Characterization Sets for the Nucleolus in
Balanced Games
1,2
Tamás Solymosi
and Balázs Sziklai
2
Corvinus University of Budapest, Department of Operations Research and
Actuarial Sciences, H-1093 Budapest F®vám tér 8. [email protected]
2
Institute of Economics, Research Center for Economic and Regional Studies,
Hungarian Academy of Sciences, H-1112, Budapest, Budaörsi út 45.
[email protected]
1
We provide a new modus operandi for the computation of the
nucleolus in cooperative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually
essential and if the game is monotonic dually saturated coalitions determine both the core and the nucleolus whenever the core is non-empty.
We show how these two sets are related to the existing characterization
sets. In particular we prove that if the grand coalition is vital then the
intersection of essential and dually essential coalitions forms a characterization set itself.
Abstract.
Keywords: Cooperative game theory, Nucleolus, Characterization sets
1
Introduction
The nucleolus, developed by Schmeidler (1969), soon became one of the most
frequently applied solution concepts of cooperative game theory. Despite its good
properties it lost some popularity in the last 20 or so years. Very much like the
Shapley-value it suers from computational diculties. While the former has an
explicit formula and various axiomatizations, the nucleolus practically can only
be computed by a sequence of linear programs and its axiomatization is less
straightforward.
N P -hard for
N P -hardness was proven for minimum cost span-
Determining the nucleolus is a notoriously hard problem, even
various classes of games. While
ning tree games (Faigle, Kern, and Kuipers, 1998), voting games (Elkind, Goldberg, Goldberg, and Wooldridge, 2009) and ow and linear production games
(Deng, Fang, and Sun, 2009), it is still unknown whether the corresponding decision problem i.e. verifying whether an allocation is the nucleolus or not belongs to
N P.
In recent years several polynomial time algorithms were proposed to nd
the nucleolus of important families of cooperative games, like standard tree, assignment, matching and bankruptcy games (Maschler, Potters, and Reijnierse,
2010; Solymosi and Raghavan, 1994; Kern and Paulusma, 2003; Aumann and
Maschler, 1985). In addition Kuipers (1996) and Arin and Inarra (1998) developed methods to compute the nucleolus for convex games.
The main breakthrough came from another direction. In their seminal paper
Maschler, Peleg, and Shapley (1979) described the geometric properties of the
nucleolus and devised a computational framework in the form of a sequence of
linear programs. Although these LPs consist of exponentially many inequalities
they can be solved eciently if one knows which constraints are redundant. Huberman (1980); Granot, Granot, and Zhu (1998); Reijnierse and Potters (1998)
provided methods to identify coalitions that correspond to non-redundant constraints.
Granot, Granot, and Zhu (1998) provided the most fruitful approach. They
introduced the concept of characterization set which is a collection of coalitions that determines the nucleolus by itself. They proved that if the size of the
characterization set is polynomially bounded in the number of players, then the
nucleolus of the game can be computed in strongly polynomial time. A collection
that characterizes the nucleolus in one game need not characterize it in another
one. Thus we are interested in a property of the characteristic function that
describes a characterization set in every game.
Huberman (1980) was the rst to show that such a property exists. He introduced the concept of essential coalitions which are coalitions that have no
weakly minorizing partition. Granot, Granot, and Zhu (1998) provided another
collection that characterize the nucleolus in cost games with non-empty cores.
Saturated coalitions contain all the players that can join the coalition without
imposing extra cost.
We introduce two new characterization sets: dually essential and dually saturated coalitions. We show that each dually inessential coalition has a weakly
minorizing overlapping decomposition which consists exclusively of dually essential coalitions. Thus dually essential coalitions determine the core, and if the core
is non-empty they determine the nucleolus as well. If every player contributes
to the value of a coalition then such coalition is called dually saturated. We
show that dually saturated coalitions also determine the core, and if the core is
non-empty, then also the nucleolus of a TU-game.
The larger a characterization set is the easier to uncover it in a particular
game class. However with smaller characterization set it comes a faster LP. Hence
there is a tradeo between the diculty in identifying the members of a characterization set and its eciency. In order to exploit this technique we analyze the
relationship of the four known characterizing properties. We prove that essential
coalitions are a subset of dually saturated coalitions in monotonic prot games
and that dually essential coalitions are a subset of saturated coalition in case of
monotonic cost games. We show that in general essential and dually essential
coalitions do not contain each other. In fact for additive games their intersection
is empty. We prove that if the grand coalition is vital then the intersection of
essential and dually essential coalitions forms a characterization set itself.
2
Game theoretical framework
cooperative game with transferable utility
is an ordered pair (N, v) consisting
N = {1, 2, . . . , n} and a characteristic function v : 2N → R with
v(∅) = 0. The value v(S) represents the worth of coalition S . No matter how
other players behave if the players of S work together they can secure themselves
v(S) amount of payo. The set N when viewed as a coalition is called the
N
grand coalition. Let P = 2 \ {∅, N } denote the family of the proper coalitions.
