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

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

1.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: 2N → 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 noS wants to deviate from the grand coalition

• Payoff vector: p=(p1, ..., p

n)

notation: p(S)= i in S pi

– pi ≥ 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 pi > 0, then p(S) < 1

Si

Multiple Coalitions

• WVGs: model only one coalition forming

• But: Why insist on players forming the grand coalition?

• Multiple coalitions multiple tasks

2

22

2 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.:

– Ui=1, ..., k

Si = N

– Si are pairwise disjoint

Payoff Distribution

• Fix CS = (S1, …, Sk). How do you distribute the payoffs from CS between agents?

• Payoff vector: p = (p1, …, pn)

– non-negative payments: pi ≥ 0 for all i = 1, …, n

– no inter-coalitional transfers:p(Sj) = v(Sj) 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:

2

2 2

2

w1 = w2 = w3 = w4 = 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=(S1, ..., S

k), can we find

a payoff vector p s.t. (CS, p) is in the CS-core?

pj ≥ 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

iLinearprogram!

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