social network games - centrum wiskunde & informaticaapt/stra13/social_networks13-sli.pdf ·...
TRANSCRIPT
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Social Network Games
Krzysztof R. Apt
CWI & and University of Amsterdam
Joint work with Sunil Simon
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Social Networks
Facebook,
Hyves,
LinkedIn,
Nasza Klasa,
. . .
Krzysztof R. Apt Social Network Games
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But also . . .
An area with links to
sociology (spread of patterns of social behaviour)
economics (effects of advertising, emergence of ‘bubbles’ in financialmarkets, . . .),
epidemiology (epidemics),
computer science (complexity analysis),
mathematics (graph theory).
Krzysztof R. Apt Social Network Games
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Example 1
(From D. Easley and J. Kleinberg, 2010).
Spread of the tuberculosis outbreak.
Krzysztof R. Apt Social Network Games
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Example 2
(From D. Easley and J. Kleinberg, 2010).
Pattern of e-mail communication among 436 employees of HP ResearchLab.
Krzysztof R. Apt Social Network Games
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Example 3
(From D. Easley and J. Kleinberg, 2010).
Collaboration of mathematicians centered on Paul Erdos.Drawing by Ron Graham.
Krzysztof R. Apt Social Network Games
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Social networks
Essential components of our model
Finite set of agents.
Influence of “friends”.
Finite product set for each agent.
Resistance level in (threshold for) adopting a product.
Krzysztof R. Apt Social Network Games
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Social networks
Essential components of our model
Finite set of agents.
Influence of “friends”.
Finite product set for each agent.
Resistance level in (threshold for) adopting a product.
4
1
3 2
0.4
0.5
0.3
0.6
Krzysztof R. Apt Social Network Games
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Social networks
Essential components of our model
Finite set of agents.
Influence of “friends”.
Finite product set for each agent.
Resistance level in (threshold for) adopting a product.
4{•}
1{•, •}
3
{•, •}
2
{•, •}
0.4
0.5
0.3
0.6
Krzysztof R. Apt Social Network Games
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Social networks
Essential components of our model
Finite set of agents.
Influence of “friends”.
Finite product set for each agent.
Resistance level in (threshold for) adopting a product.
4{•} 0.5
1 0.3{•, •}
3
{•, •}
0.2 2
{•, •}
0.4
0.4
0.5
0.3
0.6
Krzysztof R. Apt Social Network Games
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The model
Social network [Apt, Markakis 2011]
Weighted directed graph: G = (V ,→,w), whereV : a finite set of agents,wij ∈ (0, 1]: weight of the edge i → j .
Products: A finite set of products P.
Product assignment: P : V → 2P \ {∅};assigns to each agent a non-empty set of products.
Threshold function: θ(i , t) ∈ (0, 1], for each agent i and productt ∈ P(i).
Neighbours of node i : {j ∈ V | j → i}.
Source nodes: Agents with no neighbours.
Krzysztof R. Apt Social Network Games
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The associated strategic gameInteraction between agents: Each agent i can adopt a product from theset P(i) or choose not to adopt any product (t0).
Social network games
Players: Agents in the network.
Strategies: Set of strategies for player i is P(i) ∪ {t0}.
Payoff: Fix c > 0.Given a joint strategy s and an agent i ,
Krzysztof R. Apt Social Network Games
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The associated strategic gameInteraction between agents: Each agent i can adopt a product from theset P(i) or choose not to adopt any product (t0).
Social network games
Players: Agents in the network.
Strategies: Set of strategies for player i is P(i) ∪ {t0}.
Payoff: Fix c > 0.Given a joint strategy s and an agent i ,
◮ if i ∈ source(S), pi (s) =
{
0 if si = t0
c if si ∈ P(i)
Krzysztof R. Apt Social Network Games
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The associated strategic gameInteraction between agents: Each agent i can adopt a product from theset P(i) or choose not to adopt any product (t0).
Social network games
Players: Agents in the network.
Strategies: Set of strategies for player i is P(i) ∪ {t0}.