A cooperative game (N, v) is called monotonic if
A
of the player set
S ⊆ T ⊆ N ⇒ v(S) ≤ v(T ),
and
superadditive
if
(S, T ∈ P, S ∩ T = ∅) ⇒ v(S) + v(T ) ≤ v(S ∪ T ).
In a superadditive game two disjoint coalitions can always merge without
losing money. Hence we shall assume that players form the grand coalition. The
main question is then how to distribute
v(N )
among the players in some fair
way.
A solution for a cooperative game
Γ = (N, v)
x ∈ RN
is a vector
that
represents the payo of each player. For convenience, we introduce the following
P
i∈S xi for any S ⊆ N , and instead of x({i}) we simply
x(i). A solution is called ecient if x(N ) = v(N ) and individually rational
if x(i) ≥ v(i) for all i ∈ N . Ecient solutions will also be called allocations.
The imputation set I(Γ ) of the game Γ consists of the ecient and individually
notations
x(S) =
write
rational solutions, formally,
I(Γ ) = {x ∈ RN
Given a payo vector
game
Γ
| x(N ) = v(N ), x(i) ≥ v(i)
x ∈ RN ,
we dene the
for all
satisfaction
i ∈ N }.
of coalition
S
in
as
satΓ (S, x) := x(S) − v(S).
The satisfaction of the grand coalition is clearly zero at any allocation. The core
C(Γ )
of cooperative game
Γ
is the set of allocations where all the satisfaction
values are non-negative. Formally,
C(Γ ) = {x ∈ RN | satΓ (N, x) = 0, satΓ (S, x) ≥ 0
A game is called
balanced
for all
S ∈ P}.
if its core in non-empty. In this paper we will
consider only balanced games. A cooperative game
(N, v)
is called
convex
if the
characteristic function is supermodular i.e.
v(S) + v(T ) ≤ v(S ∪ T ) + v(S ∩ T ),
∀ S, T ⊆ N.
Deriving results for convex games is a less challenging task than in general due
to their nice structure. Here we only mention a result of Shapley (1971), namely
that the core of a convex game is not empty.
In many economic situations cooperation results in cost saving rather than
prot growth. For instance such situation occurs when customers would like to
gain access to some public service or public facility. The question is then, how to
share the costs of the service. Cost allocation games can be modeled in a similar
fashion as cooperative TU-games.
A cooperative cost game is an ordered pair (N, c) consisting of the player set
N = {1, 2, . . . , n} and a characteristic cost function c : 2N → R with c(∅) = 0.
The value c(S) represents the amount of cost coalition S must bear if it chooses
to act separately from the rest of the players. In most cases c is monotonic and
subadditive. That is, the more people use the service the more it costs, however
there is also an increase of eciency.
A cost game
(N, c)
is called
subadditive
if
(S, T ∈ P, S ∩ T = ∅) ⇒ c(S) + c(T ) ≥ c(S ∪ T ),
and
concave
if
c(S) + c(T ) ≥ c(S ∪ T ) + c(S ∩ T ),
It is possible to associate a prot game
the
∀ S, T ⊆ N.
(N,
P v) with a cost game (N, c), called
i∈S c(i) − c(S) for all S ⊆ N . Note
savings game, which is given by v(S) =
that a cost game is subadditive (concave) if and only if the corresponding savings
game is superadditive (convex). Similarly as we will see the relationship
is reversed in all the previously listed concepts such as individual rationality,
satisfaction, and the core. Formally, let
an arbitrary allocation. The
satisfaction
Γ = (N, c)
be a cost game and
of coalition
S
in
Γ
x ∈ RN
is dened as
satΓ (S, x) := c(S) − x(S).
As we pointed out in case of cost games the order of the characteristic function
satΓ depends on whether
satΓ (S, x) expression does
not change. It still indicates the contentment of coalition S under payo vector
x. Accordingly, also in a cost game Γ = (N, c) we say that x is an allocation if
satΓ (N, x) = 0, individually rational if satΓ (i, x) ≥ 0 for all i ∈ N , and x is a
core vector if it is an allocation and satΓ (S, x) ≥ 0 for all S ∈ P . Notice that
and the payo vector is reversed, hence the formula for
Γ
is a cost or prot game. However, the essence of the
core vectors of monotonic prot and cost games are non-negative. Indeed,
x(i) = x(N ) − x(N \ i) ≥ c(N ) − c(N \ i) ≥ 0
x and i ∈ N . For monotonic prot games this is even
x(i) ≥ v(i) ≥ v(∅) = 0.
m
We say that a vector x ∈ R
lexicographically precedes y ∈ Rm (denoted by
x y ) if either x = y or there exists a number 1 ≤ j ≤ m such that xi = yi
P
2n −2
if i < j and xj < yj . Let Γ = (N, v) be a game and let θ (x) ∈ R
be
n
the vector that contains the 2 − 2 satisfaction values satΓ (S, x), S ∈ P , in a
for any core allocation
more trivial since
non-decreasing order.
