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Anti-Coordination Games and Graph Colouring
K.S. Mallikarjuna Rao(Joint work with Arko Chatterjee)
Industrial Engineering & Operations ResearchIndian Institute of Technology Bombay
Workshop on Game Theory and Mechanism DesignIISc Bengaluru
15th January, 2016
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Graph Coloring
I G = (V ,E ) is a finite, simple and undirected graph; V is setof vertices and E is set of edges.
I A finite set C denotes the set of colours available to eachnode.
I A proper colouring of the graph G is a function c : V → Csuch that for every edge (i , j) ∈ E , c(i) 6= c(j).
I If the number of colours used in the proper colouring c of thegraph is k , then it is called proper-k-colouring.
I The minimum possible value of k such that there is aproper-k-colouring is called the chromatic number of thegraph
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Graph Coloring
I G = (V ,E ) is a finite, simple and undirected graph; V is setof vertices and E is set of edges.
I A finite set C denotes the set of colours available to eachnode.
I A proper colouring of the graph G is a function c : V → Csuch that for every edge (i , j) ∈ E , c(i) 6= c(j).
I If the number of colours used in the proper colouring c of thegraph is k , then it is called proper-k-colouring.
I The minimum possible value of k such that there is aproper-k-colouring is called the chromatic number of thegraph
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Graph Coloring
I G = (V ,E ) is a finite, simple and undirected graph; V is setof vertices and E is set of edges.
I A finite set C denotes the set of colours available to eachnode.
I A proper colouring of the graph G is a function c : V → Csuch that for every edge (i , j) ∈ E , c(i) 6= c(j).
I If the number of colours used in the proper colouring c of thegraph is k , then it is called proper-k-colouring.
I The minimum possible value of k such that there is aproper-k-colouring is called the chromatic number of thegraph
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Graph Coloring
I G = (V ,E ) is a finite, simple and undirected graph; V is setof vertices and E is set of edges.
I A finite set C denotes the set of colours available to eachnode.
I A proper colouring of the graph G is a function c : V → Csuch that for every edge (i , j) ∈ E , c(i) 6= c(j).
I If the number of colours used in the proper colouring c of thegraph is k , then it is called proper-k-colouring.
I The minimum possible value of k such that there is aproper-k-colouring is called the chromatic number of thegraph
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Graph Coloring
I G = (V ,E ) is a finite, simple and undirected graph; V is setof vertices and E is set of edges.
I A finite set C denotes the set of colours available to eachnode.
I A proper colouring of the graph G is a function c : V → Csuch that for every edge (i , j) ∈ E , c(i) 6= c(j).
I If the number of colours used in the proper colouring c of thegraph is k , then it is called proper-k-colouring.
I The minimum possible value of k such that there is aproper-k-colouring is called the chromatic number of thegraph
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Graph Coloring
I How do you find minimal colouring?
I An important problem in computer science
I Hard optimization problem
I Applications in diverse fields
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Graph Coloring
I How do you find minimal colouring?
I An important problem in computer science
I Hard optimization problem
I Applications in diverse fields
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Graph Coloring
I How do you find minimal colouring?
I An important problem in computer science
I Hard optimization problem
I Applications in diverse fields
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Graph Coloring
I How do you find minimal colouring?
I An important problem in computer science
I Hard optimization problem
I Applications in diverse fields
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Social Game or Local Interaction Game
I Let N = {1, 2, · · · , n} be the set of agents.
I Agents are embedded in a network, whose adjacency matrix isgiven by G .
I Any two partners (neighbours to each other) will play asymmetric bilateral game.
I The utility of agent i agains an agent j , in a bilateral game, isgiven by π(si , sj).
I A crucial assumption is that every player chooses the sameaction in all bilateral games.
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Social Game or Local Interaction Game
I Let N = {1, 2, · · · , n} be the set of agents.
I Agents are embedded in a network, whose adjacency matrix isgiven by G .
I Any two partners (neighbours to each other) will play asymmetric bilateral game.
I The utility of agent i agains an agent j , in a bilateral game, isgiven by π(si , sj).
I A crucial assumption is that every player chooses the sameaction in all bilateral games.
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Social Game or Local Interaction Game
I Let N = {1, 2, · · · , n} be the set of agents.
I Agents are embedded in a network, whose adjacency matrix isgiven by G .
I Any two partners (neighbours to each other) will play asymmetric bilateral game.
I The utility of agent i agains an agent j , in a bilateral game, isgiven by π(si , sj).
I A crucial assumption is that every player chooses the sameaction in all bilateral games.