Payoff: Fix c > 0.Given a joint strategy s and an agent i ,
◮ if i ∈ source(S), pi (s) =
{
0 if si = t0
c if si ∈ P(i)
◮ if i 6∈ source(S), pi (s) =
0 if si = t0∑
j∈N ti(s)
wji − θ(i , t) if si = t, for some t ∈ P(i)
N ti (s): the set of neighbours of i who adopted in s the product t.
Krzysztof R. Apt Social Network Games
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Example
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
P = {•, •, •}
Krzysztof R. Apt Social Network Games
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Example
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
P = {•, •, •}
Payoff:
p4(s) = p5(s) = p6(s) = c
Krzysztof R. Apt Social Network Games
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Example
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
P = {•, •, •}
Payoff:
p4(s) = p5(s) = p6(s) = c
p1(s) = 0.4 − 0.3 = 0.1
Krzysztof R. Apt Social Network Games
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Example
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
P = {•, •, •}
Payoff:
p4(s) = p5(s) = p6(s) = c
p1(s) = 0.4 − 0.3 = 0.1
p2(s) = 0.5 − 0.3 = 0.2
p3(s) = 0.4 − 0.3 = 0.1
Krzysztof R. Apt Social Network Games
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Social network games
Properties
Graphical game: The payoff for each player depends only on thechoices made by his neighbours.
Join the crowd property: The payoff of each player weakly increases ifmore players choose the same strategy.
Krzysztof R. Apt Social Network Games
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Does Nash equilibrium always exist?
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
Krzysztof R. Apt Social Network Games
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Does Nash equilibrium always exist?
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
Observation: No player has theincentive to choose t0.
Source nodes can ensure apayoff of c > 0.
Each player on the cycle canensure a payoff of at least 0.1.
Krzysztof R. Apt Social Network Games
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Does Nash equilibrium always exist?
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
(•, •, •)
Observation: No player has theincentive to choose t0.
Source nodes can ensure apayoff of c > 0.
Each player on the cycle canensure a payoff of at least 0.1.
Krzysztof R. Apt Social Network Games
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Does Nash equilibrium always exist?
4
{•}
1 {•, •}
3{•, •} 2 {•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Threshold is 0.3 for all the players.
Best response dynamics
(•, •, •) (•, •, •) (•, •, •)
(•, •, •)(•, •, •)(•, •, •)
Observation: No player has theincentive to choose t0.
Source nodes can ensure apayoff of c > 0.
Each player on the cycle canensure a payoff of at least 0.1.
Reason: Players keep switchingbetween the products.
Krzysztof R. Apt Social Network Games
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Nash equilibrium
Recall the network with no Nash equilibrium:
4
{•}
1 {•, •}
3{•, •}
2
{•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Krzysztof R. Apt Social Network Games
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Nash equilibrium (ctd)
Theorem. If there exists X ⊆ P where |X | ≤ 2 such that for all sourcenodes i , P(i) ∩ X 6= ∅ then S has a Nash equilibrium.
Corollary. If there are at most two products, then a Nash equilibriumalways exists.
Krzysztof R. Apt Social Network Games
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Nash equilibrium
4
{•}
1 {•, •}
3{•, •}
2
{•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Properties of the underlying graph:
Krzysztof R. Apt Social Network Games
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Nash equilibrium
4
{•}
1 {•, •}
3{•, •}
2
{•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Properties of the underlying graph:
Contains a cycle.
Krzysztof R. Apt Social Network Games
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Nash equilibrium
4
{•}
1 {•, •}
3{•, •}
2
{•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Properties of the underlying graph:
Contains a cycle.
Contains source nodes.
Krzysztof R. Apt Social Network Games
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Nash equilibrium
4
{•}
1 {•, •}
3{•, •}
2
{•, •}
6
{•}
5
{•}
0.5
0.5
0.5
0.4
0.4 0.4
Properties of the underlying graph:
Contains a cycle.
Contains source nodes.