Denition 1. The
nucleolus of a cooperative game Γ with respect to the set
X ⊆ RN is the subset of those payo vectors x ∈ X that
maximize θP (x) over X . Formally,
of payo vectors
lexicographically
N(Γ, X) = {x ∈ X | θP (y) θP (x) f or all y ∈ X}.
X is nonempty and compact then
N(Γ, X) consists of a single point.
It is well known (Schmeidler, 1969) that if
N(Γ, X) 6= ∅
and if
X
is convex as well then
Furthermore the nucleolus is a continuous function of the characteristic function.
X is chosen to be the set of allocations in Γ , we speak of the prenucleolus of
Γ , if X is the set of imputations I(Γ ) then we speak of the nucleolus of Γ .
Throughout the paper we will use the shorthand notation N(Γ ) for N(Γ, I(Γ )).
For balanced games the prenucleolus coincides with the nucleolus, and N(Γ ) ⊆
C(Γ ).
If
3
Characterization sets
The concept of characterization sets was already used by Megiddo (1974), but
somehow went unnoticed at that time. Later Granot, Granot, and Zhu (1998)
and Reijnierse and Potters (1998) re-introduced the idea almost simultaneously.
It is remarkable that two such closely related and important papers appeared
in the same year. Here we will use the formalism of Granot, Granot, and Zhu
(1998).
Denition 2. Let Γ F = (N, F, v) be a cooperative game with coalition formation restrictions, where ∅ 6= F ⊆ P consists of all coalitions deemed permissible
besides the grand coalition N . Let θF (x) ∈ R|F | be the restricted vector that
contains the satisfaction values satΓ (S, x), S ∈ F in a non-decreasingly order.
Furthermore let N(Γ F ) be dened as the set of imputations that lexicographically
maximizes θF (x). Then F is called a characterization set for the nucleolus of
the game Γ = (N, v), if N(Γ F ) = N(Γ ).
Note the dierent roles of
N
and the coalitions in
F
in the above optimiza-
tion. The satisfaction of the grand coalition is required to be constantly zero
(partly dening the feasible set), whereas the satisfactions of the smaller per-
N(Γ F ) is a singleton
N
if and only if rank({eS : S ∈ {N } ∪ F }) = n, where eS ∈ {0, 1}
denotes the
membership vector of coalition S given by (eS )i = 1 if i ∈ S and (eS )i = 0
missible coalitions form the objective function. Notice that
otherwise.
The main result of Granot, Granot, and Zhu (1998) is presented in the following theorem.
Theorem 1. (Granot, Granot, and Zhu, 1998) Let Γ be a cooperative (cost or
prot) game, F ⊆ P a non-empty collection, and x an element of the nucleolus
of Γ F . Collection F is a characterisation set for the nucleolus of Γ if for every
S ∈ P \ F there exists a non-empty subcollection FS of F , such that
i. satΓ (T, x) ≤ satΓ (S, x) for every T ∈ FS ,
ii. eS can be expressed as a linear combination of the vectors in
FS ∪ {N }}.
{eT : T ∈
Observe that the above conditions are sucient but not at all necessary.
Take for example the (superadditive and balanced) prot game with four players
N = {1, 2, 3, 4} and the following characteristic function: v(i) = 0, v(i, j) = 1,
v(i, j, k) = 3 for any i, j, k ∈ N and v(N ) = 4. Then the 2-player coalitions and
the grand coalition are sucient to determine the nucleolus, which is given by
z(i) = 1
for all
values at
z,
i ∈ N.
However, the 3-player coalitions have smaller satisfaction
thus the rst condition of Theorem 1 is violated. Notice that in
this game the 3-player coalitions and the grand coalition are also sucient to
determine the nucleolus.
In general neither the 2-player nor the 3-player coalitions (and the grand
coalition) characterize the nucleolus. The fact that in this example they did was
due to the particular choice (most importantly the symmetry) of the coalitional
function. We would like to identify properties of coalitions that characterize the
nucleolus independently of the realization of the coalitional function, at least for
a large class of games.
A straightforward corollary of Theorem 1 is that we can enlarge characterisation sets arbitrarily.
Corollary 1. Let F ⊂ P be a characterisation set that satises both conditions
of Theorem 1. Then T is a characterisation set for any F ⊂ T ⊆ P .
Now we focus on characterization sets for balanced games. As Granot, Granot, and Zhu (1998) remark, in order to apply Theorem 1 we need not nd an
allocation in the nucleolus of the restricted game
of
F
N(Γ )
ΓF
if we know a superset
such that condition (i) holds for all elements of
Y.
