31
A PPENDIX C
E FFICIENCY OF THE GRAND COALITION
Consider a coalitional game where the players within each coalition cooperate while different
coalitions compete. Given a coalitional structure ρ = {S1 , S2 , ..., Sl } and a set of players N =
{1, ..., r}, ρ is defined as a partition if ∀i 6= j, Si ∩ Sj = φ, and ∪li=1 Si = N [Saad, 2010]. In
the multi-agent control problem, the control objective of each coalition S ⊂ ρ is denoted as JS ,
given by
S
JS (K , K
−S
Z
∞
[xT (t)QS x(t) +
)=
t=0
X
uT
j (t)Rj uj (t)]dt
(48)
j∈S
where K S is the submatrix of the feedback matrix K that represents the strategy of the coalition
S and is given by the union of the submatrices K j in (11) associated with agents j ∈ S. Under
the sparsity constraint s, the Nash strategies of the coalitions in ρ are expressed as
∗
∗
∗
JS (K S , K −S ) ≤ JS (K S , K −S ) , ∀K S
s.t. cardoff (K) ≤ s
∗
∗
(49)
∗
Suppose Kρ = (K S1 , K S2 , ..., K Sl ) is the feedback matrix when the strategies of the coalitions in ρ = {S1 , ..., Sl } are at a Nash Equilibrium.
The value of a coalition S in the partition ρ is defined as the objective reduction of S, with
respect to the decoupled game, i.e.
vρ (S) =
X
JiD − JS (Kρ ).
(50)
i∈S
The above coalitional game is in partition form [31] since the value of each coalition depends
on the composition of other coalitions. It is shown in [31] that for coalitonal games in partition
form, the grand coalition N = {1, ..., r} forms when it is efficient, i.e., for any partition ρ, the
value of N is not exceeded by the combined values of the coalitions in ρ:
X
vN (N ) ≥
vρ (S), ∀ρ.
(51)
S⊂ρ
Next, suppose the matrices QS in (48) satisfy
X
QS = Q,
S⊂ρ
(52)
32
which is a coalition-level equivalent of (23). Then, for any partition ρ of N , the sum of the
values of the coalitions in ρ
X
vρ (S) = J˜D − J(Kρ ) ≤ J˜D − J(KN ) = vN (N )
(53)
S⊂ρ
where KN is the feedback matrix that satisfies the social optimization (6). To prove (53), note
P
that S⊂ρ JS (Kρ ) = J(Kρ ) when (52) holds, and thus Kρ represents a suboptimal solution
to (6) under the constraint s, resulting in J(Kρ ) ≥ J(KN ). Therefore, for any partition ρ, the
value of the grand coalition is at least as large as the sum of the values of the coalitions in ρ,
i.e., (51) holds, and the grand coalition is efficient, which guarantees the formation of the grand
coalition in the cooperative game and justifies Step 1 of Alg. 3 (social optimization) under the
assumption (52).
REFERENCE
[Saad, 2010] W. Saad, “Coalitional Game Theory for Distributed Cooperation in Next Generation Wireless Networks,” Ph.D.
dissertation, University of Oslo, 2010.