Coordination Games on Graphs Krzysztof Apt CWI & University of - - PowerPoint PPT Presentation

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 Si ⊆ M of colours available to i → each player selects a colour si ∈ Si 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 Si ⊆ M of colours available to i → each player selects a colour si ∈ Si from his colour set the payoff of player i under joint strategy s = (s1, . . . , sn) is the number of neighbours who chose the same colour: pi(s) = {j ∈ N(i) : sj = si}, where N(i) is the set of neighbours of i in G

Krzysztof Apt Coordination Games on Graphs

Example

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Krzysztof Apt Coordination Games on Graphs

Example

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Krzysztof Apt Coordination Games on Graphs

Example

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Krzysztof Apt Coordination Games on Graphs

Example

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Krzysztof Apt Coordination Games on Graphs

Motivation

Characteristics:

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join the crowd property: the payoff of each player weakly increases when more players choose his strategy

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asymmetric strategies: players may have individual strategy sets

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local dependency: the payoff of each player only depends on the choices made by his neighbors

Krzysztof Apt Coordination Games on Graphs

Motivation

Characteristics:

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join the crowd property: the payoff of each player weakly increases when more players choose his strategy

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asymmetric strategies: players may have individual strategy sets

3

local dependency: the payoff of each player only depends on the 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 pi : ×j∈N(i)∪{i}Sj → 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 pi : ×j∈N(i)∪{i}Sj → 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 pij such that pi(s) :=

• j=i

pij(si, sj) → 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: pi(s′) > pi(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: pi(s′) > pi(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: pi(s′) > pi(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: pi(s′) > pi(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

2SW(s) is an exact potential.

Krzysztof Apt Coordination Games on Graphs

Coalitional Improvement Path

A c(oalitional)-improvement path is a maximal sequence s1 → s2 → s3 → . . .

• f joint strategies such that for every k > 1 there is a coalition K such

that sk is a profitable deviation of the players in K from sk−1.

Krzysztof Apt Coordination Games on Graphs

Coalitional Improvement Path

A c(oalitional)-improvement path is a maximal sequence s1 → s2 → s3 → . . .

• f joint strategies such that for every k > 1 there is a coalition K such

that sk is a profitable deviation of the players in K from sk−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 s1 → s2 → s3 → . . .

• f joint strategies such that for every k > 1 there is a coalition K such

that sk is a profitable deviation of the players in K from sk−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 : S1 × · · · × Sn → 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 : S1 × · · · × Sn → 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

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SW(s) = 24

Krzysztof Apt Coordination Games on Graphs

Decreasing Social Welfare

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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 si = sj

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 si = sj

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

• f G[K]. Then

SW(s′) − SW(s) > 2|F ∩ E+

s | − 2|F ∩ E+ s′ |.

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 si = sj

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

• f G[K]. Then

SW(s′) − SW(s) > 2|F ∩ E+

s | − 2|F ∩ E+ s′ |.

Note: previous example shows that this bound is tight.

Krzysztof Apt Coordination Games on Graphs

Consequences

Suppose s K →s′ is a profitable deviation.

Krzysztof Apt Coordination Games on Graphs

Consequences

Suppose s K →s′ is a profitable deviation.

Corollary

If G[K] is a forest then SW(s′) > SW(s).

Krzysztof Apt Coordination Games on Graphs

Consequences

Suppose s K →s′ is a profitable deviation.

Corollary

If G[K] is a forest then SW(s′) > SW(s). → Coordination games on forests have the c-FIP .

Krzysztof Apt Coordination Games on Graphs

Consequences

Suppose s K →s′ is a profitable deviation.

Corollary

If G[K] is a forest then SW(s′) > SW(s). → Coordination games on forests have the c-FIP .

Corollary

If SW(s′) − SW(s) ≤ 0 then there is a cycle C in G[K] that is completely non-unicolor in s and unicolor in s′.