Y
Since core alloca-
tions provide non-negative satisfaction for all coalitions, the following corollary
of Theorem 1 is easily seen.
Corollary 2. Let Γ be a cooperative (cost or prot) game with a non-empty
core. The non-empty collection F ⊆ P is a characterisation set for the nucleolus
of Γ if for every S ∈ P \ F there exists a non-empty subcollection FS of F , such
that
i. satΓ (T, x) ≤ satΓ (S, x) for all x ∈ C(Γ ), whenever T ∈ FS ,
ii. eS can be expressed as a linear combination of the vectors in
FS ∪ {N }}.
{eT : T ∈
The rst characterization set for balanced games is due to Huberman (1980).
Denition 3 (Essential coalitions). Let N be a set of players, (N, v) a prot,
(N, c) a cost game. Coalition S 6= ∅ is called
essential in prot game Γ = (N, v)
.
.
if it can not be partitioned as S = S1 ∪ . . . ∪ Sk with k ≥ 2 and Sj 6= ∅ for all
1 ≤ j ≤ k such that
v(S) ≤ v(S1 ) + . . . + v(Sk ).
Similarly, coalition S 6= ∅ is called essential in cost game Γ = (N, c) if it can
not be partitioned
as
.
.
S = S1 ∪ . . . ∪ Sk with k ≥ 2 and Sj 6= ∅ for all 1 ≤ j ≤ k such that
c(S) ≥ c(S1 ) + . . . + c(Sk ).
The set of essential coalitions is denoted by E(Γ ), where Γ is either (N, v) or
(N, c). A coalition that is not essential is called inessential.
Notice that the grand coalition may or may not be essential. On the other
hand, by denition, the singleton coalitions are always essential in every game.
It is easily seen that in a prot / cost game each inessential coalition has a
weakly majorizing / weakly minorizing partition which consists exclusively of
essential coalitions. Moreover, the core is determined by the eciency equation
satΓ (N, x) = 0 and the satΓ (S, x) ≥ 0 inequalities corresponding to the essential
coalitions, all the other inequalities can be discarded from the core system.
The following important result is due to Huberman (1980).
Theorem 2. (Huberman, 1980) If the core of the game is non-empty then the
essential coalitions form a characterization set for the nucleolus.
The theorem means that in a balanced game the essential coalitions and the
grand coalition are sucient to determine the nucleolus. This observation helps
us to eliminate large coalitions which are redundant for the nucleolus. To detect
small coalitions that are unnecessary for the nucleolus, we need the concept of
dual game.
Denition 4. The dual game Γ ∗ = (N, g∗ ) of game Γ = (N, g) is dened by
the coalitional function g∗ (S) := g(N ) − g(N \ S) for all S ⊆ N , where Γ is
either (N, v) or (N, c).
Notice that
g ∗ (∅) = 0, g ∗ (N ) = g(N ),
and
(g ∗ )∗ (S) = g(S)
for all
S ⊆ N.
It
will be useful to think of the dual game of a prot game as a cost game and vice
g is monotonic if and only if g ∗ is monotonic,
∗
and g is supermodular if and only if g is submodular. However, there is no such
versa. It can be easily checked that
dualization relation between superadditivity and subadditivity.
We can identify small redundant coalitions, if we apply Huberman's argument
to the dual game.
Denition 5 (Dually essential coalitions). Let N be a set of players, (N, v)
a prot, (N, c) a cost game. Coalition S 6= ∅, N is called dually essential
in game
.
.
(N, v) if its complement can not be partitioned as N \S = (N \T1 ) ∪. . . ∪(N \Tk )
with k ≥ 2 and Tj 6= N for all 1 ≤ j ≤ k such that
v ∗ (N \ S) ≥ v ∗ (N \ T1 ) + . . . + v ∗ (N \ Tk ),
or equivalently,
v(S) ≤ v(T1 ) + . . . + v(Tk ) − (k − 1)v(N ).
Similarly, S 6= ∅, N is called dually essential in cost
game
(N, c) if its comple.
.
ment can not be partitioned as N \ S = (N \ T1 ) ∪ . . . ∪ (N \ Tk ) with k ≥ 2 and
Tj 6= N for all 1 ≤ j ≤ k such that
c∗ (N \ S) ≤ c∗ (N \ T1 ) + . . . + c∗ (N \ Tk ),
or equivalently,
c(S) ≥ c(T1 ) + . . . + c(Tk ) − (k − 1)c(N ).
The set of dually essential coalitions is denoted by DE(Γ ), where Γ is either
(N, v) or (N, c). A coalition that is not dually essential is called dually inessential.