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Social Game or Local Interaction Game
I Let N = {1, 2, · · · , n} be the set of agents.
I Agents are embedded in a network, whose adjacency matrix isgiven by G .
I Any two partners (neighbours to each other) will play asymmetric bilateral game.
I The utility of agent i agains an agent j , in a bilateral game, isgiven by π(si , sj).
I A crucial assumption is that every player chooses the sameaction in all bilateral games.
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Social Game or Local Interaction Game
I Let N = {1, 2, · · · , n} be the set of agents.
I Agents are embedded in a network, whose adjacency matrix isgiven by G .
I Any two partners (neighbours to each other) will play asymmetric bilateral game.
I The utility of agent i agains an agent j , in a bilateral game, isgiven by π(si , sj).
I A crucial assumption is that every player chooses the sameaction in all bilateral games.
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Social Game or Local Interaction Game
I Utility of an agent i in the social game is given by
πi (si , s−i ) =n∑
j=1
gijπ(si , sj).
I A profile s is a Nash equilibrium if it satisfies
∀i , ∀s ′i , πi (si , s−i ) ≥ πi (s ′i , s−i ).
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Social Game or Local Interaction Game
I Utility of an agent i in the social game is given by
πi (si , s−i ) =n∑
j=1
gijπ(si , sj).
I A profile s is a Nash equilibrium if it satisfies
∀i , ∀s ′i , πi (si , s−i ) ≥ πi (s ′i , s−i ).
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Social Game or Local Interaction Game
I The study of social interactions is a very active field.
I The literature focussed mainly on positive interactions, whenagents have an incentive to conform with what others do.
I In other words, the underlying bilateral game is a coordinationgame.
I Bramoulle is the first work to study the negative interactions.
I Many applications involving negative interactions.
I Negative interactions are modelled using anti-coordinationgames.
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Social Game or Local Interaction Game
I The study of social interactions is a very active field.
I The literature focussed mainly on positive interactions, whenagents have an incentive to conform with what others do.
I In other words, the underlying bilateral game is a coordinationgame.
I Bramoulle is the first work to study the negative interactions.
I Many applications involving negative interactions.
I Negative interactions are modelled using anti-coordinationgames.
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Social Game or Local Interaction Game
I The study of social interactions is a very active field.
I The literature focussed mainly on positive interactions, whenagents have an incentive to conform with what others do.
I In other words, the underlying bilateral game is a coordinationgame.
I Bramoulle is the first work to study the negative interactions.
I Many applications involving negative interactions.
I Negative interactions are modelled using anti-coordinationgames.
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Social Game or Local Interaction Game
I The study of social interactions is a very active field.
I The literature focussed mainly on positive interactions, whenagents have an incentive to conform with what others do.
I In other words, the underlying bilateral game is a coordinationgame.
I Bramoulle is the first work to study the negative interactions.
I Many applications involving negative interactions.
I Negative interactions are modelled using anti-coordinationgames.
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Social Game or Local Interaction Game
I The study of social interactions is a very active field.
I The literature focussed mainly on positive interactions, whenagents have an incentive to conform with what others do.
I In other words, the underlying bilateral game is a coordinationgame.
I Bramoulle is the first work to study the negative interactions.
I Many applications involving negative interactions.
I Negative interactions are modelled using anti-coordinationgames.
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Social Game or Local Interaction Game
I The study of social interactions is a very active field.
I The literature focussed mainly on positive interactions, whenagents have an incentive to conform with what others do.
I In other words, the underlying bilateral game is a coordinationgame.
I Bramoulle is the first work to study the negative interactions.
I Many applications involving negative interactions.
I Negative interactions are modelled using anti-coordinationgames.
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Anti-Coordination Games
Anti-Coordination games represent two types of situation:
I when differentiation yields mutual gains; e.g.,
(0 11 0
). The
production of positive output requires that partners adoptdifferent strategies;
I when there is a kind of predation of one strategy on the other;e.g., Hawk-Dove game and Chicken game.
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Anti-Coordination Games
Anti-Coordination games represent two types of situation:
I when differentiation yields mutual gains; e.g.,
(0 11 0
). The
production of positive output requires that partners adoptdifferent strategies;
I when there is a kind of predation of one strategy on the other;e.g., Hawk-Dove game and Chicken game.
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Bramoulle’s Model
I Each agent has two choices A and B.
I The bilateral game is anti-coordination game. It means thatthe pure strategy equilibria are (A,B) and (B,A).
I This is equivalent to saying
π(B,A) > π(A,A); π(AB) > π(B,B)
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Bramoulle’s Model
I Each agent has two choices A and B.
