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:

1

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

2

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

Results in a Nutshell

1

Do pure Nash or strong equilibria always exist?

2

What about the inefficiency of equilibria?

3

Can we compute such equilibria efficiently?

Krzysztof Apt Coordination Games on Graphs

Results in a Nutshell

1

Do pure Nash or strong equilibria always exist?

◮ pure Nash equilibria: yes ◮ strong equilibria: depends on the structure of the graph 2

What about the inefficiency of equilibria?

3

Can we compute such equilibria efficiently?

Krzysztof Apt Coordination Games on Graphs

Results in a Nutshell

1

Do pure Nash or strong equilibria always exist?

◮ pure Nash equilibria: yes ◮ strong equilibria: depends on the structure of the graph 2

What about the inefficiency of equilibria?

◮ price of anarchy ranges from ∞ for PNE to 2 for SE ◮ price of stability is 1 for many graphs 3

Can we compute such equilibria efficiently?

Krzysztof Apt Coordination Games on Graphs

Results in a Nutshell

1

Do pure Nash or strong equilibria always exist?

◮ pure Nash equilibria: yes ◮ strong equilibria: depends on the structure of the graph 2

What about the inefficiency of equilibria?

◮ price of anarchy ranges from ∞ for PNE to 2 for SE ◮ price of stability is 1 for many graphs 3

Can we compute such equilibria efficiently?

◮ 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 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

Inefficiency

Social welfare: SW(s) =

• i∈N

pi(s)

Krzysztof Apt Coordination Games on Graphs

Inefficiency

Social welfare: SW(s) =

• i∈N

pi(s) k-Price of Anarchy: maxs∈S SW(s) mins∈S, s is a k-SE SW(s)

Krzysztof Apt Coordination Games on Graphs

Inefficiency

Social welfare: SW(s) =

• i∈N

pi(s) k-Price of Anarchy: maxs∈S SW(s) mins∈S, s is a k-SE SW(s) k-Price of Stability: maxs∈S SW(s) maxs∈S, s is a k-SE SW(s)

Krzysztof Apt Coordination Games on Graphs

Exact Potentials

Assume G := (S1, . . . , Sn, p1, . . . , pn). A profitable deviation: a pair (s, s′) of joint strategies such that pi(s′) > pi(s), where s′ = (s′

i, s−i).

An exact potential for G: a function P : S1 × · · · × Sn → R such that for every profitable deviation (s, s′), where s′ = (s′

i, s−i),

P(s′) − P(s) = pi(s′) − pi(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

• f 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

• f anarchy of the coordination game is ∞.

Proof idea: (i) and (ii): P(s) := 1

2SW(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

• f anarchy of the coordination game is ∞.

Proof idea: (i) and (ii): P(s) := 1

2SW(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

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

• f anarchy of the coordination game is ∞.

Proof idea: (i) and (ii): P(s) := 1

2SW(s) is an exact potential.

(iii): Assign to each node in the graph (N, E) two colours: one private and one common. The maximal social welfare is 2|E|.

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

• f anarchy of the coordination game is ∞.

Proof idea: (i) and (ii): P(s) := 1

2SW(s) is an exact potential.

(iii): Assign to each node in the graph (N, E) two colours: one private and one common. The maximal social welfare is 2|E|. Each node choosing his private colour is a Nash equilibrium with social welfare 0.

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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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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Krzysztof Apt Coordination Games on Graphs

Strong Price of Anarchy

Theorem

The k-price of anarchy is between 2 n−1

k−1 − 1 and 2 n−1 k−1. The strong

price of anarchy is 2.

Krzysztof Apt Coordination Games on Graphs

Strong Price of Anarchy

Theorem

The k-price of anarchy is between 2 n−1

k−1 − 1 and 2 n−1 k−1. The strong

price of anarchy is 2.

Proof idea: Assume that the game has a k-equilibrium s. Let σ be a social optimum. Choose a coalition K of size k. Step 1: Show that SWK (σ) ≤ 2SWK (s) + |E+

σ ∩ δ(K)|.

Step 2: Summing over all K of size k one gets n − 1 k − 1

• SW(σ) ≤ 2

n − 1 k − 1

• SW(s) +

n − 2 k − 1

• SW(σ).

Step 3: This implies that the k-price of anarchy is at most 2 n−1

k−1.

Krzysztof Apt Coordination Games on Graphs

Complexity Results

Theorem

Consider a coordination game on a pseudoforest. Then a strong equilibrium can be computed in polynomial time.

Krzysztof Apt Coordination Games on Graphs

Complexity Results

Theorem

Consider a coordination game on a pseudoforest. Then a strong equilibrium can be computed in polynomial time. Proof idea: It is sufficient to compute a social optimum that maximizes the number of unicolor cycles. → dynamic programming + additional tricks

Krzysztof Apt Coordination Games on Graphs

Complexity Results

Theorem

Consider a coordination game on a pseudoforest. Then a strong equilibrium can be computed in polynomial time. Proof idea: It is sufficient to compute a social optimum that maximizes the number of unicolor cycles. → dynamic programming + additional tricks

Theorem

The problem of deciding whether a given joint strategy s is a k-equilibrium is co-NP-complete.

Krzysztof Apt Coordination Games on Graphs

Complexity Results

Theorem

Consider a coordination game on a pseudoforest. Then a strong equilibrium can be computed in polynomial time. Proof idea: It is sufficient to compute a social optimum that maximizes the number of unicolor cycles. → dynamic programming + additional tricks

Theorem

The problem of deciding whether a given joint strategy s is a k-equilibrium is co-NP-complete. Proof idea: reduction from CLIQUE.

Krzysztof Apt Coordination Games on Graphs

Concluding Remarks

Summary: coordination games on graphs are a natural class of games derived a partial characterization of structural properties of graphs that ensure existence of strong equilibria k-POA (almost) settled; k-POS = 1 for many graphs studied computational complexity of equilibria computation

Krzysztof Apt Coordination Games on Graphs

Concluding Remarks

Summary: coordination games on graphs are a natural class of games derived a partial characterization of structural properties of graphs that ensure existence of strong equilibria k-POA (almost) settled; k-POS = 1 for many graphs studied 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 → k-POA bound remains valid → 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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