Notice that, by denition, the grand coalition is not dually essential. On the
(n−1)-player coalitions are dually essential in any game. If S ∈ P
T1 , . . . , Tk ∈ P
expression, but every player in N \ S appears exactly k − 1 times
other hand, all
is dually inessential then it is contained in each of the coalitions
in the above
in this family. We call such a system of coalitions an
of
S.
overlapping decomposition
For a more general denition, where the complements of the overlapping
coalitions need not form a partition of the complement coalition, see e.g. Brânzei,
Solymosi, and Tijs (2005) and the references therein.
It is clear that if
S, T ∈ P
are dually inessential coalitions and
in an overlapping decomposition of
overlapping decomposition of
S,
then
S ( T
so
S
T
appears
cannot appear in an
T . Consequently, in a prot / cost game each dually
inessential coalition has a weakly majorizing / weakly minorizing overlapping
decomposition which consists exclusively of dually essential coalitions. Moreover,
Γ is also determined by the dual eciency equation satΓ ∗ (N, x) = 0
satΓ ∗ (S, x) ≥ 0 dual inequalities corresponding to the complements of
the core of
and the
the dually essential coalitions, all the other dual inequalities can be discarded
from the dual core system.
The main feature of dually essential coalitions for the nucleolus is that together with the grand coalition they are sucient to determine the nucleolus in
balanced games. The next theorem is the dual counterpart of Theorem 2.
Theorem 3. If C(Γ ) 6= ∅, then the dually essential coalitions form a characterization set for N(Γ ).
Proof.
There are various ways to derive this result. A formal proof can be ob-
tained by applying the arguments of Huberman (1980) to the dual game. Here
we pursue another way and deduce it from Corollary 2. We only give the proof
for prot games, for cost games it is analogous.
Let S ∈ P be a dually inessential coalition in the balanced prot game
Γ = (N, v). As remarked earlier, S has a weakly majorizing overlapping decomposition T1 , . . . , Tk (k ≥ 2) which consists exclusively of dually essential
coalitions. Hence ii. of Corollary 2 follows immediately. To see condition i., let
x ∈ C(Γ ).
Then
v(S) ≤ v(T1 ) + . . . + v(Tk ) − (k − 1)v(N )
v(S) − x(S) ≤ v(T1 ) + . . . + v(Tk ) − (k − 1)x(N ) − x(S)
−satΓ (S, x) ≤ −(satΓ (T1 , x) + . . . + satΓ (Tk , x))
satΓ (S, x) ≥ satΓ (T1 , x) + . . . + satΓ (Tk , x)
satΓ (S, x) ≥ satΓ (Tj , x) ≥ 0 ∀j = 1, . . . , k,
v(N ) = x(N ), while the third from
x(T
)
+
.
.
.
+
x(T
)
=
(k
−
1)x(N
) + x(S) implied by N \ S =
1
k
.
.
(N \T1 )∪. . .∪(N \Tk ), and the last one from the non-negativity of the satisfaction
where the second inequality comes from
the identity
values at any core allocation.
If we look for an as small characterisation set as possible for balanced games,
in light of Theorems 2 and 3 it seems natural to eliminate both inessential
coalitions and dually inessential coalitions at the same time. However, this might
not be possible, the intersection of essential and dually essential coalitions need
not yield a characterization set. For example, let
Γ = (N, v)
be an additive
P
i∈S v(i) for all S ⊆ N . Then only the singleton coalitions
∗
are essential, and since v = v , only the (n − 1)-player coalitions are dually
game, that is
v(S) =
essential. Thus
E(Γ ) ∩ DE(Γ ) = ∅
that cannot characterize the nucleolus in any
game. The problem lies in what we call a
when coalition
S
cycle in the decomposition 3 . This occurs
can be discarded because of a collection that contains coalition
T , and T
can also be discarded because of a collection that contains
coalition
R
S
or another
that can be eliminated because of a collection that contains
S,
etc..
As remarked after both denitions, such a cycle in the decomposition cannot
occur if we apply only essentiality or only dually essentiality. Before we show
when the two types of concepts can be mixed we need some preparation.
BS ⊆ P is P
said to be S -balanced if there exist
λT , T ∈ BS , such that
T ∈BS λT eT = eS . An N -balanced
collection is simply called balanced. A coalition S is called vital if for any S balanced collection BS and any system (λT )T ∈BS of balancing weights for BS ,
P
T ∈BS λT v(T ) < v(S). Trivially, every vital coalition is essential, but the conA collection of coalitions
positive weights
verse doesnot hold. The concept was introduced by Gillies (1959) and further
analyzed by Shellshear and Sudhölter (2009). It is easily seen that the family of vital coalitions (just like essential coalitions) and the grand coalition are
sucient to determine the core. However, Maschler, Peleg, and Shapley (1979)
demonstrated that vital coalitions (unlike essential coalitions) and the grand
coalition do not necessarily characterize the nucleolus in a balanced game.