I The bilateral game is anti-coordination game. It means thatthe pure strategy equilibria are (A,B) and (B,A).
I This is equivalent to saying
π(B,A) > π(A,A); π(AB) > π(B,B)
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Bramoulle’s Model
I Each agent has two choices A and B.
I The bilateral game is anti-coordination game. It means thatthe pure strategy equilibria are (A,B) and (B,A).
I This is equivalent to saying
π(B,A) > π(A,A); π(AB) > π(B,B)
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Bramoulle’s Model
I The bilateral game has a unique mixed equilibrium in whichthe probability of playing A is
pA =π(A,B)− π(B,B)
π(A,B)− π(B,B) + π(B,A)− π(A,A)
I A profile s is a Nash equilibrium of the social game if itsatisfies
∀i , ∀s ′i , πi (si , s−i ) ≥ πi (s ′i , s−i ).
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Bramoulle’s Model
I The bilateral game has a unique mixed equilibrium in whichthe probability of playing A is
pA =π(A,B)− π(B,B)
π(A,B)− π(B,B) + π(B,A)− π(A,A)
I A profile s is a Nash equilibrium of the social game if itsatisfies
∀i , ∀s ′i , πi (si , s−i ) ≥ πi (s ′i , s−i ).
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Bramoulle’s Model
TheoremA profile s is a Nash equilibrium if and only if for every agent i ,
ni ,A < pAni =⇒ si = A and ni ,A < pAni =⇒ si = B.
Here ni refers to the number of neighbours of i ; ni ,A refers to thenumber of neighbours of i playing A.
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Bramoulle’s Model
I Every local interaction game admits a potential function(Blume(1993) and Young(1998).
I In our case, the potential function is given by negative of thefrustration function
φ(s, πA, πB , g) = πAnBB + πBnBB
I Here πA = π(A,B)− π(B,B); πB = π(B,A)− π(A,A); nAAis the number of links between A players.
I Many results from Potential games can be applied.
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Bramoulle’s Model
I Every local interaction game admits a potential function(Blume(1993) and Young(1998).
I In our case, the potential function is given by negative of thefrustration function
φ(s, πA, πB , g) = πAnBB + πBnBB
I Here πA = π(A,B)− π(B,B); πB = π(B,A)− π(A,A); nAAis the number of links between A players.
I Many results from Potential games can be applied.
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Bramoulle’s Model
I Every local interaction game admits a potential function(Blume(1993) and Young(1998).
I In our case, the potential function is given by negative of thefrustration function
φ(s, πA, πB , g) = πAnBB + πBnBB
I Here πA = π(A,B)− π(B,B); πB = π(B,A)− π(A,A); nAAis the number of links between A players.
I Many results from Potential games can be applied.
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Bramoulle’s Model
I Every local interaction game admits a potential function(Blume(1993) and Young(1998).
I In our case, the potential function is given by negative of thefrustration function
φ(s, πA, πB , g) = πAnBB + πBnBB
I Here πA = π(A,B)− π(B,B); πB = π(B,A)− π(A,A); nAAis the number of links between A players.
I Many results from Potential games can be applied.
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Bramoulle’s Model
I Bramoulle studies the properties of frustration function and itsconnection with the welfare of the social game.
I Specific topic of our concern is the characterisation ofbipartite graphs.
TheoremA graph is bipartite if and only if there exists s, πA, πB such thatφ(s, πA, πB , g) = 0.
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Bramoulle’s Model
I Bramoulle studies the properties of frustration function and itsconnection with the welfare of the social game.
I Specific topic of our concern is the characterisation ofbipartite graphs.
TheoremA graph is bipartite if and only if there exists s, πA, πB such thatφ(s, πA, πB , g) = 0.
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![Page 38: Anti-Coordination Games and Graph Colouringmath.iisc.ernet.in/~nmi/2_KS_Mallikarjuna_Rao-talk_01.pdf · K.S. Mallikarjuna Rao (Joint work with Arko Chatterjee) Industrial Engineering](https://reader033.vdocument.in/reader033/viewer/2022051901/5ff06e8729ea7c781603ac86/html5/thumbnails/38.jpg)
Bramoulle’s Model
I Bramoulle studies the properties of frustration function and itsconnection with the welfare of the social game.
I Specific topic of our concern is the characterisation ofbipartite graphs.
TheoremA graph is bipartite if and only if there exists s, πA, πB such thatφ(s, πA, πB , g) = 0.
