In this note we introduce an egalitarian solution, called the dual egalitarian solution, that is the natural counterpart of the egalitarian solution of Dutta and Ray (1989).We prove, among others, that for a convex game the egalitarian solution coincides with the dual egalitarian solution for its dual concave game.cooperative games;egalitarianism;duality

Klijn, F.

Slikker, M.

Tijs, S.H.

English

Research Papers in Economics

Center
for
Economic Research
No. 2000-113
A DUAL EGALITARIAN SOLUTION
By Flip Klijn, Marco Slikker and Stef Tijs
November 2000
ISSN 0924-7815A Dual Egalitarian Solution
Flip Klijn
a,b,1, Marco Slikker
c,a n dS t e fT i j s
b
aDepartamento de Estadística e I.O., Universidad de Vigo, Spain
bDepartment of Econometrics and CentER, Tilburg University, The Netherlands
cDepartment of Technology Management, Technische Universiteit Eindhoven, The Netherlands
Abstract: In this note we introduce an egalitarian solution, called the dual egalitarian solution,
that is the natural counterpart of the egalitarian solution of Dutta and Ray (1989). We prove,
among others, that for a convex game the egalitarian solution coincides with the dual egalitarian
solution for its dual concave game.
Keywords: Egalitarianism; Duality; Cooperative Game Theory
JEL classication: C71, D63
1 Introduction
Often a situation involving several persons who can obtain benets by cooperating can be for-
mulated in terms of a cooperative savings game with transferable utility. Examples of such
classes of games are bankruptcy games (O’Neill (1982) and Aumann and Maschler (1985)),
sequencing games (Curiel et al. (1989)), market games (Shapley and Shubik (1969)), and linear
production games (Owen (1975)). Moreover, some interesting classes of these games men-
tioned above are convex, i.e., an agent contributes more to the benets of a coalition when the
coalition becomes larger.
A decision making situation involving several persons can sometimes also be formulated in
terms of a cooperative cost game. Among the cost games that can be found in the literature
are airport games (Littlechild and Owen (1973)) and minimum cost spanning tree games (Bird
(1976)). These games satisfy the property of concavity, i.e., the marginal burden of an agent to
the costs of a coalition decreases when the coalition becomes larger.
In this note we start with recalling the denition of the egalitarian solution, a solution con-
cept for cooperative games introduced by Dutta and Ray (1989). This solution unies the two
conicting concepts of individualistic utility maximization and the social goal of equality. The
1Corresponding author. Departamento de Estadística e I.O., Universidad de Vigo, Lagoas-Marcosende, s/n,
36310 Vigo (Pontevedra), Spain. Tel. (34) 986 812493; Fax. (34) 986 812401; e-mail: fklijn@uvigo.es
1egalitarian solution is a singleton or the empty set. For convex games Dutta and Ray (1989)
describe an algorithm to locate the unique egalitarian solution, and they show, in addition, that
it is in the core of the game. Finally, they prove that for convex games the egalitarian solution
Lorenz dominates all other allocations in the core. For an extensive discussion on the egalitarian
solution we refer to Dutta and Ray (1989).
If a game is concave but not additive, then the egalitarian solution of Dutta and Ray (1989)
is empty. Therefore, we dene, along the lines of Dutta and Ray (1989), a dual of Lorenz
domination and construct a dual egalitarian solution. We show that Lorenz domination and
dual Lorenz domination are equivalent. Furthermore, the dual egalitarian solution is either a
singleton or the empty set. After that, we study the relations with the egalitarian solution of
Dutta and Ray (1989) and the core of the game. Next, we introduce an algorithm similar to
that of Dutta and Ray (1989) to calculate the dual egalitarian solution for concave games. We
show that the dual egalitarian solution is in the dual core of the concave game and that the dual
egalitarian solution Lorenz dominates all other allocations in the dual core.