The next theorem provides a sucient condition for
E(Γ ) ∩ DE(Γ )
to be a
characterization set for the nucleolus in a balanced game.
3
Strongly essential coalitions discussed by Brânzei, Solymosi, and Tijs (2005) are not
immune to this kind of failure, hence they might not form a characterization set for
the nucleolus in a balanced game in which the grand coalition is not vital.
Theorem 4. Let Γ = (N, v) be a game with a non-empty core. If the grand
coalition is vital, the collection E(Γ ) ∩ DE(Γ ) forms a characterization set of
N(Γ ) .
Proof.
Suppose coalition
S
is redundant, that is, either inessential or dually
S1 , . . . , Sk in E(Γ ) or
DE(Γ ) that have smaller satisfaction value than S and span eS . Lexicographically maximizing the satisfactions of S1 , . . . , Sk the satisfaction of S becomes
inessential (or both). Then there is a series of coalitions
in
xed. If there is no cycle in the decomposition then there exists a decomposition
Si for i = 1, . . . , k that consist entirely of coalitions T1i , . . . , Trii that belong
i
i
to E(Γ ) ∩ DE(Γ ). Lexicographically maximizing the satisfactions of T1 , . . . , Tr
i
the satisfaction of Si becomes xed. Thus indirectly the satisfaction of S beof
comes xed. We conclude that, if there is no cycle in the decomposition, then
E(Γ ) ∩ DE(Γ )
by Theorem 1
is a characterization set.
By contradiction suppose that the grand coalition is vital, but there is a cycle
T1 , T2 , . . . , Tr
in the decomposition. Note that we may assume without loss of
generality that the series alternates between inessentiality and dual inessentiality.
If
T`
and
T`+1
were deemed redundant for the same reason (e.g. they are both
inessential) then the inequality that shows inessentiality of
by the inequality that shows inessentiality of
inessential the proof is the same if
T1
T`+1 .
T`
can be rened
Let us assume that
T1
is
is dually inessential. Thus using the
denition of essentiality and dual essentiality
v(T1 ) ≤ v(T2 ) +
k1
X
v(Sj1 )
(1)
v(Sj2 ) − k2 · v(N )
(2)
v(Sj3 )
(3)
v(Sjr ) − kr · v(N )
(4)
j=1
v(T2 ) ≤ v(T3 ) +
k2
X
j=1
v(T4 ) ≤ v(T5 ) +
k3
X
j=1
.
.
.
v(Tr ) ≤ v(T1 ) +
kr
X
j=1
In words
T1
is inessential because of the collection
T2 , S11 , . . . , Sk11
(these
are all essential coalitions, thus the inequality cannot be rened any more).
Then
T2
is dually inessential because of the collection
an overlapping decomposition of
T2
T3 , S12 , . . . , Sk22
compose
(and these are all dually essential). And so
T1 , S1r , . . . , Skrr . Note that
1
2
r
r
there may be coalitions among S1 , . . . , S1 , . . . , S1 , . . . , Sk that coincide. Using
r
indicator functions and the conditions of inessentiality and dual inessentiality.
on until nally
Tr
is deemed redundant because of
eT1 = eT2 +
k1
X
eSj1
(5)
eSj2 − k2 · eN
(6)
eSj3
(7)
eSjr − kr · eN
(8)
j=1
eT2 = eT3 +
k2
X
j=1
eT4 = eT5 +
k3
X
j=1
.
.
.
eTr = eT1 +
kr
X
j=1
Thus by summing (1)-(4) we obtain that
ki
r X
X
1
v(Sji )
v(N ) ≤
k2 + k4 + · · · + kr i=1 j=1
while from (5)-(8) we gather that
eN =
i.e. the collection
ki
r X
X
1
e i
k2 + k4 + · · · + kr i=1 j=1 Sj
S11 , . . . , S12 , . . . , S1r , . . . , Skrr
is balanced. This contradicts the
fact that the grand coalition is vital.
4
Monotonic balanced games
The next characterization set was proposed by Granot, Granot, and Zhu (1998)
for balanced monotonic cost games.
Denition 6. Let (N, c) be a monotonic cost game. A coalition S ⊆ N is said
to be saturated if c(S) = c(S ∪ {i}) implies i ∈ S .
In other words if
S
is a saturated coalition then every new member will
impose extra cost on the coalition. Let
S ∗ (Γ )
denote the set of all saturated
coalitions and
S(Γ ) = S ∗ (Γ ) ∪ {N \ i | i ∈ N } ∪ {N }.
Theorem 5. (Granot, Granot, and Zhu, 1998) Let Γ = (N, c) be a monotonic
cost game with a non-empty core. Then S(Γ ) forms a characterization set for
N(Γ ).