12 / 33
![Page 39: Anti-Coordination Games and Graph Colouringmath.iisc.ernet.in/~nmi/2_KS_Mallikarjuna_Rao-talk_01.pdf · K.S. Mallikarjuna Rao (Joint work with Arko Chatterjee) Industrial Engineering](https://reader033.vdocument.in/reader033/viewer/2022051901/5ff06e8729ea7c781603ac86/html5/thumbnails/39.jpg)
Graph Coloring: Game Theoretic View
I Consider the graph and assume that each node is a player.
I Each player interacts with each neighbors randomly.
I Players goal is to chose a colour which is different from hisopponent in these random interaction.
I The utility to the player is the expected payoff he receives inthese random interactions.
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Graph Coloring: Game Theoretic View
I Consider the graph and assume that each node is a player.
I Each player interacts with each neighbors randomly.
I Players goal is to chose a colour which is different from hisopponent in these random interaction.
I The utility to the player is the expected payoff he receives inthese random interactions.
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![Page 41: Anti-Coordination Games and Graph Colouringmath.iisc.ernet.in/~nmi/2_KS_Mallikarjuna_Rao-talk_01.pdf · K.S. Mallikarjuna Rao (Joint work with Arko Chatterjee) Industrial Engineering](https://reader033.vdocument.in/reader033/viewer/2022051901/5ff06e8729ea7c781603ac86/html5/thumbnails/41.jpg)
Graph Coloring: Game Theoretic View
I Consider the graph and assume that each node is a player.
I Each player interacts with each neighbors randomly.
I Players goal is to chose a colour which is different from hisopponent in these random interaction.
I The utility to the player is the expected payoff he receives inthese random interactions.
13 / 33
![Page 42: Anti-Coordination Games and Graph Colouringmath.iisc.ernet.in/~nmi/2_KS_Mallikarjuna_Rao-talk_01.pdf · K.S. Mallikarjuna Rao (Joint work with Arko Chatterjee) Industrial Engineering](https://reader033.vdocument.in/reader033/viewer/2022051901/5ff06e8729ea7c781603ac86/html5/thumbnails/42.jpg)
Graph Coloring: Game Theoretic View
I Consider the graph and assume that each node is a player.
I Each player interacts with each neighbors randomly.
I Players goal is to chose a colour which is different from hisopponent in these random interaction.
I The utility to the player is the expected payoff he receives inthese random interactions.
13 / 33
![Page 43: Anti-Coordination Games and Graph Colouringmath.iisc.ernet.in/~nmi/2_KS_Mallikarjuna_Rao-talk_01.pdf · K.S. Mallikarjuna Rao (Joint work with Arko Chatterjee) Industrial Engineering](https://reader033.vdocument.in/reader033/viewer/2022051901/5ff06e8729ea7c781603ac86/html5/thumbnails/43.jpg)
Graph Coloring: Game Theoretic View
I The the utility of player i is given by
πi (si , s−i ) =n∑
j=1
gijπ(si , sj).
I Here π(si , sj) = 1si 6=sj
I A profile s is a Nash equilibrium if it satisfies
∀i , ∀s ′i , πi (si , s−i ) ≥ πi (s ′i , s−i ).
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Graph Coloring: Game Theoretic View
I The the utility of player i is given by
πi (si , s−i ) =n∑
j=1
gijπ(si , sj).
I Here π(si , sj) = 1si 6=sj
I A profile s is a Nash equilibrium if it satisfies
∀i , ∀s ′i , πi (si , s−i ) ≥ πi (s ′i , s−i ).
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Graph Coloring: Game Theoretic View
I The the utility of player i is given by
πi (si , s−i ) =n∑
j=1
gijπ(si , sj).
I Here π(si , sj) = 1si 6=sj
I A profile s is a Nash equilibrium if it satisfies
∀i , ∀s ′i , πi (si , s−i ) ≥ πi (s ′i , s−i ).
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Problem with Bramoulle’s Model
Consider the network with 8 agents and the two configurations.
1 73
2
4
8
6
5
1 73
2
4
8
6
5
Both the configurations are Nash equilibrium. Note that the graphis bipartite (see the second configuration). However, the firstconfigurations is not a proper colouring. Thus the Bramoulle’smodel does not capture the anti-coordination in a stict sense.
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Problem with Bramoulle’s Model
Consider the network with 8 agents and the two configurations.
1 73
2
4
8
6
5
1 73
2
4
8
6
5
Both the configurations are Nash equilibrium. Note that the graphis bipartite (see the second configuration). However, the firstconfigurations is not a proper colouring. Thus the Bramoulle’smodel does not capture the anti-coordination in a stict sense.
