Contribution Games in Social Networks Elliot Anshelevich Rensselaer Polytechnic Institute (RPI)...

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Contribution Games in Social Networks

Elliot AnshelevichRensselaer Polytechnic Institute (RPI)

Troy, New York

Martin HoeferRWTH Aachen University

Aachen, Germany

Partitioning Effort in a Social Network

Success of Friendship/Collaboration

• Will represent “success” of relationship e by reward function:

fe(x,y) : non-negative, non-decreasing in both variables

• fe(x,y) = amount each node benefits from e

Network Contribution Game

Given: Undirected graph G=(V,E) Players: Nodes vV, each v has budget Bv of contribution Reward functions fe(x,y) for each edge e

2(x+y)

4(x+y)3(x+y)

Strategies: Node allocates its budget among incident edges:

v contributes sv(e)0 to each e, with sv(e) Bv e

2(x+y)

4(x+y)3(x+y)1 4

28151 4

Utility(v) = fe(sv(e),su(e))e=(v,u)

28151 4

Stability Concepts

Nash equilibrium?

1000xy3xy5 0

Pairwise Equilibrium

Unilateral improving move: A single player can strictly improve by changing its strategy. Bilateral improving move: A pair of players can each strictly improve their utility by changing strategies together.

Pairwise Equilibrium (PE): State s with no unilateral or bilateral improving moves.

Strong Equilibrium (SE): State s with no coalitional improving moves.

Questions of Interest

Existence: Does Pairwise Equilibrium exist?

Inefficiency: What is the price of anarchy ?

Computation: Can we compute PE efficiently?

Convergence: Can players reach PE using improvement dynamics?

OPT PE

Related WorkStable Matching “Integral” version of our game Correlated roommate problems

[Abraham et al, 07; Ackermann et al, 08]

Network Creation Games Contribution towards incident edges Rewards based on network structure

[Fabrikant et al, 03; Laoutaris et al, 08; Demaine et al, 10]

Co-Author Model [Jackson/Wolinsky, 96]

Atomic Splittable Congestion Games Mostly NE analysis and cost minimization Delay functions usually depend on x + y [Orda et al, 93; Umang et al, 10.]

Public Goods and Contribution Games Public Goods Games [Bramoulle/Kranton, 07] Contribution Games [Ballester et al, 06] Various extensions [Corbo et al, 09; Konig et al, 09]

Minimum Effort Coordination Game Simple game from experimental economics All agents get payoff based on minimum contribution [van Huyck et al, 90; Anderson et al, 01; Devetag/Ortmann, 07] Networked variants [Alos-Ferrer/Weidenholzer, 10; Bloch/Dutta, 08]

... and many more.

Main Results

Existence Price of AnarchyGeneral Convex

General Concavece(x+y)

Min-effort convexMin-effort concave

Maximum effortApproximate Equilibrium

Main Results

Existence Price of AnarchyGeneral Convex Yes (*)

(*) If fe(x,0)=0, NP-Hard otherwise

Main Results

Existence Price of AnarchyGeneral Convex Yes (*) 2

Main Results

General Concave Not always 2ce(x+y)

Main Results

General Concave Not always 2ce(x+y) Decision in P 1

Main Results

Min-effort convex Yes (**) 2 (**)Min-effort concave

(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise

Main Results

Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2

Main Results

Maximum effort Yes 2Approximate Equilibrium OPT is a 2-apx. Pairwise Equilibrium

Main Results

All Price of Anarchy upper bounds are tight

Convergence?

Main Results

Convex Reward FunctionsTheorem 1: If for all edges, fe(x,0)=0, and fe convex, then PE exists. Otherwise, PE

existence is NP-Hard to determine.

Theorem 2: If for all edges, fe is convex, then price of anarchy is 2.

Examples: 10xy, 5x2y2, 2x+y, x+4y2+7x3, polynomials with positive coefficients

Convex Reward Functions

Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge.

PE OPT/2

Main Results

Minimum Effort Games

All functions are of the form fe(x,y)=he(min(x,y)) he is concave

All functions are of the form fe(x,y)=he(min(x,y)) he is concave For general concave functions, PE may not exist:

Theorem: Pairwise equilibrium always exists in minimum effort games with continuous, piecewise differentiable, and concave he.

),min( yx

Can compute to arbitrary precision. If strictly concave, then PE is unique. Price of anarchy at most 2.

In PE, both players have matching contributions.

PE for concave-of-min2 1

PE for concave-of-min

Compute best strategy for each node v if it were able to control all other players

Derivative must equal on all edges with positive effort. Done via convex program.

182932

Compute best strategy for each node v if it were able to control all other players Fix strategy of node with highest derivative (crucial tie-breaking rule)

182932

Compute best strategy for each non-fixed node v if it were able to control all other non-fixed players

182932

Lemma: best responses consistent with fixed strategies

Compute best strategy for each non-fixed node v if it were able to control all other non-fixed players Fix strategy of node with highest derivative (crucial tie-breaking rule)

End: all strategies are fixed.

This is a Pairwise Equilibrium: in any pair, the node whose strategy was fixed first would not deviate.

Can compute to arbitrary precision. If strictly concave, then PE is unique. Price of anarchy at most 2.

In PE, both players have matching contributions.

Main Results

All Price of Anarchy upper bounds are tight

Convergence?

Extensions and Open Questions

• Other interesting classes of reward functions• Other types of dynamics• Capacity for maximum contribution on an edge

• General contribution games• Cost functions for generating contributions• Sharing reward unequally

Thank you!

Contribution Games in Social Networks Elliot Anshelevich Rensselaer Polytechnic Institute (RPI)...

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