We propose a necessary and sufficient condition for the existence of dominance core and a necessary and sufficient condition for coincidence of the core and the dominance core to the setting of multi-choice games.

Yan-An Hwang

English

Research Papers in Economics

The coincidence of the core and the dominance core on
multi−choice games 
Yan−An Hwang
game
Abstract
We propose a necessary and sufficient condition for the existence of dominance core and a
necessary and sufficient condition for coincidence of the core and the dominance core to the
setting of multi−choice games.
Citation: Hwang, Yan−An, (2005) "The coincidence of the core and the dominance core on multi−choice games." Economics
Bulletin, Vol. 3, No. 14 pp. 1−8
Submitted: January 13, 2005.  Accepted: March 18, 2005.
URL: http://www.economicsbulletin.com/2005/volume3/EB−05C70002A.pdf1 Introduction
There are two diﬀerent deﬁnitions of the core of TU games. Gillies (1959) deﬁned the
core in terms of the binary relation-domination. The other deﬁnition of the core is deﬁned
as the solution of a system of linear inequalities. We will call the former dominance core
and the latter core. Chang (2000) proposed a necessary and suﬃcient condition for the
existence of dominance core, and a necessary and suﬃcient condition for coincidence of
the core and dominance core to the setting of TU games.
A multi-choice game, was introduced by Hsiao and Raghavan (1993), is a game in
which each player has a certain number of activity levels at which he or she can choose
to play. This is formalized as follows. Let N = {1,...,n} be a set of players (n ∈ I N)
and suppose each player i ∈ N has mi + 1 ∈ I N activity levels at which he can play.
We set Mi = {0,1,...,mi} as the action space of player i ∈ N, where the action 0
means not participating, and the zero vector (0,...,0) will be denoted by θ. A function
v :
Q
i∈N
Mi → I R with v(θ) = 0 gives for each coalition s = (s1,...,sn) ∈
Q
i∈N
Mi the worth
that the players can obtain when each player i plays at level si ∈ Mi. van den Nouweland
et al. (1995) extended the core and dominance core to the setting of multi-choice games,
and introduced a notion of balancedness to generalize the Theorem of Bondareva (1963)
and Shapley (1967) to the class of multi-choice games. In this note, we will generalize
Chang’s (2000) results to the setting of multi-choice games.
2 Deﬁnitions, Notations and Facts
A multi-choice game is a triple (N,m,v), where N is the set of players, m ∈ (I N ∪ {0})N
is the vector describing the number of activity levels for all players, and v :
Q
i∈N
Mi → I R
is the characteristic function with v(θ) = 0. We will consider that mi ≥ 1 for each player
i ∈ N and if there can be no confusion we will denote a game (N,m,v) by v. We denote
the set of all multi-choice games with player set N by MCN.
A multi-choice game v is called zero-normalized if the players cannot gain anything by
working alone, i.e., v(jei) = 0 for all i ∈ N and j ∈ Mi\{0}. For an arbitrary multi-choice
game v, the zero-normalization game v0 of v is deﬁned by v0(s) = v(s) −
P
i∈N
a(siei) for
all s ∈
Q
i∈N
Mi where a(jei) = v(jei) for all i ∈ N and j ∈ Mi \ {0}.
Let (N,m,v) ∈ MCN. We deﬁne M = {(i,j) : i ∈ N,j ∈ Mi}. A ( level ) payoﬀ
vector for the game v is a function x : M → I R, where, for all i ∈ N and j ∈ Mi \{0}, xij
denotes the increase in payoﬀ to player i corresponding to a change of activity from level
j − 1 to level j by this player and xi0 = 0 for all i ∈ N. Let S ⊆ N. By eS we denote the
vector in I RN satisfying eS
i = 0 if i / ∈ S and eS
i = 1 if i ∈ S.
A payoﬀ vector is called eﬃcient if
P
i∈N
mi P
j=1
xij = v(m) and it is called level increase
rational if, for all i ∈ N and level j ∈ Mi \ {0}, xij ≥ v(jei) − v((j − 1)ei).
1Deﬁnition 2.1 A payoﬀ vector is an imputation of v if it is eﬃcient and level increase
rational.
We denote the set of imputations of the game v by I(v). It is easily seen that
I(v) 6= ∅ ⇐⇒ v(m) ≥
X
i∈N
v(mie
i)
Now let x be a payoﬀ vector for the game v. If a player i works at his jth level (j ∈ Mi),
then he obtains, according to x, the amount
j P
k=0
xik. It will often be more natural to
look at these accumulated payoﬀs. For i ∈ N and j ∈ Mi we denote Xij =
j P
k=0
xik. The
members of a coalition s ∈
Q
i∈N
Mi obtain X(s) =
P
i∈N
Xisi. Using this, we come to the
following
Deﬁnition 2.2 The core C(v) of the game v consists of all x ∈ I(v) that satisfy X(s) ≥
v(s) for all s ∈
Q
i∈N
Mi, i.e.,
C(v) = {x ∈ I(v) : X(s) ≥ v(s) for all s ∈
Y
i∈N
Mi}.
Remark 2.3 Let v be a zero-normalized game and let
C = {z ∈ I R
N
+ :
X
i∈N
zi = v(m) and
X
i∈A(s)
zi ≥ v(s), for all s ∈
Y
i∈N
Mi}.
If x is a payoﬀ vector in C(v), we can deﬁne a vector z ∈ I RN
+ by zi =
mi P
j=1
xij for all i ∈ N
such that z ∈ C. On the other hand, let a vector z ∈ C , we can also deﬁne a payoﬀ vector
x : M → I R such that x ∈ C(v) by
xij =

