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Social Distance Games Kate Larson Introduction

Social Distance Games1

The Model The Social Welfare Perspective

Kate Larson

Stability in Social Distance Games

Cheriton School of Computer Science University of Waterloo

The Stability Gap Alternative Solution Concepts

December 6, 2011

Conclusion

1

Joint work with Simina Branzei


Introduction Social Distance Games Kate Larson Introduction

The Internet was the first computational artifact that was not created by a single entity.

The Model

Arose from the strategic interactions of many.

The Social Welfare Perspective

Computer Scientists have turned to game theory for insight.

Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

The Internet is in equilibrium, we just need to identify the game. Scott Shenker.


Introduction Social Distance Games

Social networks influence all aspects of everyday life.

Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

The emergence of large online social networks (e.g. Facebook, Google+, LinkedIn,...) has enabled a much more detailed analysis of real networks. How does the structure of the network influence the behavior of agents? What structures appear in such networks? What type of equilibria arise? Which agents are influential? ... (see, for example, books by Jackson (2008), Easley and Kleinberg (2010), etc.


Introduction Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games

We were interested in settings where agents’ interactions were constrained by some underlying network agents preferred to be in groups with "similar" or "close" agents (i.e. their friends) agents exhibited homophily

The Stability Gap Alternative Solution Concepts Conclusion

Question: What groups should form? Cooperative game theory


Introduction Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games

We were interested in settings where agents’ interactions were constrained by some underlying network agents preferred to be in groups with "similar" or "close" agents (i.e. their friends) agents exhibited homophily

The Stability Gap Alternative Solution Concepts Conclusion

Question: What groups should form? Cooperative game theory


Outline Social Distance Games Kate Larson Introduction

Brief terminology break

The Model

Model for social distance games

The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Study social distance games from an efficiency (social welfare) perspective Study social distance games from a stability perspective Connections between social welfare and stability Conclusion


Coalitional Game Theory Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

We use ideas from Coalitional Game Theory to study a model of interaction. Non-Transferable Utility Games (NTU): A pair (N, v ) where N is a set of agents |S| v : 2N 7→ 2R for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjoint coalitions. Grand Coalition: The coalition which contains all agents (N).


Coalitional Game Theory Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

We use ideas from Coalitional Game Theory to study a model of interaction. Non-Transferable Utility Games (NTU): A pair (N, v ) where N is a set of agents |S| v : 2N 7→ 2R for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjoint coalitions. Grand Coalition: The coalition which contains all agents (N).


Coalitional Game Theory Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

We use ideas from Coalitional Game Theory to study a model of interaction. Non-Transferable Utility Games (NTU): A pair (N, v ) where N is a set of agents |S| v : 2N 7→ 2R for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjoint coalitions. Grand Coalition: The coalition which contains all agents (N).


Coalitional Game Theory Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

We use ideas from Coalitional Game Theory to study a model of interaction. Non-Transferable Utility Games (NTU): A pair (N, v ) where N is a set of agents |S| v : 2N 7→ 2R for each S ⊆ N.

Coalition Structure: A partition, CS, of N into disjoint coalitions. Grand Coalition: The coalition which contains all agents (N).


Modeling Social Distance Games Social Distance Games Kate Larson Introduction

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

The Model

N = {x1 , . . . , xn } is the set of agents

The Social Welfare Perspective

The utility of an agent xi in coalition C ⊆ N is

Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

u(xi , C) =

1 |C|

X xj ∈C\{xi }

1 . dC (xi , xj )

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


Example Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games

In the grand coalition u(x0 , N) = 16 (1 +

1 2

+ 3 路 13 ) =

The Stability Gap

u(x1 , N) = 16 (2 + 3 路 21 ) =

Alternative Solution Concepts

u(x2 , N) = 16 (4 + 12 ) =

Conclusion

5 12

7 12

3 4

u(x3 , N) = u(x4 , N) = u(x5 , N) = 61 (1 + 3 路

1 2

+ 13 ) =

17 36


Properties of the Utility Function Social Distance Games Kate Larson Introduction

Singletons always receive zero utility.