Our main result is that for a convex game the egalitarian solution coincides with the dual
egalitarian solution for its dual (concave) game. Similar duality results are provided by Funaki
(1994) and concern, among others, the core, the Shapley value (Shapley (1953)), and the prenu-
cleolus (Schmeidler (1969)). To this series we can also add the modied nucleolus (Sudhölter
(1997)) and the ¿-value (Tijs (1986)).
The work is organized as follows. Section 2 deals with notation and denitions regarding
cooperative games with transferable utility. Moreover, we recall the denition of the egalitarian
solution of Dutta and Ray (1989). In Section 3 we introduce the dual egalitarian solution and
present the relations with the egalitarian solution. Finally, we prove that for a convex game the
egalitarian solution coincides with the dual egalitarian solution for its dual (concave) game.
2 Preliminaries
A cooperative game with transferable utility (game, for short) is a pair (N;v),w h e r eN =
f1;:::;ng is the player set and v the characteristic function, which assigns to every subset2
S µ N av a l u ev(S),w i t hv(;)=0 .
Ag a m e(N;v) is called convex if
v(S [f ig) ¡ v(S) · v(T [f ig) ¡ v(T) for all S µ T µ N and i 2 NnT;
and concave if the reverse inequality holds.
The core of a game (N;v) is dened by
C(N;v): =fx 2 IR
N :
X
i2N
xi = v(N) and
X
i2S
xi ¸ v(S) for all S µ Ng;
and its dual core is dened by
C
¤(N;v): =fx 2 I R
N :
X
i2N
xi = v(N) and
X
i2S
xi · v(S) for all S µ Ng:
2S µ N denotes that S is a subset of N and S ½ N denotes that S is a strict subset of N.
2A game is called (dual) balanced if and only if its (dual) core is non-empty. Let S be a non-
empty subset of N. The game (S;vjS) (or (S;v), for short) is called a subgame of (N;v).A
game is called totally (dual) balanced if and only if the (dual) cores of all subgames are non-
empty.
The dual game of a game (N;v) is the game (N;v¤),w h e r ev¤ is dened by
v
¤(S): =v(N) ¡ v(NnS) for all S µ N:
It is easily shown that C(N;v)=C¤(N;v¤).
Next, we recall the denition of the egalitarian solution of Dutta and Ray (1989). For this we
need some more notation, most of which is due to Dutta and Ray (1989). Let S b ean o n - e m p t y
subset of f1;:::;ng. We denote the cardinality of S by jSj. To avoid unnecessary notational
complications we denote any x 2 IR
S by x =( x1;:::;x jSj).F o rt w ov e c t o r sx and y in IR
S,w e
write x = y if all their components are equal, and x>yif xi ¸ yi for all i =1 ;:::;jSj, with
strict inequality for some i. For any x 2 I R
S,w ed e n o t eb y^ x the vector obtained by permuting
the indices of x such that ^ x1 ¸ ^ x2 ¸¢ ¢ ¢¸^ xjSj.F o r x 2 I R
S and T µ S, we denote the
projection of x on I R
T by x(T)=( xi)i2T.
Let (N;v) be a game. An allocation x 2 IR
S is feasible for S if
P
i2S xi = v(S).T h eLorenz
map E is dened on the domain
A :=
(
A : A µ I R
k for some k; and there exists ¸ 2 I R such that
k X
i=1
xi = ¸ for all x 2 A
)
:
For each set A 2A ;E A is the set of all allocations in A that are Lorenz undominated within
A. Formally,
EA :=
n
x 2 A : there is no y 2 A such that
Pj
i=1 ^ yi ·
Pj
i=1 ^ xi for all
j =1 ;:::;k; with strict inequality for some jg:
The Lorenz core is dened inductively as follows. The Lorenz core of a singleton coalition
is dened by L(fig;v): =fv(fig)g. Now suppose that the Lorenz cores for all coalitions of
cardinality k or less have been dened, where 1 · k<n . The Lorenz core of a coalition of
size k +1is dened by
L(S;v): =
n
x 2 I R
S : x is feasible for S; and there is no T ½ S and
y 2 EL(T;v) such that y>x (T)g:
If x 2 I R
S and there is T ½ S and y 2 EL(T;v) such that y>x (T), then we say that y
Lorenz-blocks (L-blocks) x. We shall also say in this case that TL -blocks x. Furthermore, note
that C(S;v) µ L(S;v).
