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
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
Nash Equilibria Theorem Every coordination game on a graph has an exact potential. Krzysztof Apt Coordination Games on Graphs
Nash Equilibria Theorem Every coordination game on a graph has an exact potential. Proof idea: P ( s ) := 1 2 SW ( s ) is an exact potential. Krzysztof Apt Coordination Games on Graphs
Coalitional Improvement Path A c(oalitional)-improvement path is a maximal sequence s 1 → s 2 → s 3 → . . . of joint strategies such that for every k > 1 there is a coalition K such that s k is a profitable deviation of the players in K from s k − 1 . Krzysztof Apt Coordination Games on Graphs
Coalitional Improvement Path A c(oalitional)-improvement path is a maximal sequence s 1 → s 2 → s 3 → . . . of joint strategies such that for every k > 1 there is a coalition K such that s k is a profitable deviation of the players in K from s k − 1 . Definition A strategic game has the finite c -improvement path property (c-FIP) if every c -improvement path is finite. Krzysztof Apt Coordination Games on Graphs
Coalitional Improvement Path A c(oalitional)-improvement path is a maximal sequence s 1 → s 2 → s 3 → . . . of joint strategies such that for every k > 1 there is a coalition K such that s k is a profitable deviation of the players in K from s k − 1 . Definition A strategic game has the finite c -improvement path property (c-FIP) if every c -improvement path is finite. Note: c-FIP implies existence of strong equilibria. Krzysztof Apt Coordination Games on Graphs
Generalized Ordinal c-Potentials A generalized ordinal c-potential is a function P : S 1 × · · · × S n → A such that for some strict partial ordering ( P ( S ) , ≻ ) : if s K → s ′ for some K , then P ( s ′ ) ≻ P ( s ) . Krzysztof Apt Coordination Games on Graphs
Generalized Ordinal c-Potentials A generalized ordinal c-potential is a function P : S 1 × · · · × S n → A such that for some strict partial ordering ( P ( S ) , ≻ ) : if s K → s ′ for some K , then P ( s ′ ) ≻ P ( s ) . Note If a finite game has a generalized ordinal c-potential, then it has the c-FIP . Krzysztof Apt Coordination Games on Graphs
Decreasing Social Welfare { • , • } { • , • } { • } { • } 1 2 { • } { • } { • } { • } 4 3 { • } { • } { • , • } { • , • } SW ( s ) = 24 Krzysztof Apt Coordination Games on Graphs
Decreasing Social Welfare { • , • } { • , • } { • } { • } 1 2 { • } { • } { • } { • } 4 3 { • } { • } { • , • } { • , • } SW ( s ′ ) = 20 Krzysztof Apt Coordination Games on Graphs
Crucial Lemma Take a coordination game on G = ( N , E ) and a joint strategy s . E + s is the set of unicolor edges { i , j } ∈ E with s i = s j F ⊆ E is a feedback edge set of G if G \ F is acyclic G [ K ] is the subgraph of G induced by K ⊆ N Krzysztof Apt Coordination Games on Graphs
Crucial Lemma Take a coordination game on G = ( N , E ) and a joint strategy s . E + s is the set of unicolor edges { i , j } ∈ E with s i = s j F ⊆ E is a feedback edge set of G if G \ F is acyclic G [ K ] is the subgraph of G induced by K ⊆ N Lemma Suppose s K → s ′ is a profitable deviation. Let F be a feedback edge set of G [ K ] . Then SW ( s ′ ) − SW ( s ) > 2 | F ∩ E + s | − 2 | F ∩ E + s ′ | . Krzysztof Apt Coordination Games on Graphs
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