 
 
zi if i ∈ N and j = 1
0 o.w,
That is, C(v) = {x ∈ I(v) :
P
i∈A(s)
si P
j=1
xij ≥ v(s), for all s ∈
Q
i∈N
Mi} 6= ∅ if and only if
C = {z ∈ I RN
+ :
P
i∈N
zi = v(m) and
P
i∈A(s)
zi ≥ v(s), for all s ∈
Q
i∈N
Mi} 6= ∅.
Let s ∈
Q
i∈N
Mi and x, y ∈ I(v). The imputation y dominates the imputation x via
coalition s, denote y doms x, if Y (s) ≤ v(s) and Yisi > Xisi for all i ∈ A(s), where
A(s) = {i ∈ N : si > 0,s ∈
Q
i∈N
Mi} is the set of players who participate in s. We say
that the imputation y dominates the imputation x if there exists a coalition s ∈
Q
i∈N
Mi
such that y doms x.
2Deﬁnition 2.4 The dominance core DC(v) of the game v consists of all x ∈ I(v) for
which there exists no y ∈ I(v) such that y dominates x, i.e.,
DC(v) = {x ∈ I(v) :6 ∃y ∈ I(v) such that y dominates x}.
The following two Lemmas were studied by van den Nouweland et al. (1995,p.292,293).
Lemma 2.5 For each game v the core C(v) is a subset of the dominance core DC(v).
Lemma 2.6 Let v be an arbitrary game and v0 its zero-normalization. Let x be a payoﬀ
vector for this game. Deﬁne y : M → I R by yij = xij − v(jei) + v((j − 1)ei) for all i ∈ N
and j ∈ Mi \ {0}. Then we have
(1) x ∈ I(v) ⇐⇒ y ∈ I(v0)
(2) x ∈ C(v) ⇐⇒ y ∈ C(v0)
(3) x ∈ DC(v) ⇐⇒ y ∈ DC(v0).
A notion of balancedness to the setting of multi-choice games was introduced by van den
Nouweland et al. (1995) as follows.
Deﬁnition 2.7 A multi-choice game v is called balanced if for all maps λ :
Q
i∈N Mi →
I R+ satisfying
X
s∈
Q
i∈N Mi
λ(s)e
A(s) = e
N
it holds that
X
s∈
Q
i∈N Mi
λ(s)v0(s) ≤ v0(m),
where v0 is the zero-normalization of v.
The next Theorem is an extension of the Theorem of Bondareva (1963) and Shapley
(1967) to the setting of multi-choice games and gives a necessary and suﬃcient condition
for the nonemptiness of the core of a game by van den Nouweland et al. (1995,p.297).
Theorem 2.8 Let v be a multi-choice game. Then the core C(v) of v is non-empty if
and only if v is balanced.
3To end this section, we give two examples to explain that why we deﬁne such bal-
ancedness, corresponding to zero-normalization, on multi-choice games. One is that we
provide a multi-choice game v with nonempty core but it does not satisfy
X
s∈
Q
i∈N Mi
λ(s)v(s) ≤ v(m) whenever
X
s∈
Q
i∈N Mi
λ(s)e
A(s) = e
N. (2.1)
The other is that a multi-choice game v satisﬁes the condition (2.1) but it has empty core.
Example 2.9 Let (N,m,v) be a multi-choice game where N = {1,2}, m = (2,1) and