Dutta and Ray (1989) called the set of Lorenz undominated allocations in the Lorenz core
of (N;v),i . e . ,EL(N;v),t h eegalitarian solution. They proved that EL(N;v) is either empty
or a singleton. If it is non-empty, then its unique element is called the egalitarian allocation of
3the game (N;v). For convex games Dutta and Ray (1989) describe a nite algorithm to locate
the unique egalitarian allocation. Moreover, they showed that the egalitarian allocation is an
element of the core of (N;v) and that it Lorenz dominates all other core allocations.
3 The dual egalitarian solution
In this section, we will introduce a dual of the egalitarian solution of Dutta and Ray (1989),
thereby following closely the ideas of Dutta and Ray (1989) concerning egalitarianism. Theo-
rem 3.2 shows that the dual egalitarian solution is either a singleton or the empty set. After that,
we study the relations with the egalitarian solution of Dutta and Ray (1989) and the core of the
game. Next, we introduce an algorithm similar to that of Dutta and Ray (1989) to calculate the
dual egalitarian solution for concave games. We show that the dual egalitarian solution is in
the dual core of the concave game and that the dual egalitarian solution Lorenz dominates all
other allocations in the dual core. Finally, we present our main result: for a convex game the
egalitarian allocation is equal to the dual egalitarian allocation for its dual (concave) game.
For starters, we need some extra notation. Let S µf 1;:::;ng.F o ra n yx 2 I R
S, we denote
by ¹ x the vector obtained by permuting the indices of x such that ¹ x1 · ¹ x2 ·¢ ¢ ¢·¹ xjSj.L e t
(N;v) be a game. The dual Lorenz map D is dened on the same domain as the Lorenz map
E,i . e . ,
A =
(
A : A µ I R
k for some k; and there exists ¸ 2 I R such that
k X
i=1
xi = ¸ for all x 2 A
)
:
For each such set A 2A ;D A isthe set of all allocations in A that are dual Lorenz undominated
within A. Formally,
DA :=
n
x 2 A : there is no y 2 A such that
Pj
i=1 ¹ yi ¸
Pj
i=1 ¹ xi for all
j =1 ;:::;k; with strict inequality for some jg:
The dual Lorenz core of a singleton coalition is dened by L¤(fig;v): =fv(fig)g.N o w
suppose that the dual Lorenz cores for all coalitions of cardinality k or less have been dened,
where 1 · k<n . The dual Lorenz core of a coalition of size k +1is dened by
L¤(S;v): =
n
x 2 IR
S : x is feasible for S; and there is no T ½ S and
y 2 DL¤(T;v) such that y<x (T)g:
If x 2 I R
S and there is T ½ S and y 2 DL¤(T;v) such that y<x (T),t h e nw es a yt h a ty dual
Lorenz-blocks (L¤-blocks) x. We shall also say in this case that TL ¤-blocks x. Furthermore,
note that C¤(S;v) µ L¤(S;v).
We call the set of dual Lorenz undominated allocations in the dual Lorenz core of (N;v),
i.e., DL¤(N;v),t h edual egalitarian solution. Next, we will show that DL¤(N;v) is either
empty or a singleton. Hence, if DL¤(N;v) is non-empty, its unique element will be called the
dual egalitarian allocation of the game (N;v).B u trst we show that the operators D and E
coincide.
4Lemma 3.1 The operators D and E coincide.
