1. 11 The Cost of Stability in Network Flow Games Ezra Resnick Yoram Bachrach Jeffrey S. Rosenschein

2. 22 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

3. 33 Cooperative games A set of agents N A characteristic function v: 2 N → 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

4. 44 Threshold Network Flow Games (TNFGs) A TNFG is defined by a flow network and 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

5. 55 TNFG example Threshold: 3 a bts c 2 1 1 2 1 1

6. 66 TNFG winning coalition Threshold: 3 a bts c 2 1 1 2 1 1

7. 77 TNFG losing coalition Threshold: 3 a bts c 2 1 1 2 1 1

8. 88 Distributing coalitional gains Imputation: a distribution of the grand coalition’s gains among the agents p a is the payoff of agent a: is the payoff of a coalition C Solution concepts define criteria for imputations Individual rationality:

9. 99 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

10. 10 The core of a TNFG Threshold: 3 In a simple game, the core consists of imputations which divide all gains among the veto agents a bts c 2 1 1 2 1 1 0.5 00 00

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

12. 12 Supplemental payment An external party offers the grand coalition a supplemental payment Δ 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

13. 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}

14. 14 CoS in TNFG example Threshold: 2 Q. What is the CoS? a bts c 2 1 1 2 1 1 10 10 00 A. 2

15. 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|

16. 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

17. 17 CoS in connectivity games a bts c d e

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

19. 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. 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. 21 CoS in equal capacity TNFGs b = 1, r = 2, f = 3 CoS = 1.5 Threshold: 2 a bts c 1 1 1 1 1 1

22. 22 Serial TNFGs st 1 2 2 3 1 st 1 3 1 3 1 1

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

24. 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. 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. 26 CoS in bounded serial TNFGs Proof: Describe the CoS of each component TNFG as a linear program Minimize: Constraints:

27. 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. 28 TNFG super-imputation stability Threshold: b Super-imputation p gives an edge with capacity a i a payoff of v1v1 v2v2 ts vnvn … a1a1 a2a2 anan a1a1 a2a2 anan

29. 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