In fact Granot, Granot, and Zhu (1998) proved a slightly stronger statement,
namely, that the intersection of saturated and essential coalitions forms a characterization set in monotonic cost games with a non-empty core. Similarly to
the other characterization sets, S(Γ ) also induces a representation of the core
C(Γ ) as well. Let us mention here that just because a collection of coalitions determines the core it does not necessarily characterize the nucleolus of the game.
Maschler, Peleg, and Shapley (1979) presented two games with the same core,
but with dierent nucleoli.
We now convert the concept of saturatedness to monotonic prot games
based on the dualization correspondence between prot and cost games. Let
(N, v) be a monotonic prot game and S ⊆ N be an arbitrary coalition. We say
that S is dually saturated if v(S \ i) < v(S) for any i ∈ S . In other words every
member contributes to the worth of coalition S . Notice that S is saturated in the
monotonic cost game c if and only if N \ S is dually saturated in the monotonic
∗
∗
prot game c . Let DS (Γ ) denote the set of all dually saturated coalitions and
DS(Γ ) = DS ∗ (Γ ) ∪ {i | i ∈ N } ∪ {N }.
(N, v) be a
S ⊆ N a dually non-saturated coalition, then we say that
S 6= ∅ is a lower closure of S if S ⊂ S , v(S) = v(S) and S is a dually saturated
coalition. Note that if S has no lower closure, then no member contributes to the
worth of S or to any subset of S . Hence v(S) = v(i) = v(∅) = 0 for any i ∈ S .
The following denition is needed for our next theorem. Let
monotonic game and
Theorem 6. Let Γ = (N, v) be a monotonic game with a non-empty core, then
forms a characterization set for N(Γ ).
DS(Γ )
Proof.
S be a dually non-saturated coalition. If
v(S) = v(i) = 0 for any i ∈ S . From this observation
it also follows that satΓ ({i}, x) ≤ satΓ (S, x) for any i ∈ S and for any allocation
x. Since all the singleton coalitions are included in DS(Γ ) by Theorem 1, S can
be discarded. Finally let S be a lower closure of S and let S \ S = T , then
S
Again we will use Theorem 1. Let
has no lower closure then
satΓ (S, x) + x(T ) = x(S) + x(T ) − v(S) = x(S) − v(S) = satΓ (S, x).
satΓ (S, x) ≤ satΓ (S, x)
satΓ ({i}, x) ≤ satΓ (S, x) for any i ∈ T .
Since core vectors are non-negative this also means
for any
x ∈ C(Γ ).
Now we show that
satΓ ({i}, x) = x(i) − v(i) ≤ x(i) =
x(S) − x(S \ i) + v(S \ i) − v(S) =
satΓ (S, x) − satΓ (S \ i, x) ≤ satΓ (S, x)
We have shown that for any
of
DS(Γ ),
such that
F
characterization set for
S ∈ 2N \ DS(Γ )
there exist a subcollection
fullls both conditions of Theorem 1. Hence
N(Γ ).
DS(Γ )
F
is a
Next we show a relationship between dually essential and saturated coalitions.
Lemma 1. Let Γ
Proof.
= (N, c)
be a monotonic cost game, then DE(Γ ) ⊆ S(Γ )
n − 1 player coalitions are all members of
S be a non-saturated coalition with at most n − 2
players. We will show that S is dually inessential. As S is not saturated there
exists i ∈ N \ S such that c(S) = c(S ∪ i). Let S1 := S ∪ i and S2 := N \ i. Then
S1 ∪ S2 = N and S1 ∩ S2 = S therefore we can use Denition 5 since
both
The grand coalition and the
S(Γ )
and
DE(Γ ).
Let
c(N ) ≥ c(N \ i),
c(S) ≥ c(S) + c(N \ i) − c(N ),
c(S) ≥ c(S1 ) + c(S2 ) − c(N ).
In other words
S appears in an overlapping decomposition of S1 and S2 , therefore
it can not be dually essential.
The above lemma suggests that dually essential coalitions are more useful
since they dene a smaller characterization set, which in turn implies a smaller
LP. However usually it is also harder to determine whether a coalition is dually
essential or not. Saturatedness on the other hand can be checked easily. For
4 there exist at most
instance for airport games
n saturated coalitions, which can
be easily determined from the characteristic function. In fact it is enough to know
the value of the singleton coalitions to identify the saturated coalitions, which
gives us an alternative way to derive an ecient algorithm for the nucleolus.
There is a symmetrical result for essential and dually saturated coalitions.
Lemma 2. Let Γ
Proof.
= (N, v)
be a monotonic prot game, then E(Γ ) ⊆ DS(Γ ).
E(Γ ) and
S be a dually non-saturated coalition such that |S| > 1. Then
there exists i ∈ S such that v(S) = v(S \ i). By monotonicity v(i) ≥ 0, hence
v(S) ≤ v(S \ i) + v(i). Thus S is inessential.