Question: Does Nash equilibrium always exist in social networks when theunderlying graph
is acyclic?
has no source nodes?
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
A Nash equilibrium s is non-trivial if there is at least one player i suchthat si 6= t0.
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. In a DAG, a non-trivial Nash equilibrium always exist.
Procedure to generate a non-trivial Nashequilibrium
Initialise: Assigns a product for each sourcenode
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. In a DAG, a non-trivial Nash equilibrium always exist.
Procedure to generate a non-trivial Nashequilibrium
Initialise: Assigns a product for each sourcenode
Repeat until all nodes are labelled:
Pick a node which is not labelled andfor which all neighbours are labelled
Assign the product which maximises thepayoff
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. In a DAG, a non-trivial Nash equilibrium always exist.
1
{•, •}
2
{•, •}
3{•, •}
4 {•, •}
5 {•, •, •}
0.4
0.50.3
0.10.1
Threshold = 0.3
Procedure to generate a non-trivial Nashequilibrium
Initialise: Assigns a product for each sourcenode
Repeat until all nodes are labelled:
Pick a node which is not labelled andfor which all neighbours are labelled
Assign the product which maximises thepayoff
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. In a DAG, a non-trivial Nash equilibrium always exist.
1
{•, •}
2
{•, •}
3{•, •}
4 {•, •}
5 {•, •, •}
0.4
0.50.3
0.10.1
Threshold = 0.3
Procedure to generate a non-trivial Nashequilibrium
Initialise: Assigns a product for each sourcenode
Repeat until all nodes are labelled:
Pick a node which is not labelled andfor which all neighbours are labelled
Assign the product which maximises thepayoff
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. In a DAG, a non-trivial Nash equilibrium always exist.
1
{•, •}
2
{•, •}
3{•, •}
4 {•, •}
5 {•, •, •}
0.4
0.50.3
0.10.1
Threshold = 0.3
Procedure to generate a non-trivial Nashequilibrium
Initialise: Assigns a product for each sourcenode
Repeat until all nodes are labelled:
Pick a node which is not labelled andfor which all neighbours are labelled
Assign the product which maximises thepayoff
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. In a DAG, a non-trivial Nash equilibrium always exist.
1
{•, •}
2
{•, •}
3{•, •}
4 {•, •}
5 {•, •, •}
0.4
0.50.3
0.10.1
Threshold = 0.3
Procedure to generate a non-trivial Nashequilibrium
Initialise: Assigns a product for each sourcenode
Repeat until all nodes are labelled:
Pick a node which is not labelled andfor which all neighbours are labelled
Assign the product which maximises thepayoff
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. In a DAG, a non-trivial Nash equilibrium always exist.
1
{•, •}
2
{•, •}
3{•, •}
4 {•, •}
5 {•, •, •}
0.4
0.50.3
0.10.1
Threshold = 0.3
Procedure to generate a non-trivial Nashequilibrium
Initialise: Assigns a product for each sourcenode
Repeat until all nodes are labelled:
Pick a node which is not labelled andfor which all neighbours are labelled
Assign the product which maximises thepayoff
Krzysztof R. Apt Social Network Games
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Directed acyclic graphs
Theorem. A joint strategy s is a Nash equilibrium iff there is a run of thelabelling procedure such that s is defined by the labelling function.
Krzysztof R. Apt Social Network Games
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Graphs with no source nodes
1
{•, •}
3{•, •}
2
{•, •}
4
{•, •}
7{•, •}
5
{•, •}
6
{•, •}
“Circle of friends”: everyone has aneighbour.
Krzysztof R. Apt Social Network Games
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Graphs with no source nodes
1
{•, •}
3{•, •}
2
{•, •}
4
{•, •}
7{•, •}
5
{•, •}
6
{•, •}
“Circle of friends”: everyone has aneighbour.
Observation: t0 is always a Nashequilibrium.
Question: When does a non-trivial Nash equilibrium exist?
