Introduction Model and Preliminaries Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Σ i ⊆ 2 R − {∅} for i ∈ V A non-decreasing delay function d e : N → R ≥ 0 for e ∈ R Linear congestion games, i.e. d e ( x ) = a e x + b e s = ( s 1 , s 2 , ..., s n ) ∈ Σ =X i ∈ V Σ i where s i ∈ Σ i for i ∈ V A congestion s e = |{ i : e ∈ s i }| for e ∈ R A cost c i ( s ) = � e ∈ s i d e ( s e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , c i ( s − i , s ′ i ) ≥ c i ( s ) } Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Potential Games Monderer and Shapley (1996) “Potential Games” Exact potential function Φ( s ) : Σ → R such that Φ( s ) − Φ( s − i , s ′ i ) = c i ( s ) − c i ( s − i , s ′ i ) ∀ i ∈ V , s ′ i ∈ Σ i Nash dynamics converges to PNE (the directed state graph with payoff improving individual defections is a DAG) � s e For congestion games Φ( s ) = � k =1 d e ( k ) is exact e ∈ R Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Potential Games Monderer and Shapley (1996) “Potential Games” Exact potential function Φ( s ) : Σ → R such that Φ( s ) − Φ( s − i , s ′ i ) = c i ( s ) − c i ( s − i , s ′ i ) ∀ i ∈ V , s ′ i ∈ Σ i Nash dynamics converges to PNE (the directed state graph with payoff improving individual defections is a DAG) � s e For congestion games Φ( s ) = � k =1 d e ( k ) is exact e ∈ R Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Potential Games Monderer and Shapley (1996) “Potential Games” Exact potential function Φ( s ) : Σ → R such that Φ( s ) − Φ( s − i , s ′ i ) = c i ( s ) − c i ( s − i , s ′ i ) ∀ i ∈ V , s ′ i ∈ Σ i Nash dynamics converges to PNE (the directed state graph with payoff improving individual defections is a DAG) � s e For congestion games Φ( s ) = � k =1 d e ( k ) is exact e ∈ R Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Inefficiency of Pure Nash Equilibria Σ Social cost function: C ( s ) = � i ∈ V c i ( s ) OPT PNE Let configuration s opt ∈ arg min s ∈ Σ C ( s ) C ( s ) The (pure) Price of Anarchy (PoA): max s ∈ PNE C ( s opt ) C ( s ) The (pure) Price of Stability (PoS): min s ∈ PNE s opt 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Inefficiency of Pure Nash Equilibria Σ Social cost function: C ( s ) = � i ∈ V c i ( s ) OPT PNE Let configuration s opt ∈ arg min s ∈ Σ C ( s ) C ( s ) The (pure) Price of Anarchy (PoA): max s ∈ PNE C ( s opt ) C ( s ) The (pure) Price of Stability (PoS): min s ∈ PNE s opt 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Inefficiency of Pure Nash Equilibria Σ Social cost function: C ( s ) = � i ∈ V c i ( s ) OPT PNE Let configuration s opt ∈ arg min s ∈ Σ C ( s ) C ( s ) The (pure) Price of Anarchy (PoA): max s ∈ PNE C ( s opt ) C ( s ) The (pure) Price of Stability (PoS): min s ∈ PNE s opt 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Inefficiency of Pure Nash Equilibria Σ Social cost function: C ( s ) = � i ∈ V c i ( s ) OPT PNE Let configuration s opt ∈ arg min s ∈ Σ C ( s ) C ( s ) The (pure) Price of Anarchy (PoA): max s ∈ PNE C ( s opt ) C ( s ) The (pure) Price of Stability (PoS): min s ∈ PNE s opt 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Inefficiency of Pure Nash Equilibria Σ Social cost function: C ( s ) = � i ∈ V c i ( s ) OPT PNE Let configuration s opt ∈ arg min s ∈ Σ C ( s ) C ( s ) The (pure) Price of Anarchy (PoA): max s ∈ PNE C ( s opt ) C ( s ) The (pure) Price of Stability (PoS): min s ∈ PNE s opt 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Inefficiency of Pure Nash Equilibria Σ Social cost function: C ( s ) = � i ∈ V c i ( s ) OPT PNE Let configuration s opt ∈ arg min s ∈ Σ C ( s ) C ( s ) The (pure) Price of Anarchy (PoA): max s ∈ PNE C ( s opt ) C ( s ) The (pure) Price of Stability (PoS): min s ∈ PNE s opt 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 optimal 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 