v((0,1)) = v((1,1)) = v((2,1)) = 0, v((1,0)) = 1 and v((2,0)) = −1. Then the payoﬀ
vector x with x11 = 1, x12 = −1 and x21 = 0 is in C(v). For this game, we ﬁnd a
collection β = {(1,0),(0,1)} and λ((1,0)) = 1, λ((0,1)) = 1 such that
P
s∈β
λ(s)v(s) = 1 >
0 = v((2,1)).
Example 2.10 Let (N,m,v) be a multi-choice game where N = {1,2}, m = (2,1) and
v((0,1)) = v((1,0)) = v((1,1)) = −1, v((2,0)) = 1 and v((2,1)) = 0. Then v clearly satis-
ﬁes the condition (2.1). To verify that it has empty core, consider the zero-normalization
v0 of v with v0((0,1)) = v0((1,0)) = v0((2,0)) = v0((2,1)) = 0, and v0((1,1)) = 1. It is
easy to see that v0((2,1)) = 0 < 1 =
P
s∈β
λ(s)v0(s) for β = {(1,1)} and λ((1,1)) = 1, thus
C(v0) = ∅.
3 Main Results
In this section we will extend Chang’s (2000) results from TU games to multi-choice
games. It is known that the core and the dominance core are invariant under strategic
equivalence by Lemma 2.6. Hence, w.l.o.g., we assume that all multi-choice games are
zero-normalized. Besides, we will assume that v(m) ≥ 0 and thus I(v) 6= ∅.
Let (N,m,v) be a game. We deﬁne a new game by v0(s) = min{v(s),v(m)} for all
s ∈
Q
i∈N
Mi. Then v0(m) = v(m) and v0(jei) = v(jei) = 0 for all i ∈ N and j ∈ Mi \ {0}.
Hence (N,m,v0) is also with v0(m) ≥ 0 and v0(jei) = 0 for all i ∈ N and j ∈ Mi. And it
is easy to see that I(v) = I(v0).
Lemma 3.1 Let s ∈
Q
i∈N
Mi, s 6= θ, and let x, y ∈ I(v) = I(v0). Then x doms y in v0 if
and only if x doms y in v.
proof: Let s ∈
Q
i∈N
Mi, s 6= θ, and let x, y ∈ I(v) = I(v0). If x doms y in v0, then
X(s) ≤ v0(s) and Xisi > Yisi for all i ∈ A(s). Therefore X(s) ≤ v(s) and x doms y in v.
On the other hand, if x doms y in v, then X(s) ≤ v(s) and Xisi > Yisi for all i ∈ A(s).
Since x ∈ I(v), X(s) =
P
i∈N
mi P
j=1
xij−
P
i∈N
mi P
j=si+1
xij ≤ v(m). These imply that X(s) ≤ v0(s)
and x doms y in v0. Q.E.D.
4Lemma 3.2 For any game (N,m,v) ∈ MCN, DC(v) = DC(v0).
proof: It follows from Lemma 3.1. Q.E.D.
Lemma 3.3 For any game (N,m,v) ∈ MCN, C(v0) = DC(v0).
proof: According to Lemma 2.5, we know that C(v0) ⊆ DC(v0). If DC(v0) = ∅, it is easy
to see that C(v0) = DC(v0). If DC(v0) 6= ∅, it remains to show that DC(v0) ⊆ C(v0). Let
x ∈ DC(v0) and suppose that x / ∈ C(v0). Then there exists a coalition s ∈
Q
i∈N
Mi such
that X(s) < v0(s). Since v0(t) ≤ v0(m) for all t ∈
Q
i∈N
Mi, we can deﬁne a payoﬀ vector
y : M → I R by
yij =

