Contribution Games in Social Networks

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)