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Game Theoretic View: Recent Studies
I Kearns, Suri and Montfort (2006) studied experimentally froma behavioural point of view.
I Several theoretical results followed after this work.
I The results assume that the number of colours available aretwo more than chromatic number.
I Mainly these works analyse the greedy algorithm. Each time,an agent picks a colour not used by the neighbours.
I It is proved that this greedy algorithm convergences to aproper colouring. The probability of convergence is not 1.
I The model is essentially same as the model by by Bramoulle.Also, Bramoulle’s model assumes only two choices for theagents.
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Game Theoretic View: Recent Studies
I Kearns, Suri and Montfort (2006) studied experimentally froma behavioural point of view.
I Several theoretical results followed after this work.
I The results assume that the number of colours available aretwo more than chromatic number.
I Mainly these works analyse the greedy algorithm. Each time,an agent picks a colour not used by the neighbours.
I It is proved that this greedy algorithm convergences to aproper colouring. The probability of convergence is not 1.
I The model is essentially same as the model by by Bramoulle.Also, Bramoulle’s model assumes only two choices for theagents.
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Game Theoretic View: Recent Studies
I Kearns, Suri and Montfort (2006) studied experimentally froma behavioural point of view.
I Several theoretical results followed after this work.
I The results assume that the number of colours available aretwo more than chromatic number.
I Mainly these works analyse the greedy algorithm. Each time,an agent picks a colour not used by the neighbours.
I It is proved that this greedy algorithm convergences to aproper colouring. The probability of convergence is not 1.
I The model is essentially same as the model by by Bramoulle.Also, Bramoulle’s model assumes only two choices for theagents.
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Game Theoretic View: Recent Studies
I Kearns, Suri and Montfort (2006) studied experimentally froma behavioural point of view.
I Several theoretical results followed after this work.
I The results assume that the number of colours available aretwo more than chromatic number.
I Mainly these works analyse the greedy algorithm. Each time,an agent picks a colour not used by the neighbours.
I It is proved that this greedy algorithm convergences to aproper colouring. The probability of convergence is not 1.
I The model is essentially same as the model by by Bramoulle.Also, Bramoulle’s model assumes only two choices for theagents.
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Game Theoretic View: Recent Studies
I Kearns, Suri and Montfort (2006) studied experimentally froma behavioural point of view.
I Several theoretical results followed after this work.
I The results assume that the number of colours available aretwo more than chromatic number.
I Mainly these works analyse the greedy algorithm. Each time,an agent picks a colour not used by the neighbours.
I It is proved that this greedy algorithm convergences to aproper colouring. The probability of convergence is not 1.
I The model is essentially same as the model by by Bramoulle.Also, Bramoulle’s model assumes only two choices for theagents.
16 / 33
![Page 53: Anti-Coordination Games and Graph Colouringmath.iisc.ernet.in/~nmi/2_KS_Mallikarjuna_Rao-talk_01.pdf · K.S. Mallikarjuna Rao (Joint work with Arko Chatterjee) Industrial Engineering](https://reader033.vdocument.in/reader033/viewer/2022051901/5ff06e8729ea7c781603ac86/html5/thumbnails/53.jpg)
Game Theoretic View: Recent Studies
I Kearns, Suri and Montfort (2006) studied experimentally froma behavioural point of view.
I Several theoretical results followed after this work.
I The results assume that the number of colours available aretwo more than chromatic number.
I Mainly these works analyse the greedy algorithm. Each time,an agent picks a colour not used by the neighbours.
I It is proved that this greedy algorithm convergences to aproper colouring. The probability of convergence is not 1.
I The model is essentially same as the model by by Bramoulle.Also, Bramoulle’s model assumes only two choices for theagents.
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Our Model
I The utility function is given by
πi (s) = −∑j∈Ni
1si=sj︸ ︷︷ ︸Term1
+1
Ki
∑k,j∈Ni
1sk=sj︸ ︷︷ ︸Term2
, (1)
where
Ki = 2
(|Ni |
2
)
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Our Model
I The first term in the payoff counts the number of neighbourshaving the same colour and hence represents the penalty forchoosing a colour that is same as the colour of a node in theneighbourhood.
I The second term counts the number of neighbours havingsame colour and thus represents the benefit of havingminimum number of colours in the neighbourhood.
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Our Model
I The first term in the payoff counts the number of neighbourshaving the same colour and hence represents the penalty forchoosing a colour that is same as the colour of a node in theneighbourhood.