  
  
xij +
v0(s)−X(s) P
k∈N sk if i ∈ N,j ∈ {1,2,...,si}
v0(m)−v0(s) P
k∈N(mk−sk) if i ∈ N,j ∈ {si + 1,...,mi}.
Then yij > xij ≥ 0 and
Y (m) =
P
i∈N
mi P
j=1
yij
=
P
i∈N
{
si P
j=1
yij +
mi P
j=si+1
yij}
= (
P
i∈N
si P
j=1
xij +
P
i∈N
si
v0(s)−X(s) P
k∈N sk ) +
P
i∈N
(mi − si)
v0(m)−v0(s) P
k∈N(mk−sk)
= X(s) + v0(s) − X(s) + v0(m) − v0(s)
= v0(m).
Hence y ∈ I(v0). Since Yisi > X isi and Y (s) =
P
i∈N
si P
j=1
yij = v0(s), y doms x in v0. This
contradicts the assumption. Hence x ∈ C(v0) and DC(v0) ⊆ C(v0). Q.E.D.
Lemma 3.4 For any game (N,m,v) ∈ MCN, DC(v) = C(v0).
proof: It follows from Lemmas 3.2 and 3.3. Q.E.D
Lemma 3.5 For any game (N,m,v) ∈ MCN,DC(v) 6= ∅ if and only if (N,m,v0) is
balanced.
proof: It follows from Theorem 2.8 and Lemma 3.4. Q.E.D.
Lemma 3.6 For any game (N,m,v) ∈ MCN with C(v) 6= ∅, C(v) = C(v0).
5proof: Using Lemmas 2.5 and 3.4, we know that C(v) ⊆ C(v0). It remains to show that
C(v0) ⊆ C(v). Let x ∈ C(v0), then x ∈ I(v0) = I(v) and X(s) ≥ v0(s) for all s ∈
Q
i∈N
Mi.
Now we will show that v(s) ≤ v(m) for all s ∈
Q
i∈N
Mi. Since C(v) 6= ∅, there exists an
y ∈ C(v) such that
Y (s) ≥ v(s) for all s ∈
Q
i∈N
Mi and
Y (s) =
P
i∈N
mi P
j=1
yij −
P
i∈N
mi P
j=si+1
yij ≤ v(m).
Hence v(s) ≤ v(m). Therefore X(s) ≥ v0(s) = v(s) for all s ∈
Q
i∈N
Mi and x ∈ C(v). This
completes the proof. Q.E.D.
Lemma 3.7 For any game (N,m,v) ∈ MCN, C(v) = C(v0) if and only if (N,m,v) is
balanced or (N,m,v0) is not balanced.
proof: For any game (N,m,v) ∈ MCN. If C(v) = C(v0), then either both C(v) and
C(v0) are empty or both are nonempty. If both C(v) and C(v0) are empty, then (N,m,v0)
is not balanced. If both C(v) and C(v0) are nonempty, then (N,m,v) is balanced. On the
other hand, if (N,m,v0) is not balanced, C(v) ⊆ C(v0) = ∅. This implies C(v) = C(v0). If
(N,m,v) is balanced, C(v) 6= ∅. Using Lemma 3.6, we have C(v) = C(v0). Q.E.D.
Theorem 3.8 For any game (N,m,v) ∈ MCN, C(v) = DC(v) if and only if (N,m,v)
is balanced or (N,m,v0) is not balanced.
proof: Since we have known that DC(v) = C(v0) for any game (N,m,v) ∈ MCN by
Lemma 3.4, it suﬃces to show C(v) = C(v0) if and only if (N,m,v) is balanced or
(N,m,v0) is not balanced. Then, using Lemma 3.7, we obtain C(v) = DC(v) if and only
if (N,m,v) is balanced or (N,m,v0) is not balanced. Q.E.D.
References
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terly, 47, pp. 455-458
[3] Gillies D (1959): “Solutions to general non-zero-sum games,” Annals of Math. Stud-
ies, 40, pp. 47-85
[4] Hsiao C, Raghavan T (1993): “Shapley value for multi-choice cooperative games (I),”
Games and Economic Behavior, 5, pp. 240-256
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7

On balanced set and cores,”

Remarks on theory of the core,”

Shapley value for multi-choice cooperative games

Solutions to general non-zero-sum games,”

Some applications of linear programming methods to theory of cooperative games,”

The coincidence of the core and the dominance core on multi-choice games

http://www.accessecon.com/pubs/EB/2005/Volume3/EB-05C70002A.pdf

The coincidence of the core and the dominance core on multi-choice games

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