Proof. Let A 2A . Then there is a coalition S = f1;:::;kg and a number ¸ 2 I R such
that
Pk
i=1 zi = ¸ for all z 2 A.L e t z 2 A. Consider the vectors ^ z and ¹ z. Recall that
^ z1 ¸ ^ z2 ¸¢¢¢¸^ zk and ¹ z1 · ¹ z2 ·¢¢¢· ¹ zk. It is easily veried that for all i =1 ;:::;kwe
have ¹ zi =^ zk¡i+1.
Let x;y 2 A. From the observation above and
Pk
i=1 xi = ¸ =
Pk
i=1 yi it follows that the
following conditions are equivalent:
j X
i=1
¹ yi ¸
j X
i=1
¹ xi for all j =1 ;:::;kwith a strict inequality for some j ,
j X
i=1
¹ yi ¸
j X
i=1
¹ xi for all j =1 ;:::;k¡ 1 with a strict inequality for some j ,
k X
i=j+1
¹ yi ·
k X
i=j+1
¹ xi for all j =1 ;:::;k¡ 1 with a strict inequality for some j ,
k X
i=j+1
^ yk¡i+1 ·
k X
i=j+1
^ xk¡i+1 for all j =1 ;:::;k¡ 1 with a strict inequality for some j ,
j X
i=1
^ yi ·
j X
i=1
^ xi for all j =1 ;:::;k¡ 1 with a strict inequality for some j ,
j X
i=1
^ yi ·
j X
i=1
^ xi for all j =1 ;:::;kwith a strict inequality for some j
From this it immediately follows that DA = EA. 2
From Lemma 3.1 it follows that dual Lorenz domination coincides with Lorenz domination.
Henceforth, we will use the notation E instead of D. The proof of the next Theorem is omitted
since it runs completely analogously to the proof of Theorem 1 of Dutta and Ray (1989).
Theorem 3.2 There is at most one dual egalitarian allocation.
The following examples from Dutta and Ray (1989)give an idea about the relations between
the egalitarian solution, the dual egalitarian solution, and the core of the game.
The game (N;v) in the rst example (Example 1 from Dutta and Ray (1989)) is totally
balanced game with EL(N;v)=;,b u tEL¤(N;v¤) 6= ;.
Example 3.3 Let N = f1;2;3g;v (fig)=0for all i 2 N; v(f1;2g)=v(f1;3g)=v(N)=1 ;
and v(f2;3g)=0 . Dutta and Ray (1989) showed that EL(N;v)=;.U s i n gt h a tL¤(f1;2g;v¤)
=( 1 ;0), L¤(f1;3g;v¤)=( 1 ;0),a n dL¤(f2;3g;v¤)=; one easily veries that L¤(N;v¤)=
(1;0;0) and, hence, EL¤(N;v¤)=f(1;0;0)g. Note that the game (N;v) is not convex, since
v(f1;2g) ¡ v(f1g)=1> 0=v(f1;2;3g) ¡ v(f1;3g). ¦
5The game (N;v) in the second example (Example 2 from Dutta and Ray (1989)) is not
convex. It holds that EL(N;v) 6= ;, EL¤(N;v¤)=;,a n dC(N;v)=;.
Example 3.4 Let N = f1;2;3g;v(f1g)=0 ;v(f2g)=v(f3g)=1 ;v (f1;2g)=v(f1;3g)=
1:4,a n dv(f2;3g)=v(N)=2 :2. Dutta and Ray (1989) observed that EL(N;v)=f(0;1:1;
1:1)g and C(N;v)=;. It is easy to verify that EL¤(f1g;v¤)=f(0)g and EL¤(f2g;v¤)=
EL¤(f3g;v¤)=f(0:8)g.S i n c ev¤(N)=2 :2,i tf o l l o w st h a tEL¤(N;v¤)=;. ¦
Finally, we consider Example 3 from Dutta and Ray (1989). In this example we have a
non-convex game (N;v) with EL(N;v)=EL¤(N;v¤) 6= ;, C(N;v) 6= ;,a n dEL(N;v) \
C(N;v)=;,
Example 3.5 Let N = f1;2;3;4g;v(fig)=0for all i 2 N, v(N)=2 ;v(f2;3g)=1 :05;
v(f3;4g)=1 :9; and for all other S, v(S) is the minimal superadditive function compatible
with these values. Dutta and Ray (1989) observed that EL(N;v)=f(0:05;0:05;0:95;0:95)g6 2
C(N;v) 6= ;. One can verify that EL¤(N;v¤)=EL(N;v). ¦
Next, we describe an algorithm for locating the (unique) dual egalitarian allocation in a
concavegame. Thealgorithmisanalogous to thealgorithm ofDuttaandRay(1989)forlocating
the egalitarian allocation in a convex game. Denote the average worth of coalition S with
respect to a characteristic function v0 by
a(S;v
0): =
v0(S)
jSj
:
Let (N;w) b eac o n c a v eg a m e .D e ne N1 := N and w1 := w.
