social distance games

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Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion Social Distance Games 1 Kate Larson Cheriton School of Computer Science University of Waterloo December 6, 2011 1 Joint work with Simina Branzei

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Professor Kate Larson speaks for WICI on December 6th, 2011

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Page 1: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Social Distance Games1

Kate Larson

Cheriton School of Computer ScienceUniversity of Waterloo

December 6, 2011

1Joint work with Simina Branzei

Page 2: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Introduction

The Internet was the first computational artifact that was notcreated by a single entity.

Arose from the strategic interactions of many.Computer Scientists have turned to game theory forinsight.

The Internet is in equilibrium, we just need to identify thegame.

Scott Shenker.

Page 3: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Introduction

Social networks influence all aspects of everyday life.

The emergence of large online social networks (e.g.Facebook, Google+, LinkedIn,...) has enabled a much moredetailed analysis of real networks.

How does the structure of the network influence thebehavior of agents?What structures appear in such networks?What type of equilibria arise?Which agents are influential?... (see, for example, books by Jackson (2008), Easleyand Kleinberg (2010), etc.

Page 4: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Introduction

We were interested in settings whereagents’ interactions were constrained by someunderlying networkagents preferred to be in groups with "similar" or "close"agents (i.e. their friends)

agents exhibited homophily

Question: What groups should form?Cooperative game theory

Page 5: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Introduction

We were interested in settings whereagents’ interactions were constrained by someunderlying networkagents preferred to be in groups with "similar" or "close"agents (i.e. their friends)

agents exhibited homophily

Question: What groups should form?Cooperative game theory

Page 6: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Outline

Brief terminology breakModel for social distance gamesStudy social distance games from an efficiency (socialwelfare) perspectiveStudy social distance games from a stabilityperspectiveConnections between social welfare and stabilityConclusion

Page 7: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Coalitional Game Theory

We use ideas from Coalitional Game Theory to study amodel of interaction.

Non-Transferable Utility Games (NTU): A pair (N, v)where

N is a set of agentsv : 2N 7→ 2R|S|

for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjointcoalitions.Grand Coalition: The coalition which contains allagents (N).

Page 8: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Coalitional Game Theory

We use ideas from Coalitional Game Theory to study amodel of interaction.

Non-Transferable Utility Games (NTU): A pair (N, v)where

N is a set of agentsv : 2N 7→ 2R|S|

for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjointcoalitions.Grand Coalition: The coalition which contains allagents (N).

Page 9: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Coalitional Game Theory

We use ideas from Coalitional Game Theory to study amodel of interaction.

Non-Transferable Utility Games (NTU): A pair (N, v)where

N is a set of agentsv : 2N 7→ 2R|S|

for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjointcoalitions.Grand Coalition: The coalition which contains allagents (N).

Page 10: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Coalitional Game Theory

We use ideas from Coalitional Game Theory to study amodel of interaction.

Non-Transferable Utility Games (NTU): A pair (N, v)where

N is a set of agentsv : 2N 7→ 2R|S|

for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjointcoalitions.Grand Coalition: The coalition which contains allagents (N).

Page 11: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Modeling Social Distance Games

A social distance game is represented by an unweightedgraph G = (N,E) where

N = {x1, . . . , xn} is the set of agentsThe utility of an agent xi in coalition C ⊆ N is

u(xi ,C) =1|C|

∑xj∈C\{xi}

1dC(xi , xj)

.

where dC(xi , xj) is the shortest path distance betweenxi and xj in the subgraph induced by C. If xi and xj aredisconnected in C then dC(xi , xj) =∞.

Page 12: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Example

In the grand coalitionu(x0,N) = 1

6(1 + 12 + 3 · 1

3) = 512

u(x1,N) = 16(2 + 3 · 1

2) = 712

u(x2,N) = 16(4 + 1

2) = 34

u(x3,N) = u(x4,N) = u(x5,N) = 16(1 + 3 · 1

2 + 13) = 17

36

Page 13: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Properties of the Utility Function

Singletons always receive zero utility.An agent prefers direct connections over indirect ones.Adding a close connection positively affects an agent’sutility.Adding a distant connection negatively affects anagent’s utility.All things being equal, agents favor larger coalitions.

Page 14: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Properties of the Utility Function

Singletons always receive zero utility.An agent prefers direct connections over indirect ones.Adding a close connection positively affects an agent’sutility.Adding a distant connection negatively affects anagent’s utility.All things being equal, agents favor larger coalitions.

Page 15: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Properties of the Utility Function

Singletons always receive zero utility.An agent prefers direct connections over indirect ones.Adding a close connection positively affects an agent’sutility.Adding a distant connection negatively affects anagent’s utility.All things being equal, agents favor larger coalitions.

