Coalition Structures in Weighted Voting Games Georgios Chalkiadakis Edith Elkind Nicholas R. Jennings

What is this paper all about? 1.We introduce WVGs with coalition structures 2.We define the CS-core for such games  Show correspondences between core/CS-core elements & existence for specific classes of games [see paper] 3.We obtain various complexity results for CS-core non-emptiness and membership  NP-hard as opposed to normal WVG setting 4.We propose algorithms and tractable heuristics to check CS-core membership and non-emptiness

Coalitional Games Non-cooperative games: players choose actions to obtain outcomes that maximize individual utility Cooperative (coalitional TU) games: players form coalitions and distribute payoffs resulting from coalitional actions… …but still selfish…

Coalitional Games: Formal Setup Set of agents N, |N|=n Characteristic function v: 2 N → R –v(S): value of coalition S –intuition: agents in S can collaborate to achieve v(S) How should (selfish) agents distribute payoffs?

Stability Core: distribute the value of N so that no S wants to deviate from the grand coalition Payoff vector: p=(p 1,..., p n ) notation: p(S)=  i in S p i –p i ≥ 0 for all i = 1,..., n –p(N) = v(N) p is in the core if p(S) ≥ v(S) for all S

Weighted Voting Games Intuition: –agents possess resources –need a certain amount of resources to complete a task Model: –agents have weights: w 1,..., w n –threshold T –a coalition S is winning ( v(S) = 1) if w(S) ≥ T and losing ( v(S) = 0 ) otherwise

Stability in WVGs? Given a WVG G = (w; T), can we find its core? Yes, but it may be empty... Claim: G has an empty core unless there is a veto player [EGGW2007] if p i > 0, then p(S) < 1 S i

Multiple Coalitions WVGs: model only one coalition forming But: Why insist on players forming the grand coalition? Multiple coalitions  multiple tasks T = 4

Coalition Structures Need to formally model several coalitions forming simultaneously  CSs arise (see, e.g., [Aumann&Dreze74]) Given a game G=(N, v), a coalition structure CS is a partition of N into S 1,..., S k, i.e.: –U i=1,..., k S i = N –S i are pairwise disjoint

Payoff Distribution Fix C S = (S 1, …, S k ). How do you distribute the payoffs from CS between agents? Payoff vector: p = (p 1, …, p n ) –non-negative payments: p i ≥ 0 for all i = 1, …, n –no inter-coalitional transfers: p(S j ) = v(S j ) for j = 1, …, k

WVGs with Coalition Structures We use the coalition structures framework in WVGs, and study stability Core with coalition structures (CS, p) is in the CS-core iff p(S) ≥ v(S) for all S  Definition 3. The CS-core of a WVG game G = (N ; w; T ) with coalition structures is the set of outcomes (CS, p) such that ∀ S ⊆ N, w(S) ≥ T ⇒ p(S) ≥ 1 and ∀ C ∈ CS it holds p(C) = v(C) In WVGs :

WVGs with Coalition Structures Different nature of stability, more payoff to distribute: w 1 = w 2 = w 3 = w 4 = 2, T = 4 Core is empty, but CS-core is not: ({1, 2}, {3, 4}), ( ½, ½, ½, ½) is in the CS-core It is thus “easier” to ensure stability …while serving multiple tasks

Stability in WVGs: Computational Issues Standard representation: weights are integers given in binary Checking non-emptiness of the core: –core: poly-time –CS-core: NP-hard (this paper) Checking if an outcome is in the core: –core: poly-time –CS-core: coNP-complete (this paper)

Stability in WVGs: Small Weights Suppose all weights are polynomial in n –alternatively, given in unary –Realistic in many scenarios Can we check if (CS, p) is in the CS-core? Is there an S s.t. w(S) ≥ T, p(S) < 1? …thus, reducible to Knapsack => poly-time solvable for small weights

Can We Make a Given CS Stable? Given a CS=(S 1,..., S k ), can we find a payoff vector p s.t. (CS, p) is in the CS-core? p j ≥ 0 for j = 1,..., n  j in S p j = v(S i ) for i = 1,..., k  j in T p j ≥ v(T) for all T  N Exponentially many constraints, but has a separation oracle - ellipsoid method i Linear program!

Back to Checking Non-emptiness of the CS-core Can we check non-emptiness of the CS-core for small weights? Seems hard....(and actually is!) Can try all coalition structures and check if there is a matching payoff vector –exponential in n –…but can prune using heuristics (see paper)

Conclusions WVGs with coalition structures: a richer model for resource allocation than ordinary WVGs Unlike in ordinary WVGs, checking stability is hard For small weights, can check if an outcome is in the CS-core Exp-time algorithm with heuristic improvements for checking non-emptiness of the CS-core