Coordination Games on Graphs Krzysztof Apt CWI & University of Amsterdam joint work with: Guido Sch¨ afer, Mona Rahn, Sunil Simon
Coordination Games on Graphs Setting: undirected graph G = ( N , E ) with n nodes → each node i ∈ N corresponds to a strategic player Krzysztof Apt Coordination Games on Graphs
Coordination Games on Graphs Setting: undirected graph G = ( N , E ) with n nodes → each node i ∈ N corresponds to a strategic player set of colours M Krzysztof Apt Coordination Games on Graphs
Coordination Games on Graphs Setting: undirected graph G = ( N , E ) with n nodes → each node i ∈ N corresponds to a strategic player set of colours M for each node i ∈ N : set S i ⊆ M of colours available to i → each player selects a colour s i ∈ S i from his colour set Krzysztof Apt Coordination Games on Graphs
Coordination Games on Graphs Setting: undirected graph G = ( N , E ) with n nodes → each node i ∈ N corresponds to a strategic player set of colours M for each node i ∈ N : set S i ⊆ M of colours available to i → each player selects a colour s i ∈ S i from his colour set the payoff of player i under joint strategy s = ( s 1 , . . . , s n ) is the number of neighbours who chose the same colour: p i ( s ) = { j ∈ N ( i ) : s j = s i } , where N ( i ) is the set of neighbours of i in G Krzysztof Apt Coordination Games on Graphs
Example { • , • } { • , • } { • , • } 2 4 6 { • , • } { • , • } 1 8 3 5 7 { • , • } { • , • } { • , • } Krzysztof Apt Coordination Games on Graphs
Example { • , • } { • , • } { • , • } 2 4 6 { • , • } { • , • } 1 8 5 3 7 { • , • } { • , • } { • , • } Krzysztof Apt Coordination Games on Graphs
Example { • , • } { • , • } { • , • } 2 4 6 { • , • } { • , • } 1 8 5 3 7 { • , • } { • , • } { • , • } Krzysztof Apt Coordination Games on Graphs
Example { • , • } { • , • } { • , • } 2 4 6 { • , • } { • , • } 1 8 5 3 7 { • , • } { • , • } { • , • } Krzysztof Apt Coordination Games on Graphs
Motivation Characteristics: join the crowd property: the payoff of each player weakly 1 increases when more players choose his strategy asymmetric strategies: players may have individual strategy sets 2 local dependency: the payoff of each player only depends on the 3 choices made by his neighbors Krzysztof Apt Coordination Games on Graphs
Motivation Characteristics: join the crowd property: the payoff of each player weakly 1 increases when more players choose his strategy asymmetric strategies: players may have individual strategy sets 2 local dependency: the payoff of each player only depends on the 3 choices made by his neighbors Applications: choose between multiple competing providers offering the same service or product peer-to-peer networks social networks photo sharing platforms mobile phone providers, etc. Krzysztof Apt Coordination Games on Graphs
Results in a Nutshell Do pure Nash or strong equilibria always exist? 1 What about the inefficiency of equilibria? 2 Can we compute such equilibria efficiently? 3 Krzysztof Apt Coordination Games on Graphs
Results in a Nutshell Do pure Nash or strong equilibria always exist? 1 ◮ pure Nash equilibria: yes ◮ strong equilibria: depends on the structure of the graph What about the inefficiency of equilibria? 2 Can we compute such equilibria efficiently? 3 Krzysztof Apt Coordination Games on Graphs
Results in a Nutshell Do pure Nash or strong equilibria always exist? 1 ◮ pure Nash equilibria: yes ◮ strong equilibria: depends on the structure of the graph What about the inefficiency of equilibria? 2 ◮ price of anarchy ranges from ∞ for PNE to 2 for SE ◮ price of stability is 1 for many graphs Can we compute such equilibria efficiently? 3 Krzysztof Apt Coordination Games on Graphs
Results in a Nutshell Do pure Nash or strong equilibria always exist? 1 ◮ pure Nash equilibria: yes ◮ strong equilibria: depends on the structure of the graph What about the inefficiency of equilibria? 2 ◮ price of anarchy ranges from ∞ for PNE to 2 for SE ◮ price of stability is 1 for many graphs Can we compute such equilibria efficiently? 3 ◮ pure Nash equilibria: yes ◮ strong equilibria in pseudoforests: yes ◮ decision problem in general: co-NP-complete Krzysztof Apt Coordination Games on Graphs
Related Classes of Games Graphical Games: [Kearns, Littman, Singh ’01] given a graph G = ( N , E ) on player set N , the payoff of player i is a function p i : × j ∈ N ( i ) ∪{ i } S j → R → The payoff of each player depends only on his strategy and the strategies of its neighbours. Krzysztof Apt Coordination Games on Graphs