Page 16: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Properties of the Utility Function

Singletons always receive zero utility.An agent prefers direct connections over indirect ones.Adding a close connection positively affects an agent’sutility.Adding a distant connection negatively affects anagent’s utility.All things being equal, agents favor larger coalitions.

Page 17: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Properties of the Utility Function

Singletons always receive zero utility.An agent prefers direct connections over indirect ones.Adding a close connection positively affects an agent’sutility.Adding a distant connection negatively affects anagent’s utility.All things being equal, agents favor larger coalitions.

Page 18: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Properties of the Utility Function

Singletons always receive zero utility.An agent prefers direct connections over indirect ones.Adding a close connection positively affects an agent’sutility.Adding a distant connection negatively affects anagent’s utility.All things being equal, agents favor larger coalitions.

Page 19: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Social Welfare

The social welfare of coalition structure CS = (C1, . . . ,Ck )is

SW (CS) =k∑

i=1

∑xj∈Ci

u(xj ,Ci).

Page 20: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Example

SW (N) = 316

SW ({x0, x1}, {x2, x3, x4, x5}) = 314 .

We are interested in social welfare maximizing coalitionstructures since these can be viewed as the best outcomefor society overall.

Page 21: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Characterization of SW Maximizing CS

Observation: On complete graphs the unique SWmaximizing structure is the grand coalition.Observation: The SW of any coalition structure isbounded by n − 1.

This upper bound is only obtained by the grand coalitionon complete graphs.

Observation: The grand coalition maximizes socialwelfare on complete bipartite graphs (e.g. on stars).

It also guarantees utility of at least 12 to each agent.

Page 22: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Characterization of SW Maximizing CS

Observation: On complete graphs the unique SWmaximizing structure is the grand coalition.Observation: The SW of any coalition structure isbounded by n − 1.

This upper bound is only obtained by the grand coalitionon complete graphs.

Observation: The grand coalition maximizes socialwelfare on complete bipartite graphs (e.g. on stars).

It also guarantees utility of at least 12 to each agent.

Page 23: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Characterization of SW Maximizing CS

Observation: On complete graphs the unique SWmaximizing structure is the grand coalition.Observation: The SW of any coalition structure isbounded by n − 1.

This upper bound is only obtained by the grand coalitionon complete graphs.

Observation: The grand coalition maximizes socialwelfare on complete bipartite graphs (e.g. on stars).

It also guarantees utility of at least 12 to each agent.

Page 24: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Approximating Social Welfare

Finding the optimal social welfare partition is NP-hard.

TheoremDiameter two decompositions guarantee to each agent atleast utility 1

2 .

CorollaryWe can approximate optimal social welfare within a factor oftwo using a two-decomposition.

Page 25: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Approximating Social Welfare

Finding the optimal social welfare partition is NP-hard.

TheoremDiameter two decompositions guarantee to each agent atleast utility 1

2 .

CorollaryWe can approximate optimal social welfare within a factor oftwo using a two-decomposition.

Page 26: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Approximating Social Welfare

Finding the optimal social welfare partition is NP-hard.

TheoremDiameter two decompositions guarantee to each agent atleast utility 1

2 .

CorollaryWe can approximate optimal social welfare within a factor oftwo using a two-decomposition.

Page 27: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Example: Approximating SW

Compute Minimum Spanning Tree, TIdentify deepest leaf node xi and its parent Parent(xi)

Put xi , Parent(xi) and the children of Parent(xi) intocoalition Ci . Remove all agents in Ci from TRepeat previous two steps until done, handling the rootof T as necessary.

Page 28: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Stability in Social Distance Games

Lack of stability can threaten coalition structures.

Definition (Core)

A coalition structure, CS = (C1, . . . ,Ck ) is in the core ifthere is no coalition B ⊆ N such that ∀x ∈ B,u(x ,B) ≥ u(x ,CS) and for some y ∈ B the inequality isstrict.

B is called a blocking coalition.

Page 29: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Existence of Stable Games

For some games, the core is empty.

The grand coalition is blocked by {x2, x3, x4, x5}({x0, x1}, {x2, x3, x4, x5}) is blocked by {x1, x2, x3, x4, x5}

Page 30: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Existence of Stable Games

Observation: In complete graphs, the grand coalition is theonly core stable coalition structure.

Observation: If the graph is a tree, then thetwo-decomposition algorithm returns a core coalitionstructure.

Page 31: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Core Coalition Structures are Small Worlds

Small World Network: Most nodes can be reached fromany other node using a small number of steps throughintermediate nodes.