Krzysztof Apt Coordination Games on Graphs

Consequences

Suppose s K →s′ is a profitable deviation.

Corollary

If G[K] is a forest then SW(s′) > SW(s). → Coordination games on forests have the c-FIP .

Corollary

If SW(s′) − SW(s) ≤ 0 then there is a cycle C in G[K] that is completely non-unicolor in s and unicolor in s′. → Coordination games on colour forests have the c-FIP .

Krzysztof Apt Coordination Games on Graphs

Strong Equilibria for Pseudoforests

A pseudoforest is a graph in which each connected component contains at most one cycle.

Krzysztof Apt Coordination Games on Graphs

Strong Equilibria for Pseudoforests

A pseudoforest is a graph in which each connected component contains at most one cycle.

Theorem

Consider a coordination game on a graph that is a pseudoforest. Then the game has the c-FIP .

Krzysztof Apt Coordination Games on Graphs

Strong Equilibria for Pseudoforests

A pseudoforest is a graph in which each connected component contains at most one cycle.

Theorem

Consider a coordination game on a graph that is a pseudoforest. Then the game has the c-FIP . Proof idea: P(s) := (SW(s), |{C : C is a unicolor cycle in G under s}|) is a generalized ordinal c-potential when we take the lexicographic

• rdering.

Krzysztof Apt Coordination Games on Graphs

Other Positive Results

Consider the subgraph Gx of G induced by all nodes that can choose colour x.

Krzysztof Apt Coordination Games on Graphs

Other Positive Results

Consider the subgraph Gx of G induced by all nodes that can choose colour x. Call G colour complete if Gx is complete for every colour x ∈ M.

Krzysztof Apt Coordination Games on Graphs

Other Positive Results

Consider the subgraph Gx of G induced by all nodes that can choose colour x. Call G colour complete if Gx is complete for every colour x ∈ M.

Theorem

Every coordination game whose underlying graph is colour complete has the c-FIP .

Krzysztof Apt Coordination Games on Graphs

Other Positive Results

Consider the subgraph Gx of G induced by all nodes that can choose colour x. Call G colour complete if Gx is complete for every colour x ∈ M.

Theorem

Every coordination game whose underlying graph is colour complete has the c-FIP . Proof idea: Let P(s) := (p1(s), . . . , pn(s))∗ be the vector of payoffs, ordered from largest to smallest.

Krzysztof Apt Coordination Games on Graphs

Other Positive Results

Consider the subgraph Gx of G induced by all nodes that can choose colour x. Call G colour complete if Gx is complete for every colour x ∈ M.

Theorem

Every coordination game whose underlying graph is colour complete has the c-FIP . Proof idea: Let P(s) := (p1(s), . . . , pn(s))∗ be the vector of payoffs, ordered from largest to smallest. Then P is a generalized ordinal c-potential when we take the lexicographic ordering.

Krzysztof Apt Coordination Games on Graphs

Non-Existence of Strong Equilibria

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Non-Existence of Strong Equilibria

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Non-Existence of Strong Equilibria

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Non-Existence of Strong Equilibria

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Non-Existence of Strong Equilibria

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Non-Existence of Strong Equilibria

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Krzysztof Apt Coordination Games on Graphs

Concluding Remarks

Omitted Results on (strong) price of anarchy and price of stability. Computational complexity of equilibria computation.

Krzysztof Apt Coordination Games on Graphs

Concluding Remarks

Omitted Results on (strong) price of anarchy and price of stability. Computational complexity of equilibria computation. Extensions and Open Problems

• pen: characterize which graphs admit strong equilibria
• pen: existence of k-equilibria for k = 3, 4

super-strong equilibria do not exist, even for paths consider weighted version: w{i,j} for each edge {i, j} ∈ E → Nash equilibria always exist. → 2-equilibria do not need to exist.

• pen: consider more general strategy sets Si ⊆ 2M.

Krzysztof Apt Coordination Games on Graphs

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