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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
Partitioning Effort in a Social Network
1
Partitioning Effort in a Social Network
0.6
0.2
0.2
Success of Friendship/Collaboration
Success of Friendship/Collaboration
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
10 8
5
2(x+y)
4(x+y)3(x+y)
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
Strategies: Node allocates its budget among incident edges:
v contributes sv(e)0 to each e, with sv(e) Bv e
10 8
5
2(x+y)
4(x+y)3(x+y)1 4
46
35
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
Strategies: Node allocates its budget among incident edges:
v contributes sv(e)0 to each e, with sv(e) Bv e
10 8
5
22
28151 4
46
35
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)
10 8
5
22
28151 4
46
35
Stability Concepts
Nash equilibrium?
10 8
5
2xy
1000xy3xy5 0
100
08
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 (*)
General Concavece(x+y)
Min-effort convexMin-effort concave
Maximum effortApproximate Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concavece(x+y)
Min-effort convexMin-effort concave
Maximum effortApproximate Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y)
Min-effort convexMin-effort concave
Maximum effortApproximate Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convexMin-effort concave
Maximum effortApproximate Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave
Maximum effortApproximate Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effortApproximate Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effort Yes 2Approximate Equilibrium OPT is a 2-apx. Pairwise Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effort Yes 2Approximate Equilibrium OPT is a 2-apx. Pairwise Equilibrium
All Price of Anarchy upper bounds are tight
Convergence?
?
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effort Yes 2Approximate Equilibrium OPT is a 2-apx. Pairwise Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise
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
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.
10 8
5
50
100 0
8
6
60
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.
10 8
5
3
3
7
50
100 0
8
6
60
3
5
33
2
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.
10 8
5
3
3
7
50
100 0
8
6
60
3
5
33
2
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.
10 8
5
3
3
7
50
100 0
8
6
60
3
5
33
2
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.
10 8
5
3
3
7
50
100 0
8
6
60
3
5
33
2
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.
10 8
5
3
3
7
50
100 0
8
6
60
3
5
33
2
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.
10 8
5
3
3
7
50
100 0
8
6
60
3
5
33
2
PE OPT/2
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effort Yes 2Approximate Equilibrium OPT is a 2-apx. Pairwise Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effort Yes 2Approximate Equilibrium OPT is a 2-apx. Pairwise Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise
Minimum Effort Games
All functions are of the form fe(x,y)=he(min(x,y)) he is concave
Minimum Effort Games
All functions are of the form fe(x,y)=he(min(x,y)) he is concave For general concave functions, PE may not exist:
1 1
1
xy
xy
xy
Minimum Effort Games
Theorem: Pairwise equilibrium always exists in minimum effort games with continuous, piecewise differentiable, and concave he.
1 1
1
),min( yx
),min( yx
),min( yx
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.
PE for concave-of-min2 1
132
2x
4x3x
4x
3x
x
3x
PE for concave-of-min
Compute best strategy for each node v if it were able to control all other players
2 1
132
2x
4x3x
4x
3x
x
3x
PE for concave-of-min
Compute best strategy for each node v if it were able to control all other players
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
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.
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
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.
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
910
2743
110
2743
2743
4843
1
1
829
182932
29
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)
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
910
2743
110
2743
2743
4843
1
1
829
182932
29
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
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
910
2743
110
2743
2743
4843
1
1
829
182932
29
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
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
910
23
110
23
231
1
413
913
1
1
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
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
910
23
110
23
231
1
413
913
1
1
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)
2 1
132
2x
4x3x
4x
3x
x
3x
414
114
914
910
23
110
23
231
1
413
913
1
1
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)
2 1
132
2x
4x3x
4x
3x
x
3x
415
115
23
23
23
13
23
231
1
13
23
1
1
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.
2 1
132
2x
4x3x
4x
3x
x
3x
415
115
23
23
23
115
23
231
1
415
23
1
1
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.
2 1
132
2x
4x3x
4x
3x
x
3x
415
115
23
23
23
115
23
231
1
415
23
1
1
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.
2 1
132
2x
4x3x
4x
3x
x
3x
415
115
23
23
23
115
23
231
1
415
23
1
1
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.
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effort Yes 2Approximate Equilibrium OPT is a 2-apx. Pairwise Equilibrium
(*) If fe(x,0)=0, NP-Hard otherwise(**) If budgets are uniform, NP-Hard otherwise
Main Results
Existence Price of AnarchyGeneral Convex Yes (*) 2
General Concave Not always 2ce(x+y) Decision in P 1
Min-effort convex Yes (**) 2 (**)Min-effort concave Yes 2
Maximum effort Yes 2Approximate Equilibrium OPT 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!