STEP 1: Let T1 be the largest coalition with the lowest average worth in the game (N;w1).
Dene
x
¤
i(N;w): =a(T1;w 1) for all i 2 T1: (1)
STEP k: Suppose that T1;:::;T k¡1 have been dened recursively and T1 [¢¢¢[Tk¡1 6= N.
Dene a new game with player set Nk := Nn(T1[¢¢¢[Tk¡1). For all subcoalitions T µ
Nk,d e ne wk(T): =wk¡1(Tk¡1[T)¡wk¡1(Tk¡1). The game (Nk;w k) is concave since
(Nk¡1;w k¡1) is concave. Dene Tk to be the largest coalition with the lowest average
w o r t hi nt h i sg a m e .D e ne
x
¤
i(N;w): =a(Tk;w k) for all i 2 Tk: (2)
We remark that the concavity of the game (Nk;w k) ensures that there is a largest coalition
with the lowest average worth in (Nk;w k).L e t x¤ be the allocation dened by equations (1)
and (2).
Theorem 3.6 Inaconcavegame(N;w),x¤ asconstructedbythealgorithmaboveistheunique
dual egalitarian allocation. Moreover, x¤ is in the dual core C¤(N;w) and Lorenz dominates
every allocation in the dual core.
6The proof of Theorem 3.6 is omitted since it runs completely analogously to the proofs of
Theorem 2 and 3 of Dutta and Ray (1989).
From Examples 3.3, 3.4, and 3.5 it follows that EL(N;v) does not need to coincide with
EL¤(N;v¤). For convex games, however, they do coincide. This is our main result:
Theorem 3.7 For a convex game (N;v) it holds that EL(N;v)=EL¤(N;v¤).
Proof. Let (N;v) be a convex game. Dutta and Ray (1989) showed that EL(N;v) µ C(N;v)
and that the unique egalitarian solution Lorenz dominates all other core allocations in C(N;v).
Note also that (N;v¤) is a concave game, and hence, according to Theorem 3.6, we have that
EL¤(N;v) µ C¤(N;v¤). Moreover, the unique dual egalitarian solution Lorenz dominates all
other dual core allocations in C¤(N;v). Now the Theorem follows from the observation that
C(N;v)=C¤(N;v¤). 2
Given Theorem 3.7 it is not difcult to axiomatically characterize the dual egalitarian solu-
tion on the class of concave games. The procedure runs along the lines of Funaki (1994) and
consists of dualizing the axioms that characterize the egalitarian solution. We refer to Funaki
(1994) for the details. Thus, dualizing the axioms in Dutta (1990) and Klijn et al. (2000) yields
characterizations of the dual egalitarian solution. Since this procedure is straightforward we
have omitted it.
Weconcludewiththefollowingopenquestion: ifforagame(N;v)itholdsthatEL(N;v) 6=
; and EL¤(N;v¤) 6= ;, then EL(N;v)=EL¤(N;v¤)? We know by Theorem 3.7 that if (N;v)
is convex this is true. Note also that by Example 3.5 EL(N;v) and EL¤(N;v¤) can be both
non-empty for non-convex games (N;v).
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A Dual Egalitarian Solution

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