Krzysztof R. Apt Social Network Games
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Graphs with no source nodes
1
{•, •}
3{•, •}
2
{•, •}
4
{•, •}
7{•, •}
5
{•, •}
6
{•, •}
0.4
0.4
0.5
0.30.2
0.1
0.2
0.1
0.4
0.2
Threshold=0.3
Self sustaining subgraph
A subgraph Ct is self sustaining forproduct t if it is strongly connected andfor all i in Ct ,
t ∈ P(i)
∑
j∈N (i)∩Ct
wji ≥ θ(i , t)
Krzysztof R. Apt Social Network Games
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Graphs with no source nodes
1
{•, •}
3{•, •}
2
{•, •}
4
{•, •}
7{•, •}
5
{•, •}
6
{•, •}
0.4
0.4
0.5
0.30.2
0.1
0.2
0.1
0.4
0.2
Threshold=0.3
Self sustaining subgraph
A subgraph Ct is self sustaining forproduct t if it is strongly connected andfor all i in Ct ,
t ∈ P(i)
∑
j∈N (i)∩Ct
wji ≥ θ(i , t)
Krzysztof R. Apt Social Network Games
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Graphs with no source nodes
1
{•, •}
3{•, •}
2
{•, •}
4
{•, •}
7{•, •}
5
{•, •}
6
{•, •}
0.4
0.4
0.5
0.30.2
0.1
0.2
0.1
0.4
0.2
Threshold=0.3
Self sustaining subgraph
A subgraph Ct is self sustaining forproduct t if it is strongly connected andfor all i in Ct ,
t ∈ P(i)
∑
j∈N (i)∩Ct
wji ≥ θ(i , t)
Theorem. There is a non-trivial Nash equilibrium iff there exists a productt and a self sustaining subgraph Ct for t.
Krzysztof R. Apt Social Network Games
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Graphs with no source nodes
For a product t,
X 0t := {i ∈ V | t ∈ P(i)}
Xm+1t := {i ∈ V |
∑
j∈N (i)∩Xmj
wji ≥ θ(i , t)}
Xt :=⋂
m∈NXm
t
Theorem. There is a non-trivial Nash equilibrium iff there exists a productt such that Xt 6= ∅.
Krzysztof R. Apt Social Network Games
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Finite Improvement Property
Fix a game.
Profitable deviation: a pair (s, s ′) such that s ′ = (s ′i , s−i) for some s ′iand pi(s
′) > pi (s).
Improvement path: a maximal sequence of profitable deviations.
A game has the FIP if all improvement paths are finite.
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FIP
Theorem. Every two players social network game has the FIP.
Krzysztof R. Apt Social Network Games
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FIP
Theorem. Every two players social network game has the FIP.Proof.
Consider an improvement path ρ.
Krzysztof R. Apt Social Network Games
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FIP
Theorem. Every two players social network game has the FIP.Proof.
Consider an improvement path ρ.
We can assume that the players alternate their moves in ρ.
Krzysztof R. Apt Social Network Games
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FIP
Theorem. Every two players social network game has the FIP.Proof.
Consider an improvement path ρ.
We can assume that the players alternate their moves in ρ.
A match: an element of ρ of the type (t, t) or (t, t).
Krzysztof R. Apt Social Network Games
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FIP
Theorem. Every two players social network game has the FIP.Proof.
Consider an improvement path ρ.
We can assume that the players alternate their moves in ρ.
A match: an element of ρ of the type (t, t) or (t, t).
Consider two successive matches in ρ.
Krzysztof R. Apt Social Network Games
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FIP
Theorem. Every two players social network game has the FIP.Proof.
Consider an improvement path ρ.
We can assume that the players alternate their moves in ρ.
A match: an element of ρ of the type (t, t) or (t, t).
Consider two successive matches in ρ.
The corresponding segment of ρ is of one of the following types.Type 1. (t, t) ⇒∗ (t1, t1).Type 2. (t, t) ⇒∗ (t1, t1).Type 3. (t, t) ⇒∗ (t1, t1).Type 4. (t, t) ⇒∗ (t1, t1).