Observe that the singleton coalitions are all members of both
DS(Γ ).
5
Let
Verifying the nucleolus
Faigle, Kern, and Kuipers (1998) conjecture the decision problem: "given
R
N
, is
x
∗
the nucleolus?" to be
N P -hard
x∗ ∈
in general. The most resourceful tool
regarding this problem is the criterion developed by Kohlberg (1971).
Theorem 7. (Kohlberg, 1971) Let Γ = (N, v) be a game with non-empty core
and let x ∈ I(Γ ). Then x = N(Γ ) if and only if for all y ∈ R the collection
{∅ =
6 S ⊂ N | satΓ (S, x) ≤ y} is balanced or empty.
4
This class of games was introduced by Littlechild and Owen (1973).
The Kohlberg-criterion is often used to verify the nucleolus in practical computation when the size of the player set is not too large. As the following LP
shows it is easy to tell whether a given collection of coalitions is balanced or
S1 , . . . , Sm be the collection
q ∈ [0, 1]m . For k = 1, . . . , m let
not. Let
which balancedness is in question and let
p∗k = max qk
m
X
qi eSi = eN
(9)
i=1
q1 , . . . , qm ≥ 0.
Lemma 3. The collection S1 , . . . , Sm is balanced if and only if p∗k > 0 for each
k = 1, . . . , m.
Proof.
p∗k > 0
is a necessary condition. Let a1 , . . . , am ∈ [0, 1] be ark
a1 + · · · + am = 1. Furthermore
let q be an optimal
Pm
th
i
solution of the k
LP. We claim that λ =
vector of balancing
i=1 ai q is aP
Pm
m
j
i
∗
i
weights. Note that λj =
i=1 ai qj > 0 as qj = pj > 0 and
j=1 qj eSj = eN for
all i = 1, . . . , m by (9). Then
Trivially
bitrary reals such that
m
X
m
m X
X
λj eSj =
ai qji eSj
=
m
X
i=1
j=1 i=1
j=1
ai
m
X
qji eSj = (a1 + · · · + am )eN = eN .
j=1
Sobolev (1975) extended Theorem 7 to the prenucleolus (where instead of
x ∈ I(Γ )
we only require
x
to be an allocation). A direct consequence of the
Sobolev-criterion is that the prenucleolus of monotonic games is non-negative.
Theorem 8. Let Γ = (N, v) be a monotonic game and let
cleolus, then z(i) ≥ 0 for all i ∈ N .
Proof.
z
denote its prenu-
z(i) < 0 for some i ∈ N . Let B0 contain the
z . By the Sobolev-criterion
every S ∈ B0 , i ∈ S otherwise S ∪ {i} would
By contradiction suppose that
coalitions with the smallest satisfaction values under
B0
is a balanced collection. For
have an even smaller satisfaction due to the monotonicity of the characteristic
function and the fact that
As
i∈S
z(i) < 0.
P
S ∈ B0 this also means that
S ∈ B0 , S must contain j . Thus the
for all
and for all
P
B0 , S∈B0 λS eS = eN .
S∈B0 λS = 1. Then for all j 6= i
only coalition in B0 is the grand
By balancedness of
coalition. However the grand coalition has zero satisfaction under any allocation,
while coalition
{i}
has negative satisfaction under
z,
which contradicts that
B0
contains the coalitions with the smallest satisfaction values.
Reijnierse and Potters (1998) proved that for every game
a collection with at most
2(n − 1)
(N, v)
there exists
coalitions that determine the nucleolus. How-
ever, they remarked that nding these coalitions is as hard as computing the
nucleolus itself. Unfortunately this result does not make the verication of the
nucleolus belong to
faction values of the
2(n − 1)
N P . Even if somehow we could
2n coalition in increasing order.
eortlessly put the satisIt can happen that these
coalitions are scattered among the dierent balanced coalition arrays
and we have to evaluate exponential many of them before we could conrm that
the given allocation is indeed the nucleolus.
The Kohlberg-criterion applied to games with coalition formation restrictions
yields the following theorem.
Theorem 9. (Maschler, Potters, and Tijs, 1992) Let F be a characterization
set and x be an imputation of the game Γ with C(Γ ) 6= ∅. Then x = N(Γ ) if
and only if for all y ∈ R the collection {S ∈ F | satΓ (S, x) ≤ y} is balanced or
empty.
A similar criterion appears in (Groote Schaarsberg, Borm, Hamers, and Reijnierse, 2013). With the help of the Kohlberg-criterion the problem of nding
the nucleolus is reduced to nding the right characterization set.
Acknowledgements
The authors would like to thank Prof. Tamás Kis of Institute for Computer
Science and Control (MTA SZTAKI) for his valuable comments. Research was
funded by OTKA grants K101224 and K108383 and by the Hungarian Academy
of Sciences under its Momentum Programme (LD-004/2010).
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