Contribution Games in Social Networks Elliot Anshelevich Rensselaer Polytechnic Institute (RPI) Troy, New York Martin Hoefer RWTH Aachen University Aachen, Germany

Partitioning Effort in a Social Network

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Success of Friendship/Collaboration

Will represent “success” of relationship e by reward function: f e (x,y) : non-negative, non-decreasing in both variables f e (x,y) = amount each node benefits from e

Network Contribution Game Given:  Undirected graph G=(V,E)  Players: Nodes v  V, each v has budget B v of contribution  Reward functions f e (x,y) for each edge e (x+y) 4(x+y)3(x+y)

Network Contribution Game Given:  Undirected graph G=(V,E)  Players: Nodes v  V, each v has budget B v of contribution  Reward functions f e (x,y) for each edge e Strategies: Node allocates its budget among incident edges: v contributes s v (e)  0 to each e, with  s v (e)  B v e (x+y) 4(x+y)3(x+y)

Network Contribution Game Given:  Undirected graph G=(V,E)  Players: Nodes v  V, each v has budget B v of contribution  Reward functions f e (x,y) for each edge e Strategies: Node allocates its budget among incident edges: v contributes s v (e)  0 to each e, with  s v (e)  B v e

Network Contribution Game Strategies: Node allocates its budget among incident edges: v contributes s v (e)  0 to each e, with  s v (e)  B v e Utility(v) =  f e (s v (e),s u (e)) e=(v,u)

Stability Concepts Nash equilibrium? xy 1000xy3xy

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 Work Stable 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 ExistencePrice of Anarchy General Convex General Concave c e (x+y) Min-effort convex Min-effort concave Maximum effort Approximate Equilibrium

Main Results ExistencePrice of Anarchy General ConvexYes (*) General Concave c e (x+y) Min-effort convex Min-effort concave Maximum effort Approximate Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General Concave c e (x+y) Min-effort convex Min-effort concave Maximum effort Approximate Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y) Min-effort convex Min-effort concave Maximum effort Approximate Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convex Min-effort concave Maximum effort Approximate Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concave Maximum effort Approximate Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise (**) If budgets are uniform, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effort Approximate Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise (**) If budgets are uniform, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effortYes2 Approximate EquilibriumOPT is a 2-apx. Pairwise Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise (**) If budgets are uniform, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effortYes2 Approximate EquilibriumOPT is a 2-apx. Pairwise Equilibrium All Price of Anarchy upper bounds are tight Convergence? ?

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effortYes2 Approximate EquilibriumOPT is a 2-apx. Pairwise Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise (**) If budgets are uniform, NP-Hard otherwise

Convex Reward Functions Theorem 1: If for all edges, f e (x,0)=0, and f e convex, then PE exists. Otherwise, PE existence is NP-Hard to determine. Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Examples: 10xy, 5x 2 y 2, 2 x+y, x+4y 2 +7x 3, polynomials with positive coefficients

Convex Reward Functions Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge

Convex Reward Functions Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge

Convex Reward Functions Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge

Convex Reward Functions Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge

Convex Reward Functions Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge 

Convex Reward Functions Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge   

Convex Reward Functions Theorem 2: If for all edges, f e is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge    PE  OPT/2

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effortYes2 Approximate EquilibriumOPT is a 2-apx. Pairwise Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise (**) If budgets are uniform, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effortYes2 Approximate EquilibriumOPT is a 2-apx. Pairwise Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise (**) If budgets are uniform, NP-Hard otherwise

Minimum Effort Games  All functions are of the form f e (x,y)=h e (min(x,y))  h e is concave

Minimum Effort Games  All functions are of the form f e (x,y)=h e (min(x,y))  h e is concave  For general concave functions, PE may not exist: 11 1

Minimum Effort Games Theorem: Pairwise equilibrium always exists in minimum effort games with continuous, piecewise differentiable, and concave h e. 11 1

Minimum Effort Games Theorem: Pairwise equilibrium always exists in minimum effort games with continuous, piecewise differentiable, and concave h e.  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-min x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

 Compute best strategy for each node v if it were able to control all other players x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  Compute best strategy for each node v if it were able to control all other players x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

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 x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

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 x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  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) x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  Compute best strategy for each non-fixed node v if it were able to control all other non-fixed players x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  Compute best strategy for each non-fixed node v if it were able to control all other non-fixed players x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  Compute best strategy for each non-fixed node v if it were able to control all other non-fixed players Lemma:  best responses consistent with fixed strategies x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  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) x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  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) x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  End: all strategies are fixed. This is a Pairwise Equilibrium: in any pair, the node whose strategy was fixed first would not deviate x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  End: all strategies are fixed. This is a Pairwise Equilibrium: in any pair, the node whose strategy was fixed first would not deviate x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

PE for concave-of-min  End: all strategies are fixed. This is a Pairwise Equilibrium: in any pair, the node whose strategy was fixed first would not deviate x2x 4x4x 3x3x 4x4x 3x3x xx 3x3x

Minimum Effort Games Theorem: Pairwise equilibrium always exists in minimum effort games with continuous, piecewise differentiable, and concave h e.  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 ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effortYes2 Approximate EquilibriumOPT is a 2-apx. Pairwise Equilibrium (*) If f e (x,0)=0, NP-Hard otherwise (**) If budgets are uniform, NP-Hard otherwise

Main Results ExistencePrice of Anarchy General ConvexYes (*)2 General ConcaveNot always2 c e (x+y)Decision in P1 Min-effort convexYes (**)2 (**) Min-effort concaveYes2 Maximum effortYes2 Approximate EquilibriumOPT is a 2-apx. Pairwise Equilibrium 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!