Page 32: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Core Coalition Structures are Small Worlds

Small World Network: Most nodes can be reached fromany other node using a small number of steps throughintermediate nodes.

If coalition structure, CS, is inthe core, then for anyCi ∈ CS the diameter of Ci isbounded by 14.

Page 33: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Stability and Social Welfare

Social welfare maximizing coalition structures are notalways stable (i.e. when the core is empty).

Page 34: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Stability and Social Welfare

Stable coalition structures do not always maximize socialwelfare.

X0 X2X1

X3 X4

The core is({x0, x1, x2, x3, x4})

Social welfare is maximizedby either({x0, x1, x3}, {x2, x4}) or({x0, x3}, {x1, x2, x4})

Page 35: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

The Stability Gap

LetG be an arbitrary graph for a social distance game,CS∗ be a social welfare maximizing coalition structure,CSC be a member of the core induced by G.

The stability gap, Gap(G) is

Gap(G) =SW (CS∗)

minCSC∈Core(G) SW (CSC).

Page 36: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

The Stability Gap: The General Case

TheoremLet G = (N,E) be a game with non-empty core. ThenGap(G) is, in the worst case, Θ(

√n).

Page 37: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

The Stability Gap: Special Cases

For dense graphs the stability gap is small.

TheoremThe stability gap of every graph with m edges where

m ≥(

1− ε2

2

)n2 −

(1− ε

2

)n

is at most 41−ε where 0 < ε < 1.

TheoremThe expected stability gap of graphs generated under theErdos-Renyi G(n,p) graph model is bounded by 4

1−2 log(n)/nwhenever p ≥ 1/2.

Page 38: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

The Stability Gap: Special Cases

For dense graphs the stability gap is small.

TheoremThe stability gap of every graph with m edges where

m ≥(

1− ε2

2

)n2 −

(1− ε

2

)n

is at most 41−ε where 0 < ε < 1.

TheoremThe expected stability gap of graphs generated under theErdos-Renyi G(n,p) graph model is bounded by 4

1−2 log(n)/nwhenever p ≥ 1/2.

Page 39: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Alternative Solution Concepts

Observation: For general games, a stable coalitionstructure can come at a high cost in social welfare.

Question: Can we develop reasonable variations of thecore solution concept, which provide improved socialsupport?

Page 40: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Stability Threshold

Assume that after achieving utility kk+1 for some k > 1 an

agent is satisfied.Stop seeking improvements once they have achieved aminimum value.Reasonable in situations with diminishing returns.

TheoremLet G = (N,E) be a game with stability threshold k/(k + 1).If the core with stability threshold is non-empty thenGap(G) ≤ 4 if k = 1 and Gap(G) ≤ 2k if k ≥ 2.

Page 41: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Stability Threshold

Assume that after achieving utility kk+1 for some k > 1 an

agent is satisfied.Stop seeking improvements once they have achieved aminimum value.Reasonable in situations with diminishing returns.

TheoremLet G = (N,E) be a game with stability threshold k/(k + 1).If the core with stability threshold is non-empty thenGap(G) ≤ 4 if k = 1 and Gap(G) ≤ 2k if k ≥ 2.

Page 42: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

"No Man Left Behind"

Observation: The low social welfare sometimes seen inmembers of the core comes from isolated agents.

No Man Left Behind Policy: As new coalition forms,agents can not be isolated.

TheoremLet G = (N,E) be a game that is stable under the "No ManLeft Behind" policy. Then Gap(G) < 4.

Page 43: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

"No Man Left Behind"

Observation: The low social welfare sometimes seen inmembers of the core comes from isolated agents.

No Man Left Behind Policy: As new coalition forms,agents can not be isolated.

TheoremLet G = (N,E) be a game that is stable under the "No ManLeft Behind" policy. Then Gap(G) < 4.

Page 44: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Summary

This work is a step in the direction of understanding networkinteractions from the perspective of coalitional game theory.

Proposed a mathematical modelAnalyzed the model’s welfare and stability propertiesProposed two solution concepts with improved socialwelfare guarantees

Page 45: Social Distance Games

SocialDistanceGames

Kate Larson

Introduction

The Model

The SocialWelfarePerspective

Stability inSocialDistanceGames

The StabilityGap

AlternativeSolutionConcepts

Conclusion

Future work

Characterization of the extent an agent contributes tothe social welfare or stabilizes the game.Understand how the degree and position of agents inthe network correspond with its welfare in equilibrium.Are stable structures small worlds under general utilityfunctions that reflect homophily?Empirical analysis.