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: 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 no
S wants to deviate from the grand coalition
• Payoff vector: p=(p1, ..., pn)
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: w1, ..., wn
– 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
i
S
Multiple Coalitions
• WVGs: model only one coalition forming
• But: Why insist on players forming the grand
coalition?
• Multiple coalitions  multiple tasks
2
2
2
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 S1, ..., Sk, 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  N
In WVGs :
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)
WVGs with Coalition Structures
• Different nature of stability, more payoff to
distribute:
w1 = w2 = w3 = w4 = 2, T = 4
2
2
2
2
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, ..., Sk), can we find
a payoff vector p s.t.
(CS, p) is in the CS-core?
pj ≥ 0 for j = 1, ..., n
Linear
 j in S i pj = v(Si) for i = 1, ..., k
program!
 j in T pj ≥ v(T) for all T  N
• Exponentially many constraints,
but has a separation oracle - ellipsoid method
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