Optimization and Stability in Games with Restricted Interactions

1. Optimization and Stability in Games with Restricted Interactions Reshef Meir, YairZick and Jeffrey S. Rosenschein CoopMAS 2012

2. Lecture content • Coalitional (TU) games • Restricted cooperation • The Cost of Stability • Main result: a bound on the CoS • Discussion

3. TU Games - Notations • Agents: N= (1,…,n) • Coalition: S µ N • Characteristic function: v: 2I→ R • A TU game is simple, if every coalition either wins or loses, i.e. v: 2I→ {0,1} • A TU game is monotone, if the value of a coalition can only increase by adding more agents to it

4. TU Games – notations (2) • A TU game is superadditive (SA), if there is positive synergy. That is, v(S[ T) ≥ v(S) + v(T) for disjoint S,T. • No need to consider coalition structures • Results can be generalized • Every game has a superadditive cover

5. Weighted Voting Games (WVG) A class of simple TU games Each agent has a weight wi2 R A game has a quota q 2 R G = [w1,w2,…,wn;q] A coalition Swins if Σi2Swi¸q

6. Payoffs • Agents may freely distribute profits. • An imputation is a vector x = (x1,…,xn) such that Σi2Nxi= v(N) • Individual rationality: each agent gets at least what she can make on her own: xi≥v({i}) • The payoff of a coalition x(S) is the sum of payments to its agents.

7. The Core • The coreis the set of all stable imputations: for all S µ N we haveΣi2S xi¸ v(S) • May be empty in many games: • No stable imputations • Example: G = [2,2,3;4] • Computational questions: • Is the core empty? • Is the vector x in the core?

8. Restricted cooperation • Some coalitions may be impossible or unlikely due to practical reasons • an underlying communication network (Myerson’77). • agents are nodes. • A coalition can form only if its agents are connected. 1 2 4 9 11 3 5 6 10 8 12 7

9. Restricted cooperation - example • The coalition {2,9,10,12} is allowed • The coalition {3,6,7,8} is not allowed 1 2 4 9 11 3 5 6 10 8 12 7

10. Restricted cooperation increases stability Theorem [Demange’04] : If the underlying communication network H is a tree, then the core is non-empty. Moreover, a core imputation can be computed efficiently. 1 2 4 9 11 3 5 6 10 8 12 7

11. What if the core is empty? • A solution: subsidies • Sometimes an external partyis interested in the stability of a specific outcome • Willing to spend money to increase stability

12. External Payments v(N) • Originally, we divided v(N)between the agents. • We increase the value of v(N), creating a “superimputation”: • Division of the incremented value v’(N)= α∙v(N) • Create a new game G(α)

13. The Cost Of Stability (CoS) (Bachrach et al., SAGT’09) • Observation: With a big enough payment, every game can be stabilized • α ≤ n • The Cost of Stability (CoS) is the minimal subsidy αthat stabilizes the grand coalition i.e. allows a non-empty core in G(α) • Can also stabilize coalition structures

14. Back to our example • G = [2,2,3;4] (core is empty) • By distributing a total payoff of 1½ (rather than 1), the core of G(1½) is non-empty. • x = (½, ½, ½) is a stable super-imputation. • Thus CoS(G) ≤ 1½ • Is this bound tight? • A lower payment cannot stabilize the game • Thus CoS(G) = 1½

15. Conceptual Issues • How do properties of the game affect the CoS? • Superadditivity, restricted cooperation, convexity… • Can we stabilize other outcomes? • A particular coalition, coalition structures…

16. Computational Issues • How hard is computing the optimal coalitional structure? • How hard is computing the CoS? • How hard is checking whether a specific super-imputation is stable? • The answer depends on game representation • We assume oracle access to v(S)

17. Previous work Bounds on the CoS • In the general case can be as high as n • For example, the WVG [1,1,1,…,1; 1] • If G is superadditive, CoS(G)≤√n • Easier to achieve cooperation • If G is superadditive and symmetric, CoS(G) ≤ 2 Bachrach et al., SAGT’09 Meir et al., SAGT ‘10

18. Previous work CoS with restricted cooperation • Recall that by [Demange’04] : if H is a tree, then the core is non-empty (i.e. CoS= 0). • Sparse graphs  lower subsidies? • Sparse graphs  easier computation? Theorem: If Hcontains a single cycle, then CoS(G) ≤ 2, and this is tight Meir et al., IJCAI ‘11

19. Graphs and tree-width • Combinatorial measures to the “cyclicity” of a graph: • Degree • Path-width • Tree-width • … • Many NP-hard combinatorial problems become easy when the tree-width is bounded. 1 2 4 9 11 3 5 10 6 8 7 1,2,3 2,4 2,5,9 5,9,10 5,6,8 6,7,8 9,11 5,8,10

20. Bounding the CoS Conjecture [MRM’11]: Let d be the maximal degree in H, then CoS(G) ≤ d There are games on a 3-dimensional grid (d = 6) with unbounded CoS Conjecture (fixed): Let k be the tree-width of H, then CoS(G) ≤ k

21. Main result Theorem: Let G be a superadditivegame, then CoS(G) ≤ (TW(H) + 1) ∙ log(n) Also, a stable payoff vector can be found efficiently

22. Proof a b x y z … a b c d a b e f c d i j b c k a d l m

23. Proof a b x y z … a b c d a b e f c d i j b c k a d l m (k+1)v(N)

24. Proof a b x y z … a b c d a b e f c d i j b c k a d l m (k+1)v(N)

25. Proof a b x y z … a b c d a b e f c d i j b c k a d l m f x z i j … (k+1)v(N) + (k+1)(v(S1) + v(S2) + …) ≤ (k+1)v(N) + (k+1)v(N) + …

26. Proof a b x y z … a b c d a b e f c d i j b c k a d l m f x z i j … We pay at most (k+1)v(N) at each iteration

27. Proof a b x y z … a b c d a b e f c d i j b c k a d l m f x z i j … We repeat at mostlog(|T |) ≤ log(n) times

28. Discussion • The CoS depends on the tree-width of the underlying graph • New results… • Bounded tree-width does not facilitate computations (e.g. Greco et al.’11)

29. Thank You For more information: http://www.huji.ac.il/~reshef24