Krzysztof R. Apt Social Network Games
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Proof, ctd
Type p1 p2
1 increases decreasesby > w21 by < w12
2, 3 increases increases
4 decreases increasesby < w21 by > w12
Table: Changes in p1 and p2
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Proof, ctd
Type p1 p2
1 increases decreasesby > w21 by < w12
2, 3 increases increases
4 decreases increasesby < w21 by > w12
Table: Changes in p1 and p2
Suppose (t, t) ⇒∗ (t1, t1) in ρ.Ti : the number of internal segments of type i .
Krzysztof R. Apt Social Network Games
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Proof, ctd
Type p1 p2
1 increases decreasesby > w21 by < w12
2, 3 increases increases
4 decreases increasesby < w21 by > w12
Table: Changes in p1 and p2
Suppose (t, t) ⇒∗ (t1, t1) in ρ.Ti : the number of internal segments of type i .
Case 1. T1 ≥ T4.Then p1(t) < p1(t1).
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Proof, ctd
Type p1 p2
1 increases decreasesby > w21 by < w12
2, 3 increases increases
4 decreases increasesby < w21 by > w12
Table: Changes in p1 and p2
Suppose (t, t) ⇒∗ (t1, t1) in ρ.Ti : the number of internal segments of type i .
Case 1. T1 ≥ T4.Then p1(t) < p1(t1).
Case 2. T1 < T4.Then p2(t) < p2(t1).
Krzysztof R. Apt Social Network Games
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Proof, ctd
Type p1 p2
1 increases decreasesby > w21 by < w12
2, 3 increases increases
4 decreases increasesby < w21 by > w12
Table: Changes in p1 and p2
Suppose (t, t) ⇒∗ (t1, t1) in ρ.Ti : the number of internal segments of type i .
Case 1. T1 ≥ T4.Then p1(t) < p1(t1).
Case 2. T1 < T4.Then p2(t) < p2(t1).
Conclusion: t 6= t1. So each match occurs in ρ at most once.Krzysztof R. Apt Social Network Games
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Proof, ctd
So from some moment on in ρ no matches occur.
Krzysztof R. Apt Social Network Games
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Proof, ctd
So from some moment on in ρ no matches occur.
So from that moment on the social welfare keeps increasing.
Krzysztof R. Apt Social Network Games
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Proof, ctd
So from some moment on in ρ no matches occur.
So from that moment on the social welfare keeps increasing.
Hence ρ is finite.
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A generalization: two player coordination games
Theorem. Consider a finite two players game G such that
pi(s) := fi (si) + ai(si = s−i),where fi : Si → R, ai > 0 and
(si = s−i ) :=
{
1 if si = s−i
0 otherwise
Then G has the FIP.
Intuition: ai is a bonus for player i for coordinating with his opponent.
Krzysztof R. Apt Social Network Games
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Price of Anarchy and Price of Stability
Theorem. The price of anarchy and the price of stability for the gamesassociated with the networks whose underlying graph is a DAG or a simplecycle is unbounded.
Krzysztof R. Apt Social Network Games
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ProofFor a simple cycle.
Choose arbitrary r > 0 and ǫ such that ǫ < min(14 , 1
2(r+1) ).
Then 1 − 2ǫ > 2ǫ and 1−2ǫ
2ǫ> r .
Consider the network
1)){t1,t2}
2ii{t1,t2}
Assumew12 − θ(2, t2) = 1 − ǫ, w21 − θ(1, t2) = −ǫ,
w12 − θ(2, t1) = ǫ, w21 − θ(1, t1) = ǫ.
Social optimum: (t2, t2) with social welfare 1 − 2ǫ.
There are two Nash equilibria, (t1, t1) and (t0, t0) with the socialwelfare 2ǫ and 0.
Price of anarchy: 1−2ǫ
0 . We interpret it as ∞.
Price of stability: 1−2ǫ
2ǫ> r .
Krzysztof R. Apt Social Network Games