Related Classes of Games Graphical Games: [Kearns, Littman, Singh ’01] given a graph G = ( N , E ) on player set N , the payoff of player i is a function p i : × j ∈ N ( i ) ∪{ i } S j → R → The payoff of each player depends only on his strategy and the strategies of its neighbours. Polymatrix Games: [Janovskaya ’68] for every pair of players i and j there exists a partial payoff function p ij such that � p ij ( s i , s j ) p i ( s ) := j � = i → Each pair of players plays a separate game and the payoff is the sum of the payoffs in these separate games. Krzysztof Apt Coordination Games on Graphs
Strong Equilibrium Notation: coalition: non-empty subset K ⊆ N of players coalition K can profitably deviate from s if there is some s ′ = ( s ′ K , s − K ) such that for every i ∈ K : p i ( s ′ ) > p i ( s ) Krzysztof Apt Coordination Games on Graphs
Strong Equilibrium Notation: coalition: non-empty subset K ⊆ N of players coalition K can profitably deviate from s if there is some s ′ = ( s ′ K , s − K ) such that for every i ∈ K : p i ( s ′ ) > p i ( s ) → every player of the coalition strictly improves his payoff → write: s K → s ′ Krzysztof Apt Coordination Games on Graphs
Strong Equilibrium Notation: coalition: non-empty subset K ⊆ N of players coalition K can profitably deviate from s if there is some s ′ = ( s ′ K , s − K ) such that for every i ∈ K : p i ( s ′ ) > p i ( s ) → every player of the coalition strictly improves his payoff → write: s K → s ′ Definition A joint strategy s is a k -equilibrium if there is no coalition of at most k players that can profitably deviate. Krzysztof Apt Coordination Games on Graphs
Strong Equilibrium Notation: coalition: non-empty subset K ⊆ N of players coalition K can profitably deviate from s if there is some s ′ = ( s ′ K , s − K ) such that for every i ∈ K : p i ( s ′ ) > p i ( s ) → every player of the coalition strictly improves his payoff → write: s K → s ′ Definition A joint strategy s is a k -equilibrium if there is no coalition of at most k players that can profitably deviate. Note: 1-equilibrium is a pure Nash equilibrium n -equilibrium is a strong equilibrium Krzysztof Apt Coordination Games on Graphs
Inefficiency Social welfare: � SW ( s ) = p i ( s ) i ∈ N Krzysztof Apt Coordination Games on Graphs
Inefficiency Social welfare: � SW ( s ) = p i ( s ) i ∈ N k -Price of Anarchy: max s ∈ S SW ( s ) min s ∈ S , s is a k -SE SW ( s ) Krzysztof Apt Coordination Games on Graphs
Inefficiency Social welfare: � SW ( s ) = p i ( s ) i ∈ N k -Price of Anarchy: max s ∈ S SW ( s ) min s ∈ S , s is a k -SE SW ( s ) k -Price of Stability: max s ∈ S SW ( s ) max s ∈ S , s is a k -SE SW ( s ) Krzysztof Apt Coordination Games on Graphs
Exact Potentials Assume G := ( S 1 , . . . , S n , p 1 , . . . , p n ) . A profitable deviation: a pair ( s , s ′ ) of joint strategies such that p i ( s ′ ) > p i ( s ) , where s ′ = ( s ′ i , s − i ) . An exact potential for G : a function P : S 1 × · · · × S n → R such that for every profitable deviation ( s , s ′ ) , where s ′ = ( s ′ i , s − i ) , P ( s ′ ) − P ( s ) = p i ( s ′ ) − p i ( s ) . Note Every finite game with an exact potential has a Nash equilibrium. Krzysztof Apt Coordination Games on Graphs
Pure Nash Equilibria Theorem (i) Every coordination game on a graph has an exact potential. (ii) The price of stability is 1. (iii) For every graph there is a colour assignment such that the price of anarchy of the coordination game is ∞ . Krzysztof Apt Coordination Games on Graphs
Pure Nash Equilibria Theorem (i) Every coordination game on a graph has an exact potential. (ii) The price of stability is 1. (iii) For every graph there is a colour assignment such that the price of anarchy of the coordination game is ∞ . Proof idea: (i) and (ii): P ( s ) := 1 2 SW ( s ) is an exact potential. Krzysztof Apt Coordination Games on Graphs
Pure Nash Equilibria Theorem (i) Every coordination game on a graph has an exact potential. (ii) The price of stability is 1. (iii) For every graph there is a colour assignment such that the price of anarchy of the coordination game is ∞ . Proof idea: (i) and (ii): P ( s ) := 1 2 SW ( s ) is an exact potential. (iii): Assign to each node in the graph ( N , E ) two colours: one private and one common. Krzysztof Apt Coordination Games on Graphs
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