V pi(s)

Lemma For any configuration s, C(s) ≤ α(G)P(s) where α(G) is the independence number of the social graph G Proof sketch: For any resource e ∈ R let Ve(s) = {i ∈ V : e ∈ si} and Ge = (Ve(s), Ee(s)) be the graph induced by these players. It suffices to show Ce(s)≤ α(Ge)Pe(s) We first prove that this is true for the case of unweighted players. We then reduce the case of weighted players to the unweighted.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation

Total actual cost of s: C(s) =

i∈V ci(s)= e∈R Ce(s)

Total presumed cost of s: P(s) =

i∈V pi(s)= e∈R Pe(s)

Lemma For any configuration s, C(s) ≤ α(G)P(s) where α(G) is the independence number of the social graph G Proof sketch: For any resource e ∈ R let Ve(s) = {i ∈ V : e ∈ si} and Ge = (Ve(s), Ee(s)) be the graph induced by these players. It suffices to show Ce(s)≤ α(Ge)Pe(s) We first prove that this is true for the case of unweighted players. We then reduce the case of weighted players to the unweighted.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation

Total actual cost of s: C(s) =

i∈V ci(s)= e∈R Ce(s)

Total presumed cost of s: P(s) =

i∈V pi(s)= e∈R Pe(s)

Lemma For any configuration s, C(s) ≤ α(G)P(s) where α(G) is the independence number of the social graph G Proof sketch: For any resource e ∈ R let Ve(s) = {i ∈ V : e ∈ si} and Ge = (Ve(s), Ee(s)) be the graph induced by these players. It suffices to show Ce(s)≤ α(Ge)Pe(s) We first prove that this is true for the case of unweighted players. We then reduce the case of weighted players to the unweighted.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation

Total actual cost of s: C(s) =

i∈V ci(s)= e∈R Ce(s)

Total presumed cost of s: P(s) =

i∈V pi(s)= e∈R Pe(s)

Lemma For any configuration s, C(s) ≤ α(G)P(s) where α(G) is the independence number of the social graph G Proof sketch: For any resource e ∈ R let Ve(s) = {i ∈ V : e ∈ si} and Ge = (Ve(s), Ee(s)) be the graph induced by these players. It suffices to show Ce(s)≤ α(Ge)Pe(s) We first prove that this is true for the case of unweighted players. We then reduce the case of weighted players to the unweighted.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

Ce(s) = aen2

e + bene and Pe(s) = ae(2me + ne) + bene

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge Let ne = |Ve(s)|, me = |Ee(s)|, k =

ne α(Ge) and r = k − ⌊k⌋.

We get that me ≥ (k − r)(k + r − 1)α(Ge)/2

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics Potential Function Cost Approximation

Cost Approximation Proof

For any configuration s and resource e, Ce(s)≤ α(Ge)Pe(s) where α(Ge) is the independence number of Ge me ≥ (k − r)(k + r − 1)α(Ge)/2 ⇒ Pe(s)≥ ae [(k − r)(k + r + 1) + k] α(Ge) + bekα(Ge) ⇒ α(Ge)Pe(s)≥ aek2α2(Ge) + bekα(Ge) =Ce(s) For weighted players: We create a new graph G ′

e of unweighted players as follows:

Replace each player i of weight wi with a clique Qi of size wi For {i, j} ∈ Ee(s), connect all players of Qi and Qj i) Ce(s) and Pe(s) are not affected and ii) α(Ge) = α(G ′

e)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics The Price of Anarchy The Price of Stability

Bounds for PoA

n/2 n n

α(G) PoA Price of Anarchy Bounds for n=100

Θ(α2(G)) Θ(α(G)(n−α(G)))

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics The Price of Anarchy The Price of Stability

Lower bound for α(G) ≤ n/2

The blue circles are players The rectangles are resources with delay function de(x) = x Cost in sopt for each player is 1 Cost in equilibrium for each player is α(G)(α(G) + 1)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics The Price of Anarchy The Price of Stability

Lower bound for α(G) ≤ n/2

The blue circles are players The rectangles are resources with delay function de(x) = x Cost in sopt for each player is 1 Cost in equilibrium for each player is α(G)(α(G) + 1)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics The Price of Anarchy The Price of Stability

Lower bound for α(G) ≤ n/2

The blue circles are players The rectangles are resources with delay function de(x) = x Cost in sopt for each player is 1 Cost in equilibrium for each player is α(G)(α(G) + 1)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics The Price of Anarchy The Price of Stability

Lower bound for α(G) ≤ n/2

The blue circles are players The rectangles are resources with delay function de(x) = x Cost in sopt for each player is 1 Cost in equilibrium for each player is α(G)(α(G) + 1)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics The Price of Anarchy The Price of Stability

Lower bound for α(G) ≤ n/2

The blue circles are players The rectangles are resources with delay function de(x) = x Cost in sopt for each player is 1 Cost in equilibrium for each player is α(G)(α(G) + 1)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics The Price of Anarchy The Price of Stability

Bounds for PoS

n/2 n n

α(G) PoS Price of Stability Bounds for n=100

(2nα(G))/(n+α(G)) α(G) 2α(G)

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Convergence to Near Optimal Configurations

For n weighted players, let s∗ be a minimizer of the potential function Φ and 1

8 ≥ δ ≥ ǫ > 0.

Theorem Starting from a configuration s0, the largest improvement ǫ-Nash dynamics reaches a configuration s with C(s) ≤ ρ(1 + 8δ)C(OPT) in O(n

δ log Φ(s0) Φ(s∗)) steps.

For symmetric strategies and n unweighted players, s∗ be a minimizer of the potential function Φ, and ǫ ∈ (0, 1). Theorem Starting from a configuration s0, the largest improvement ǫ-Nash dynamics converges in O(n2

ǫ log Φ(s0) Φ(s∗)) steps.

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Summary and Conclusion

For weighted graphical congestion games, α(G) is a natural parameter for characterizing inefficiency and dynamics Allowing players to be weighted doesn’t make things worse We showed that:

These games admit a potential function For unweighted players: PoA ≤ 3α(G)+7

3α(G)+1α2(G)

if α(G) < n

2,

2n(n − α(G) + 1) if α(G) ≥ n

2

For weighted players:

PoA≤

α(G)(α(G)+2+√ α2(G)+4α(G)) 2

< α(G)(α(G) + 2) α(G) ≤PoS≤ 2α(G)

The techniques of Chien et al. and Awerbuch et al. apply

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games

Introduction Model and Preliminaries Potential Function and Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ-Nash Dynamics

Thank you!

Thank you!

Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games