The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Properties of the Utility Function Social Distance Games Kate Larson Introduction

Singletons always receive zero utility.

The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Properties of the Utility Function Social Distance Games Kate Larson Introduction

Singletons always receive zero utility.

The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Properties of the Utility Function Social Distance Games Kate Larson Introduction

Singletons always receive zero utility.

The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Properties of the Utility Function Social Distance Games Kate Larson Introduction

Singletons always receive zero utility.

The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Properties of the Utility Function Social Distance Games Kate Larson Introduction

Singletons always receive zero utility.

The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Social Welfare Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

The social welfare of coalition structure CS = (C1 , . . . , Ck ) is k X X SW (CS) = u(xj , Ci ). i=1 xj ∈Ci


Example Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

SW (N) = 3 16 SW ({x0 , x1 }, {x2 , x3 , x4 , x5 }) = 3 41 . We are interested in social welfare maximizing coalition structures since these can be viewed as the best outcome for society overall.


Characterization of SW Maximizing CS Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Observation: On complete graphs the unique SW maximizing structure is the grand coalition. Observation: The SW of any coalition structure is bounded by n − 1. This upper bound is only obtained by the grand coalition on complete graphs.

Observation: The grand coalition maximizes social welfare on complete bipartite graphs (e.g. on stars). It also guarantees utility of at least

1 2

to each agent.


Characterization of SW Maximizing CS Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Observation: On complete graphs the unique SW maximizing structure is the grand coalition. Observation: The SW of any coalition structure is bounded by n − 1. This upper bound is only obtained by the grand coalition on complete graphs.

Observation: The grand coalition maximizes social welfare on complete bipartite graphs (e.g. on stars). It also guarantees utility of at least

1 2

to each agent.


Characterization of SW Maximizing CS Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Observation: On complete graphs the unique SW maximizing structure is the grand coalition. Observation: The SW of any coalition structure is bounded by n − 1. This upper bound is only obtained by the grand coalition on complete graphs.

Observation: The grand coalition maximizes social welfare on complete bipartite graphs (e.g. on stars). It also guarantees utility of at least

1 2

to each agent.


Approximating Social Welfare Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Finding the optimal social welfare partition is NP-hard. Theorem Diameter two decompositions guarantee to each agent at least utility 21 . Corollary We can approximate optimal social welfare within a factor of two using a two-decomposition.


Approximating Social Welfare Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Finding the optimal social welfare partition is NP-hard. Theorem Diameter two decompositions guarantee to each agent at least utility 21 . Corollary We can approximate optimal social welfare within a factor of two using a two-decomposition.


Approximating Social Welfare Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Finding the optimal social welfare partition is NP-hard. Theorem Diameter two decompositions guarantee to each agent at least utility 21 . Corollary We can approximate optimal social welfare within a factor of two using a two-decomposition.


Example: Approximating SW Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Compute Minimum Spanning Tree, T Identify deepest leaf node xi and its parent Parent(xi ) Put xi , Parent(xi ) and the children of Parent(xi ) into coalition Ci . Remove all agents in Ci from T Repeat previous two steps until done, handling the root of T as necessary.


Stability in Social Distance Games Social Distance Games Kate Larson Introduction

Lack of stability can threaten coalition structures.

The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Definition (Core) A coalition structure, CS = (C1 , . . . , Ck ) is in the core if there is no coalition B ⊆ N such that ∀x ∈ B, u(x, B) ≥ u(x, CS) and for some y ∈ B the inequality is strict. B is called a blocking coalition.


Existence of Stable Games Social Distance Games Kate Larson

For some games, the core is empty.

Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Existence of Stable Games Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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

Observation: If the graph is a tree, then the two-decomposition algorithm returns a core coalition structure.


Core Coalition Structures are Small Worlds Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Small World Network: Most nodes can be reached from any other node using a small number of steps through intermediate nodes.


