The Cost of Stability in Network Flow Games

1. The Cost of Stability in Network Flow Games Ezra Resnick Yoram Bachrach Jeffrey S. Rosenschein 1

2. Overview • Goal: In cooperative games, distribute the grand coalition’s gains among the agents in a stable manner • This is not always possible (empty core) • Stabilize the game using an external payment • Cost of Stability: minimal necessary external payment to stabilize the game • Focus on Threshold Network Flow Games 2

3. Cooperative games • A set of agents N • A characteristic function v: 2N → R • the utility achievable by each coalition of agents • Example: • N = {1,2,3} • v(Φ) = v(1) = v(2) = v(3) = 0 • v(1,2) = v(1,3) = v(2,3) = 2 • v(1,2,3) = 3 3

4. Threshold Network Flow Games (TNFGs) • A TNFG is defined by a flow networkand a threshold value • Each agent controls an edge • The utility of a coalition is 1 if the flow it allows from source to sink reaches the threshold, 0 otherwise • TNFGs are simple, increasing games 4

5. TNFG example Threshold: 3 a 2 2 1 1 s b t 1 1 c 5

6. TNFG winning coalition Threshold: 3 a 2 2 1 1 s b t 1 1 c 6

7. TNFG losing coalition Threshold: 3 a 2 2 1 1 s b t 1 1 c 7

8. Distributing coalitional gains • Imputation: a distribution of the grand coalition’s gains among the agents • pa is the payoff of agent a: • is the payoff of a coalition C • Solution concepts define criteria for imputations • Individual rationality: 8

9. The core • Coalitional rationality • A coalition C blocks an imputation p if • An imputation p is stable if it is not blocked by any coalition: • The core is the set of all stable imputations 9

10. The core of a TNFG a 2 2 0.5 0.5 1 1 s b t 0 0 1 1 0 c 0 Threshold: 3 In a simple game, the core consists of imputations which divide all gains among the veto agents 10

11. A TNFG with an empty core a 2 2 1 1 s b t 1 1 c Threshold: 2 If a simple game has no veto agents then the core is empty 11

12. Supplemental payment • An external party offers the grand coalition a supplemental paymentΔ if all agents cooperate • This produces an adjusted game • v(N) + Δ are the adjusted gains • A distribution of the adjusted gains is a super-imputation 12

13. The Cost of Stability (CoS) • The core of the adjusted game may be nonempty – if Δ is large enough • The Cost of Stability:CoS = min {v(N) + Δ : the core of the adjusted game is nonempty} 13

14. CoS in TNFG example a 2 2 1 0 1 1 s b t 1 0 1 1 0 c 0 Threshold: 2 Q. What is the CoS? A. 2 14

15. CoS in simple games • Theorem: If a simple game contains m pairwise-disjoint winning coalitions, then CoS ≥ m • Theorem: In a simple game, if there exists a subset of agents S such that every winning coalition contains at least one agent from S, then CoS ≤ |S| 15

16. Connectivity games • A connectivity game is a TNFG where all capacities are 1 and the threshold is 1 • A coalition wins iff it contains a path from source to sink • Theorem: The CoS of a connectivity game equals the min-cut (and max-flow) of the network 16

17. CoS in connectivity games a d s b t e c 17

18. CoS in connectivity games a d s b t e c CoS = min-cut = max-flow = 2 18

19. CoS in TNFG – upper bound • Theorem: If the threshold of a TNFG is k and the max-flow of the network is f, then CoS≤ f/k • Proof: Find a min-cut, and pay each c-capacity edge in the cut c/k • This gives a stable super-imputation with adjusted gains of f/k • f/k can serve as an approximation of the CoS (useful if the ratio f/k is small) 19

20. CoS in equal capacity TNFGs • Theorem: If all edge capacities in a TNFG equal b, and the threshold is rb(r ∈ N), and f is the max-flow of the network, then CoS = f/rb • Connectivity games are a special case (r = b = 1) • Proof: We already know that CoS ≤ f/rb, so it suffices to prove CoS ≥ f/rb… 20

21. CoS in equal capacity TNFGs b = 1, r = 2, f = 3 CoS = 1.5 a 1 1 1 1 s b t 1 1 c Threshold: 2

22. Serial TNFGs 1 1 1 1 2 s t s t 1 2 3 3 3 1

23. Serial TNFGs 1 1 1 1 2 s t 1 2 3 3 3 1

24. CoS in serial TNFGs • Theorem: The CoS of a serial TNFG equals the minimal CoS of any of the component TNFGs • Proof: Show that a super-imputation which is stable and optimal in the component with the minimal CoS is also a stable and optimal super-imputation for the entire series

25. CoS in bounded serial TNFGs • Theorem: If the number of edges in each component TNFG is bounded, then the CoS of a serial TNFG can be computed in polynomial time • Runtime will be linear in the number of components, but exponential in the number of edges in each component

26. CoS in bounded serial TNFGs • Proof: Describe the CoS of each component TNFG as a linear programMinimize:Constraints:

27. TNFG super-imputation stability • TNFG-SIS: Given a TNFG, a supplemental payment, and a super-imputation p in the adjusted game, determine whether p is stable • Theorem: TNFG-SIS is coNP-complete • Proof: Reduction from SUBSET-SUM

28. TNFG super-imputation stability • Threshold: b • Super-imputation p gives an edge with capacity ai a payoff of a1 v1 a1 a2 a2 v2 s t an an … vn

29. Summary • CoS defined for any cooperative game • coNP-complete to determine whether a super-imputation in a TNFG is stable • For any TNFG, CoS ≤ max-flow/threshold • CoS in special TNFGs: • Connectivity games • Equal capacity TNFGs • Serial TNFGs