I The second term counts the number of neighbours havingsame colour and thus represents the benefit of havingminimum number of colours in the neighbourhood.
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Our Model and Results
I Term 2 is independent of the colour of player i , and henceunilateral deviation by player i will not effect this term. Whenconsidering unilateral deviations, only Term 1 matters.
I Term 1 represents the number of neighbours having the samecolour as the player i with a negative sign.
I Thus Term 1 will be higher if no neighbour of player i hassame colour as the player i .
I In other words, proper colouring will always be a Nashequilibrium.
I In fact, we have the following result: A pure strategy is aNash equilibrium if and only if it is proper colouring.
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Our Model and Results
I Term 2 is independent of the colour of player i , and henceunilateral deviation by player i will not effect this term. Whenconsidering unilateral deviations, only Term 1 matters.
I Term 1 represents the number of neighbours having the samecolour as the player i with a negative sign.
I Thus Term 1 will be higher if no neighbour of player i hassame colour as the player i .
I In other words, proper colouring will always be a Nashequilibrium.
I In fact, we have the following result: A pure strategy is aNash equilibrium if and only if it is proper colouring.
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Our Model and Results
I Term 2 is independent of the colour of player i , and henceunilateral deviation by player i will not effect this term. Whenconsidering unilateral deviations, only Term 1 matters.
I Term 1 represents the number of neighbours having the samecolour as the player i with a negative sign.
I Thus Term 1 will be higher if no neighbour of player i hassame colour as the player i .
I In other words, proper colouring will always be a Nashequilibrium.
I In fact, we have the following result: A pure strategy is aNash equilibrium if and only if it is proper colouring.
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Our Model and Results
I Term 2 is independent of the colour of player i , and henceunilateral deviation by player i will not effect this term. Whenconsidering unilateral deviations, only Term 1 matters.
I Term 1 represents the number of neighbours having the samecolour as the player i with a negative sign.
I Thus Term 1 will be higher if no neighbour of player i hassame colour as the player i .
I In other words, proper colouring will always be a Nashequilibrium.
I In fact, we have the following result: A pure strategy is aNash equilibrium if and only if it is proper colouring.
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Our Model and Results
I Term 2 is independent of the colour of player i , and henceunilateral deviation by player i will not effect this term. Whenconsidering unilateral deviations, only Term 1 matters.
I Term 1 represents the number of neighbours having the samecolour as the player i with a negative sign.
I Thus Term 1 will be higher if no neighbour of player i hassame colour as the player i .
I In other words, proper colouring will always be a Nashequilibrium.
I In fact, we have the following result: A pure strategy is aNash equilibrium if and only if it is proper colouring.
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Our Model and Results
I We can also prove: A Pareto equilibrium is always a minimalcolouring.
I The converse is not true.
1
2
3
4
5
1
2
3
4
5
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Our Model and Results
I We can also prove: A Pareto equilibrium is always a minimalcolouring.
I The converse is not true.
1
2
3
4
5
1
2
3
4
5
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Our Model and Results
I We can also prove: A Pareto equilibrium is always a minimalcolouring.
I The converse is not true.
1
2
3
4
5
1
2
3
4
5
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Our Model and Results
I We can also prove: A Pareto equilibrium is always a minimalcolouring.
I The converse is not true.
1
2
3
4
5
1
2
3
4
5
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Our Model and Results
I Given a proper colouring, for a player i and herneighbourhood N(i), define the neighbourhood conflictcount (NCC) of player i as the number of pairs of agentsbelonging to N(i) that have different colours.
I Each such pair of agents in the neighbourhood of i whosecolours are different, is termed as a neighbourhood conflictof player i .
I Pareto equilibria correspond to minal “neighborhoodconflicting” profiles.
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Our Model and Results
I Given a proper colouring, for a player i and herneighbourhood N(i), define the neighbourhood conflictcount (NCC) of player i as the number of pairs of agentsbelonging to N(i) that have different colours.
I Each such pair of agents in the neighbourhood of i whosecolours are different, is termed as a neighbourhood conflictof player i .
I Pareto equilibria correspond to minal “neighborhoodconflicting” profiles.
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Our Model and Results
I Given a proper colouring, for a player i and herneighbourhood N(i), define the neighbourhood conflictcount (NCC) of player i as the number of pairs of agentsbelonging to N(i) that have different colours.
I Each such pair of agents in the neighbourhood of i whosecolours are different, is termed as a neighbourhood conflictof player i .
I Pareto equilibria correspond to minal “neighborhoodconflicting” profiles.