Core Coalition Structures are Small Worlds Social Distance Games Kate Larson Introduction The Model

Small World Network: Most nodes can be reached from any other node using a small number of steps through intermediate nodes.

The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

If coalition structure, CS, is in the core, then for any Ci ∈ CS the diameter of Ci is bounded by 14.


Stability and Social Welfare 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 welfare maximizing coalition structures are not always stable (i.e. when the core is empty).


Stability and Social Welfare Social Distance Games Kate Larson

Stable coalition structures do not always maximize social welfare.

Introduction The Model The Social Welfare Perspective

X0

Stability in Social Distance Games

X3

The Stability Gap Alternative Solution Concepts Conclusion

X1

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

X2

X4

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


The Stability Gap Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Let G be an arbitrary graph for a social distance game, CS ∗ be a social welfare maximizing coalition structure, CS C be a member of the core induced by G. The stability gap, Gap(G) is Gap(G) =

SW (CS ∗ ) . minCS C ∈Core(G) SW (CS C )


The Stability Gap: The General Case Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Theorem Let G = (N, E) be a game with non-empty core. Then √ Gap(G) is, in the worst case, Θ( n).


The Stability Gap: Special Cases Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

For dense graphs the stability gap is small. Theorem The stability gap of every graph with m edges where     1− 1 − 2 2 n − n m≼ 2 2 is at most

4 1−

where 0 <  < 1.

Theorem The expected stability gap of graphs generated under the 4 Erdos-Renyi G(n, p) graph model is bounded by 1â&#x2C6;&#x2019;2 log(n)/n whenever p â&#x2030;Ľ 1/2.


The Stability Gap: Special Cases Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

For dense graphs the stability gap is small. Theorem The stability gap of every graph with m edges where     1â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; 2 2 n â&#x2C6;&#x2019; n mâ&#x2030;Ľ 2 2 is at most

4 1â&#x2C6;&#x2019;

where 0 <  < 1.

Theorem The expected stability gap of graphs generated under the 4 Erdos-Renyi G(n, p) graph model is bounded by 1â&#x2C6;&#x2019;2 log(n)/n whenever p â&#x2030;Ľ 1/2.


Alternative Solution Concepts Social Distance Games Kate Larson Introduction The Model

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

The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Question: Can we develop reasonable variations of the core solution concept, which provide improved social support?


Stability Threshold Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Assume that after achieving utility agent is satisfied.

k k +1

for some k > 1 an

Stop seeking improvements once they have achieved a minimum value. Reasonable in situations with diminishing returns. Theorem Let G = (N, E) be a game with stability threshold k /(k + 1). If the core with stability threshold is non-empty then Gap(G) ≤ 4 if k = 1 and Gap(G) ≤ 2k if k ≥ 2.


Stability Threshold Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

Assume that after achieving utility agent is satisfied.

k k +1

for some k > 1 an

Stop seeking improvements once they have achieved a minimum value. Reasonable in situations with diminishing returns. Theorem Let G = (N, E) be a game with stability threshold k /(k + 1). If the core with stability threshold is non-empty then Gap(G) ≤ 4 if k = 1 and Gap(G) ≤ 2k if k ≥ 2.


"No Man Left Behind" Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective

Observation: The low social welfare sometimes seen in members of the core comes from isolated agents. No Man Left Behind Policy: As new coalition forms, agents can not be isolated.

Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


"No Man Left Behind" Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective

Observation: The low social welfare sometimes seen in members of the core comes from isolated agents. No Man Left Behind Policy: As new coalition forms, agents can not be isolated.

Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Summary Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

This work is a step in the direction of understanding network interactions from the perspective of coalitional game theory. Proposed a mathematical model Analyzed the modelâ&#x20AC;&#x2122;s welfare and stability properties Proposed two solution concepts with improved social welfare guarantees


Future work Social Distance Games Kate Larson Introduction The Model The Social Welfare Perspective Stability in Social Distance Games The Stability Gap Alternative Solution Concepts Conclusion

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


Social Distance Games