optimal 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 optimal 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 optimal 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Ignorance in Congestion Games Bil` o 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 | ( deg max ( G ) + 1)) where deg max ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Motivation and Previous Work Potential Function and Cost Approximation Contribution Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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: � 3 α ( G )+7 3 α ( G )+1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1) 2 For weighted players: α ( G )( α ( G )+2+ √ α 2 ( G )+4 α ( G )) PoA ≤ < α ( G )( α ( G ) + 2) 2 α ( G ) ≤ PoS ≤ 2 α ( 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 Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Linear Congestion Games A set V = { 1 , 2 , ..., n } of selfish players A positive integer weight w i for i ∈ V A set R of resources A strategy space Σ i ⊆ 2 R − {∅} for i ∈ V A delay function d e ( x ) = a e x + b e for e ∈ R A configuration s = ( s 1 , s 2 , ..., s n ) ∈ Σ A congestion s e = � i : e ∈ s i w i for e ∈ R A cost c i ( s ) = w i � e ∈ s i d e ( s e ) for i ∈ V Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Graphical Congestion Games An undirected social graph G = ( V , E ) A neighborhood Γ i = { j ∈ V : { i , j } ∈ E } An actual congestion s e = � j ∈ V : e ∈ s j w j for e ∈ R A presumed congestion s i e = w i + � j ∈ Γ i : e ∈ s j w j for e ∈ R An actual cost c i ( s ) = � e ∈ s i w i ( a e s e + b e ) for i ∈ V e ∈ s i w i ( a e s i A presumed cost p i ( s ) = � e + b e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , p i ( s − i , s ′ i ) ≥ p i ( s ) } Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Graphical Congestion Games An undirected social graph G = ( V , E ) A neighborhood Γ i = { j ∈ V : { i , j } ∈ E } An actual congestion s e = � j ∈ V : e ∈ s j w j for e ∈ R A presumed congestion s i e = w i + � j ∈ Γ i : e ∈ s j w j for e ∈ R An actual cost c i ( s ) = � e ∈ s i w i ( a e s e + b e ) for i ∈ V e ∈ s i w i ( a e s i A presumed cost p i ( s ) = � e + b e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , p i ( s − i , s ′ i ) ≥ p i ( s ) } Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Graphical Congestion Games An undirected social graph G = ( V , E ) A neighborhood Γ i = { j ∈ V : { i , j } ∈ E } An actual congestion s e = � j ∈ V : e ∈ s j w j for e ∈ R A presumed congestion s i e = w i + � j ∈ Γ i : e ∈ s j w j for e ∈ R An actual cost c i ( s ) = � e ∈ s i w i ( a e s e + b e ) for i ∈ V e ∈ s i w i ( a e s i A presumed cost p i ( s ) = � e + b e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , p i ( s − i , s ′ i ) ≥ p i ( s ) } Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Graphical Congestion Games An undirected social graph G = ( V , E ) A neighborhood Γ i = { j ∈ V : { i , j } ∈ E } An actual congestion s e = � j ∈ V : e ∈ s j w j for e ∈ R A presumed congestion s i e = w i + � j ∈ Γ i : e ∈ s j w j for e ∈ R An actual cost c i ( s ) = � e ∈ s i w i ( a e s e + b e ) for i ∈ V e ∈ s i w i ( a e s i A presumed cost p i ( s ) = � e + b e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , p i ( s − i , s ′ i ) ≥ p i ( s ) } Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Graphical Congestion Games An undirected social graph G = ( V , E ) A neighborhood Γ i = { j ∈ V : { i , j } ∈ E } An actual congestion s e = � j ∈ V : e ∈ s j w j for e ∈ R A presumed congestion s i e = w i + � j ∈ Γ i : e ∈ s j w j for e ∈ R An actual cost c i ( s ) = � e ∈ s i w i ( a e s e + b e ) for i ∈ V e ∈ s i w i ( a e s i A presumed cost p i ( s ) = � e + b e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , p i ( s − i , s ′ i ) ≥ p i ( s ) } Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Graphical Congestion Games An undirected social graph G = ( V , E ) A neighborhood Γ i = { j ∈ V : { i , j } ∈ E } An actual congestion s e = � j ∈ V : e ∈ s j w j for e ∈ R A presumed congestion s i e = w i + � j ∈ Γ i : e ∈ s j w j for e ∈ R An actual cost c i ( s ) = � e ∈ s i w i ( a e s e + b e ) for i ∈ V e ∈ s i w i ( a e s i A presumed cost p i ( s ) = � e + b e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , p i ( s − i , s ′ i ) ≥ p i ( s ) } Σ OPT PNE Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Weighted Graphical Congestion Games An undirected social graph G = ( V , E ) A neighborhood Γ i = { j ∈ V : { i , j } ∈ E } An actual congestion s e = � j ∈ V : e ∈ s j w j for e ∈ R A presumed congestion s i e = w i + � j ∈ Γ i : e ∈ s j w j for e ∈ R An actual cost c i ( s ) = � e ∈ s i w i ( a e s e + b e ) for i ∈ V e ∈ s i w i ( a e s i A presumed cost p i ( s ) = � e + b e ) for i ∈ V PNE= { s ∈ Σ : ∀ i ∈ V , ∀ s ′ i ∈ Σ i , p i ( s − i , s ′ i ) ≥ p i ( s ) } Σ PNE OPT Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example Social graph G = ( V , E ), resource set R and a configuration s . 2 3 4 3 2 1 2 2 1 1 4 2 3 Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example Let e be some resource with d e ( x ) = a e x + b e . 2 3 4 3 2 1 2 2 1 1 4 2 3 e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example The set V e ( s ) of players using it induces subgraph G e . 2 3 4 3 2 1 2 2 1 1 4 2 3 e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example The actual congestion s e of resource e in s is 20 . 2 3 4 3 2 1 2 2 1 1 4 2 3 e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example This player’s presumed congestion for resource e was 14 . 2 3 4 3 2 1 2 2 1 1 4 2 3 e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example This player’s presumed congestion for resource e was 3 . 2 3 4 3 2 1 2 2 1 1 4 2 3 e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example C e ( s )= 20( a e 20 + b e ) and P e ( s )= a e 164 + 20 b e . 2 3 4 3 2 1 2 2 1 1 4 2 3 e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Weighted Linear Congestion Games Potential Function and Cost Approximation Weighted Graphical Congestion Games Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Social Graph Example We will show that C e ( s ) ≤ α ( G e ) P e ( s ) for all e ∈ R and s ∈ Σ. 2 3 4 3 2 1 2 2 1 1 4 2 3 e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 � e ∈ s i w i ( a e w i + b e ) is a potential function. i =1 Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics 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 � e ∈ s i w i ( a e w i + b e ) is a potential function. i =1 Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Total actual cost of s : C ( s ) = � i ∈ V c i ( s ) Total presumed cost of s : P ( s ) = � i ∈ V p i ( 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 V e ( s ) = { i ∈ V : e ∈ s i } and G e = ( V e ( s ) , E e ( s )) be the graph induced by these players. It suffices to show C e ( s ) ≤ α ( G e ) P e ( 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 Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Total actual cost of s : C ( s ) = � i ∈ V c i ( s ) Total presumed cost of s : P ( s ) = � i ∈ V p i ( 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 V e ( s ) = { i ∈ V : e ∈ s i } and G e = ( V e ( s ) , E e ( s )) be the graph induced by these players. It suffices to show C e ( s ) ≤ α ( G e ) P e ( 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 Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Total actual cost of s : C ( s ) = � i ∈ V c i ( s ) Total presumed cost of s : P ( s ) = � i ∈ V p i ( 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 V e ( s ) = { i ∈ V : e ∈ s i } and G e = ( V e ( s ) , E e ( s )) be the graph induced by these players. It suffices to show C e ( s ) ≤ α ( G e ) P e ( 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 Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Total actual cost of s : C ( s ) = � i ∈ V c i ( s ) Total presumed cost of s : P ( s ) = � i ∈ V p i ( 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 V e ( s ) = { i ∈ V : e ∈ s i } and G e = ( V e ( s ) , E e ( s )) be the graph induced by these players. It suffices to show C e ( s ) ≤ α ( G e ) P e ( 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 Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Total actual cost of s : C ( s ) = � i ∈ V c i ( s )= � e ∈ R C e ( s ) Total presumed cost of s : P ( s ) = � i ∈ V p i ( s )= � e ∈ R P e ( 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 V e ( s ) = { i ∈ V : e ∈ s i } and G e = ( V e ( s ) , E e ( s )) be the graph induced by these players. It suffices to show C e ( s ) ≤ α ( G e ) P e ( 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 Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Total actual cost of s : C ( s ) = � i ∈ V c i ( s )= � e ∈ R C e ( s ) Total presumed cost of s : P ( s ) = � i ∈ V p i ( s )= � e ∈ R P e ( 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 V e ( s ) = { i ∈ V : e ∈ s i } and G e = ( V e ( s ) , E e ( s )) be the graph induced by these players. It suffices to show C e ( s ) ≤ α ( G e ) P e ( 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 Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Total actual cost of s : C ( s ) = � i ∈ V c i ( s )= � e ∈ R C e ( s ) Total presumed cost of s : P ( s ) = � i ∈ V p i ( s )= � e ∈ R P e ( 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 V e ( s ) = { i ∈ V : e ∈ s i } and G e = ( V e ( s ) , E e ( s )) be the graph induced by these players. It suffices to show C e ( s ) ≤ α ( G e ) P e ( 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 Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . C e ( s ) = a e n 2 e + b e n e and P e ( s ) = a e (2 m e + n e ) + b e n e Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e n e Let n e = | V e ( s ) | , m e = | E e ( s ) | , k = α ( G e ) and r = k − ⌊ k ⌋ . We get that m e ≥ ( k − r )( k + r − 1) α ( G e ) / 2 Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e m e ≥ ( k − r )( k + r − 1) α ( G e ) / 2 ⇒ P e ( s ) ≥ a e [( k − r )( k + r + 1) + k ] α ( G e ) + b e k α ( G e ) ⇒ α ( G e ) P e ( s ) ≥ a e k 2 α 2 ( G e ) + b e k α ( G e ) = C e ( s ) For weighted players: We create a new graph G ′ e of unweighted players as follows: Replace each player i of weight w i with a clique Q i of size w i For { i , j } ∈ E e ( s ), connect all players of Q i and Q j i) C e ( s ) and P e ( s ) are not affected and ii) α ( G e ) = α ( G ′ e ) Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
Introduction Model and Preliminaries Potential Function Potential Function and Cost Approximation Cost Approximation Inefficiency of Pure Nash Equilibria Convergence of ǫ -Nash Dynamics Cost Approximation Proof For any configuration s and resource e , C e ( s ) ≤ α ( G e ) P e ( s ) where α ( G e ) is the independence number of G e m e ≥ ( k − r )( k + r − 1) α ( G e ) / 2 ⇒ P e ( s ) ≥ a e [( k − r )( k + r + 1) + k ] α ( G e ) + b e k α ( G e ) ⇒ α ( G e ) P e ( s ) ≥ a e k 2 α 2 ( G e ) + b e k α ( G e ) = C e ( s ) For weighted players: We create a new graph G ′ e of unweighted players as follows: Replace each player i of weight w i with a clique Q i of size w i For { i , j } ∈ E e ( s ), connect all players of Q i and Q j i) C e ( s ) and P e ( s ) are not affected and ii) α ( G e ) = α ( G ′ e ) Vasilis Gkatzelis Impact of Social Ignorance on Weighted Congestion Games
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