social distance games
DESCRIPTION
Professor Kate Larson speaks for WICI on December 6th, 2011TRANSCRIPT
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
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.
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.
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
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
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
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).
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).
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).
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).
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) =∞.
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
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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}
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.
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.
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.
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).
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})
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).
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).
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.
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.
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?
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.
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.
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.
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.
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
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.