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Our Model and Results
I Pareto equilibria need not be unique.
1
2
3 4
56
7
8 9
10
1
2
3 4
56
7
8 9
10
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Question
How do we obtain minimal colouring?
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Difficulties
I The game has too many Nash equilibrium.
I The game is not Potential.
I Note that our game is not a local interaction game (in thesense of Blume). It should be understood as a game withnetworked agents.
I To get the minimal colouring, we consider a modification ofthe game.
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Difficulties
I The game has too many Nash equilibrium.
I The game is not Potential.
I Note that our game is not a local interaction game (in thesense of Blume). It should be understood as a game withnetworked agents.
I To get the minimal colouring, we consider a modification ofthe game.
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Difficulties
I The game has too many Nash equilibrium.
I The game is not Potential.
I Note that our game is not a local interaction game (in thesense of Blume). It should be understood as a game withnetworked agents.
I To get the minimal colouring, we consider a modification ofthe game.
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Difficulties
I The game has too many Nash equilibrium.
I The game is not Potential.
I Note that our game is not a local interaction game (in thesense of Blume). It should be understood as a game withnetworked agents.
I To get the minimal colouring, we consider a modification ofthe game.
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Modified Game
I The payoff function is defined by
v i (a) = ui (a) +1
|N(i)|∑
j∈N(i)
uj(a).
I This requires a 2-hop neighbourhood information.
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Modified Game
I The payoff function is defined by
v i (a) = ui (a) +1
|N(i)|∑
j∈N(i)
uj(a).
I This requires a 2-hop neighbourhood information.
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Modified Game: Main Result
TheoremEvery Nash equilibrium is a Pareto and hence it is minimal.
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Best Response Dynamics
I The modified game can be analysed using best responsedynamics.
I Any improvement by a player gives a strict increment in thepayoff.
I This increment is lower bounded by a positive constant.
I The payoffs of the game are bounded.
I Hence the best response dynamics gives minimal colouring.
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Best Response Dynamics
I The modified game can be analysed using best responsedynamics.
I Any improvement by a player gives a strict increment in thepayoff.
I This increment is lower bounded by a positive constant.
I The payoffs of the game are bounded.
I Hence the best response dynamics gives minimal colouring.
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Best Response Dynamics
I The modified game can be analysed using best responsedynamics.
I Any improvement by a player gives a strict increment in thepayoff.
I This increment is lower bounded by a positive constant.
I The payoffs of the game are bounded.
I Hence the best response dynamics gives minimal colouring.
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Best Response Dynamics
I The modified game can be analysed using best responsedynamics.
I Any improvement by a player gives a strict increment in thepayoff.
I This increment is lower bounded by a positive constant.
I The payoffs of the game are bounded.
I Hence the best response dynamics gives minimal colouring.
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Best Response Dynamics
I The modified game can be analysed using best responsedynamics.
I Any improvement by a player gives a strict increment in thepayoff.
I This increment is lower bounded by a positive constant.
I The payoffs of the game are bounded.
I Hence the best response dynamics gives minimal colouring.
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A Simple Learning Scheme
I Consider a repeated interaction.
I At each round of interaction, pick an agen i uniformly.
I The agent i will pick a neighbour j uniformly.
I The agent i will ask j about his neighbours’ colours.
I He picks the colour which is picked by most of j ’s neighbours.
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A Simple Learning Scheme
I Consider a repeated interaction.
I At each round of interaction, pick an agen i uniformly.
I The agent i will pick a neighbour j uniformly.
I The agent i will ask j about his neighbours’ colours.
I He picks the colour which is picked by most of j ’s neighbours.
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A Simple Learning Scheme
I Consider a repeated interaction.
I At each round of interaction, pick an agen i uniformly.
I The agent i will pick a neighbour j uniformly.
I The agent i will ask j about his neighbours’ colours.
I He picks the colour which is picked by most of j ’s neighbours.
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A Simple Learning Scheme
I Consider a repeated interaction.
I At each round of interaction, pick an agen i uniformly.
I The agent i will pick a neighbour j uniformly.
I The agent i will ask j about his neighbours’ colours.
I He picks the colour which is picked by most of j ’s neighbours.
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A Simple Learning Scheme
I Consider a repeated interaction.
I At each round of interaction, pick an agen i uniformly.
I The agent i will pick a neighbour j uniformly.
I The agent i will ask j about his neighbours’ colours.
I He picks the colour which is picked by most of j ’s neighbours.
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A Simple Learning Scheme: Convergence
I There is an irreducible Markov chain in the back ground.
