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Contribution Games in Social Networks

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. Partitioning Effort in a Social Network. 1.

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Contribution Games in Social Networks

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

  2. Partitioning Effort in a Social Network

  3. Partitioning Effort in a Social Network 1

  4. Partitioning Effort in a Social Network 0.2 0.6 0.2

  5. Success of Friendship/Collaboration

  6. Success of Friendship/Collaboration

  7. 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

  8. Network Contribution Game • Given: • Undirected graph G=(V,E) • Players: Nodes vV, each v has budget Bv of contribution • Reward functions fe(x,y) for each edge e 5 3(x+y) 4(x+y) 2(x+y) 10 8

  9. Network Contribution Game • Given: • Undirected graph G=(V,E) • Players: Nodes vV, each v has budget Bv of contribution • Reward functions fe(x,y) for each edge e • Strategies: Node allocates its budget among incident edges: • v contributes sv(e)0 to each e, with sv(e)  Bv e 5 1 4 3(x+y) 4(x+y) 4 3 6 5 2(x+y) 10 8

  10. Network Contribution Game • Given: • Undirected graph G=(V,E) • Players: Nodes vV, each v has budget Bv of contribution • Reward functions fe(x,y) for each edge e • Strategies: Node allocates its budget among incident edges: • v contributes sv(e)0 to each e, with sv(e)  Bv e 5 1 4 15 28 4 3 6 5 22 10 8

  11. Network Contribution Game Strategies: Node allocates its budget among incident edges: v contributes sv(e)0 to each e, with sv(e)  Bv e Utility(v) =  fe(sv(e),su(e)) e=(v,u) 5 1 4 15 28 4 3 6 5 22 10 8

  12. Stability Concepts • Nash equilibrium? 5 5 0 3xy 1000xy 10 0 0 8 2xy 10 8

  13. 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.

  14. 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

  15. Related Work • Public Goods and Contribution Games • Public Goods Games [Bramoulle/Kranton, 07] • Contribution Games [Ballester et al, 06] • Various extensions [Corbo et al, 09; Koniget 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. 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.]

  16. Main Results

  17. Main Results (*) If fe(x,0)=0, NP-Hard otherwise

  18. Main Results (*) If fe(x,0)=0, NP-Hard otherwise

  19. Main Results (*) If fe(x,0)=0, NP-Hard otherwise

  20. Main Results (*) If fe(x,0)=0, NP-Hard otherwise

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

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

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

  24. Main Results  ?  All Price of Anarchy upper bounds are tight Convergence?

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

  26. Convex Reward Functions Theorem 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

  27. Convex Reward Functions Theorem 2: If for all edges,fe is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge. 6 5 0 0 5 6 0 0 10 8 10 8

  28. Convex Reward Functions Theorem 2: If for all edges,fe is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge. 3 3 6 5 0 0 5 3 2 6 7 5 0 0 10 8 3 3 10 8

  29. Convex Reward Functions Theorem 2: If for all edges,fe is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge. 3 3 6 5 0 0 5 3 2 6 7 5 0 0 10 8 3 3 10 8

  30. Convex Reward Functions Theorem 2: If for all edges,fe is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge. 3 3 6 5 0 0 5 3 2 6 7 5 0 0 10 8 3 3 10 8

  31. Convex Reward Functions Theorem 2: If for all edges,fe is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge. 3 3 6  5 0 0 5 3 2 6 7 5 0 0 10 8 3 3 10 8

  32. Convex Reward Functions Theorem 2: If for all edges,fe is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge. 3 3  6 5 0 0 5 3 2 6  7 5 0 0  10 8 3 3 10 8

  33. Convex Reward Functions Theorem 2: If for all edges,fe is convex, then price of anarchy is 2. Claim: Always exists “integral” optimal solution: each player spends entire budget on one edge. 3 3  6 5 0 0 5 3 2 6  7 5 0 0  10 8 3 3 10 8 PE OPT/2 

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

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

  36. Minimum Effort Games • All functions are of the form fe(x,y)=he(min(x,y)) • heis concave

  37. Minimum Effort Games • All functions are of the form fe(x,y)=he(min(x,y)) • heis concave • For general concave functions, PE may not exist: 1 1 1

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

  39. Minimum Effort Games • Theorem: Pairwise equilibrium always exists in minimum effort games with continuous, piecewise differentiable, and concave he. • 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.

  40. PE for concave-of-min 2 1 2x 3x 4x x 3x 4x 3x 2 3 1

  41. PE for concave-of-min 2 1 2x 3x 4x x 3x 4x 3x 2 3 1 • Compute best strategy for each node v • if it were able to control all other players

  42. PE for concave-of-min 2 1 4 14 1 14 2x 9 14 3x 4x x 3x 4x 3x 2 3 1 • Compute best strategy for each node v • if it were able to control all other players

  43. PE for concave-of-min 2 1 4 14 1 14 2x 9 14 3x 4x x 3x 4x 3x 2 3 1 • 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.

  44. PE for concave-of-min 2 1 8 29 4 14 1 14 2x 18 29 9 14 32 29 3x 4x x 3x 27 43 1 10 27 43 1 48 43 27 43 9 10 1 4x 3x 2 3 1 • 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.

  45. PE for concave-of-min 2 1 8 29 4 14 1 14 2x 18 29 9 14 32 29 3x 4x x 3x 27 43 1 10 27 43 1 48 43 27 43 9 10 1 4x 3x 2 3 1 • 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)

  46. PE for concave-of-min 2 1 8 29 4 14 1 14 2x 18 29 9 14 32 29 3x 4x x 3x 27 43 1 10 27 43 1 48 43 27 43 9 10 1 4x 3x 2 3 1 • Compute best strategy for each non-fixed node v • if it were able to control all other non-fixed players

  47. PE for concave-of-min 2 1 4 13 4 14 1 14 2x 9 13 9 14 1 3x 4x x 3x 2 3 1 10 2 3 1 2 3 9 10 1 1 4x 3x 2 3 1 • Compute best strategy for each non-fixed node v • if it were able to control all other non-fixed players

  48. PE for concave-of-min 2 1 4 13 4 14 1 14 2x 9 13 9 14 1 3x 4x x 3x 2 3 1 10 2 3 1 2 3 9 10 1 1 4x 3x 2 3 1 • 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

  49. PE for concave-of-min 2 1 4 13 4 14 1 14 2x 9 13 9 14 1 3x 4x x 3x 2 3 1 10 2 3 1 2 3 9 10 1 1 4x 3x 2 3 1 • 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)

  50. PE for concave-of-min 2 1 1 3 4 15 1 15 2x 2 3 2 3 1 3x 4x x 3x 2 3 1 3 2 3 1 2 3 2 3 1 1 4x 3x 2 3 1 • 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)

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