I Each best response improvement iterate will happen.
I So, the algorithm converges.
I In fact, the algorithm will reach the steady state in finite timewith probability 1.
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A Simple Learning Scheme: Convergence
I There is an irreducible Markov chain in the back ground.
I Each best response improvement iterate will happen.
I So, the algorithm converges.
I In fact, the algorithm will reach the steady state in finite timewith probability 1.
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A Simple Learning Scheme: Convergence
I There is an irreducible Markov chain in the back ground.
I Each best response improvement iterate will happen.
I So, the algorithm converges.
I In fact, the algorithm will reach the steady state in finite timewith probability 1.
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A Simple Learning Scheme: Convergence
I There is an irreducible Markov chain in the back ground.
I Each best response improvement iterate will happen.
I So, the algorithm converges.
I In fact, the algorithm will reach the steady state in finite timewith probability 1.
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Some Remarks
I Our model can be studied for any number of colours (can beless than the chromatic number), in which case it captures themodel of Bramoulle.
I The learning scheme works irrespective of the number ofcolours.
I We can handle general anti-coordination games.
I There is no clear definition for anti-coordination games withmany players. Graph colouring is one way of defininganti-coordination game.
I The idea of the modified game can help in studying sociallyoptimal equilibrium in general games.
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Some Remarks
I Our model can be studied for any number of colours (can beless than the chromatic number), in which case it captures themodel of Bramoulle.
I The learning scheme works irrespective of the number ofcolours.
I We can handle general anti-coordination games.
I There is no clear definition for anti-coordination games withmany players. Graph colouring is one way of defininganti-coordination game.
I The idea of the modified game can help in studying sociallyoptimal equilibrium in general games.
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Some Remarks
I Our model can be studied for any number of colours (can beless than the chromatic number), in which case it captures themodel of Bramoulle.
I The learning scheme works irrespective of the number ofcolours.
I We can handle general anti-coordination games.
I There is no clear definition for anti-coordination games withmany players. Graph colouring is one way of defininganti-coordination game.
I The idea of the modified game can help in studying sociallyoptimal equilibrium in general games.
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Some Remarks
I Our model can be studied for any number of colours (can beless than the chromatic number), in which case it captures themodel of Bramoulle.
I The learning scheme works irrespective of the number ofcolours.
I We can handle general anti-coordination games.
I There is no clear definition for anti-coordination games withmany players. Graph colouring is one way of defininganti-coordination game.
I The idea of the modified game can help in studying sociallyoptimal equilibrium in general games.
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Some Remarks
I Our model can be studied for any number of colours (can beless than the chromatic number), in which case it captures themodel of Bramoulle.
I The learning scheme works irrespective of the number ofcolours.
I We can handle general anti-coordination games.
I There is no clear definition for anti-coordination games withmany players. Graph colouring is one way of defininganti-coordination game.
I The idea of the modified game can help in studying sociallyoptimal equilibrium in general games.
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Some References
I Blume, The statistical mechanics of strategic interaction,Games and Economic Behavior, 1993.
I Bramoulle, Anti-coordination and social interactions, Gamesand Economic Behavior, 2007.
I Kearns, Suri and Montfort, An experimental study of thecolouring problem on human subject networks, Science, 2006.
I Young, Individual Strategy and Social Structure, PrincetonUniversity Press, 1998.
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Some References
I Blume, The statistical mechanics of strategic interaction,Games and Economic Behavior, 1993.
I Bramoulle, Anti-coordination and social interactions, Gamesand Economic Behavior, 2007.
I Kearns, Suri and Montfort, An experimental study of thecolouring problem on human subject networks, Science, 2006.
I Young, Individual Strategy and Social Structure, PrincetonUniversity Press, 1998.
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Some References
I Blume, The statistical mechanics of strategic interaction,Games and Economic Behavior, 1993.
I Bramoulle, Anti-coordination and social interactions, Gamesand Economic Behavior, 2007.
I Kearns, Suri and Montfort, An experimental study of thecolouring problem on human subject networks, Science, 2006.
I Young, Individual Strategy and Social Structure, PrincetonUniversity Press, 1998.
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Some References
I Blume, The statistical mechanics of strategic interaction,Games and Economic Behavior, 1993.
I Bramoulle, Anti-coordination and social interactions, Gamesand Economic Behavior, 2007.
I Kearns, Suri and Montfort, An experimental study of thecolouring problem on human subject networks, Science, 2006.
I Young, Individual Strategy and Social Structure, PrincetonUniversity Press, 1998.
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Questions, Comments?
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Thank You
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