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On the existence of the Dutta-Ray’s
egalitarian solution
Francesc Llerena
Llúcia Mauri

Document de treball n.14- 2016

DEPARTAMENT D’ECONOMIA – CREIP
Facultat d’Economia i Empresa

UNIVERSITAT
ROVIRA I VIRGILI
DEPARTAMENT D’ECONOMIA 

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DEPARTAMENT D’ECONOMIA – CREIP
Facultat d’Economia i Empresa

On the existence of the Dutta-Ray’s egalitarian
solution
Francesc Llerena and Llúcia Mauri∗

Abstract
A class of balanced games, called exact partition games, is introduced.
Within this class, it is shown that the egalitarian solution of Dutta and
Ray (1989) behaves as in the class of convex games. Moreover, we provide two axiomatic characterization by means of suitable properties such as
consistency, rationality and Lorenz-fairness. As a by-product, alternative
characterizations of the egalitarian solution over the class of convex games
are obtained.

1

Introduction

On the domain of transferable utility coalitional game (TU-games, for short), several solution concepts have been motivated by the idea of egalitarianism. One of
the best known is the weak constrained egalitarian solution (WCES, for short),
introduced by Dutta and Ray (1989). This solution is defined in a setting where
agents believe in equality as a desirable social goal, but their individual preferences dictate selfish behavior. The WCES yields, whenever it exists, the unique
Lorenz-maximal imputation within the Lorenz core, which is a proper extension
of the core. Although this is a sharp result because the Lorenz domination generates a partial ranquing, this solution lacks general existence properties. In fact,
the class of convex games (Shapley, 1971) is the only standard class of TU-games
where its existence is guaranteed. On this domain, Dutta and Ray (1989) describe
an algorithm for finding their egalitarian allocation and show that it belongs to
the core and Lorenz dominates every other core element. Unfortunatly, several
examples in the same paper show that, in a general domain, these assertions are
not true: there are games with a nonempty core where the WCES does not exist,
and vice-versa; games where both the core and the WCES exist but the latter does
not lie in the core, or games where the WCES belongs to the core but does not
∗
Dep. de Gestió d’Empreses, Universitat Rovira i Virgili-CREIP,
e-mail: francesc.llerena@urv.cat (Francesc Llerena), llucia.mauri@urv.cat (Llúcia Mauri).

Lorenz dominate every other core element. On the domain of balanced games, an
alternative route, already suggested by Dutta and Ray (1989) and latter adopted
by Arin and Iñarra (2001) and Hougaard et al. (2001), is to focus on the Lorenz
maximal allocations within the core. A problem with this solution concept is that
it is not single-valued. To overcome this drawback, Arin and Iñarra (2001) and
Arin et al. (2003) propose single-valued solutions which are derived from the application of the Rawlsian criterion on the core. On the domain of convex games
all these solution concepts produce the same outcome.
A review of the proofs of Theorems 2 and 3 in Dutta and Ray (1989) shows that
weaker conditions than convexity are enough to guarantee that their egalitarian
solution behaves as in convex games. With this objective, in Section 3 we introduce
a subclasses of balanced games called exact partition games. This class of games
is rich enough to include convex games and dominant diagonal assignment games
(Solymosi and Raghavan, 2001), but also nonsuperadditive games. Within this
class, in Section 4 we provide two axiomatic characterization of the WCES by
means of suitable properties such as consistency (à la Davis and Maschler, 1965),
rationality and two new properties inspired by the notion of stable sets of von
Neumann and Morgensten, but changing the usual order in RN by the Lorenz
order. As particular cases, we obtain alternative characterizations of the WCES
over the domain of convex games, and of the set of Lorenz maximal allocation
within the core over the domain of balanced games. Some final remarks conclude
the paper. We begin with notation and terminology.

2

Notation and terminology

The set of natural numbers N denotes the universe of potential players. A coalition is a non-empty finite subset of N and let N := {N | ∅ = N ⊆ N, |N | < ∞}
denote the set of all coalitions of N. A TU-game (a game) is a pair (N, v) where
N ∈ N is the set of players and v : 2N −→ R is the characteristic function that
assigns to each coalition S ⊆ N a real number v(S), with the convention v(∅) = 0.
Given S, T ∈ N , we use S ⊂ T to indicate strict inclusion, that is, S ⊆ T but
S = T . By |S| we denote the cardinality of the coalition S ∈ N . By Γ we denote
the class of all games.
Given N ∈ N , let RN stand for the space of real-valued vectors indexed by N ,
x = (xi )i∈N , and for all S ⊆ N , x(S) = i∈S xi , with the convention x(∅) = 0. For
each x ∈ RN and T ⊆ N , x|T denotes the restriction of x to T : x|T = (xi )i∈T ∈ RT .
Given two vectors x, y ∈ RN , x ≥ y if xi ≥ yi , for all i ∈ N . We say that
x > y if x ≥ y and for some j ∈ N , xj > yj . Let (N, v) be a game and S ⊆ N ,
S = ∅. A coalition S is an equity coalition of (N, v) if S ∈ Argmax∅=R⊆N v(R) .
|R|
In addition, S is a maximal (w.r.t. inclusion) equity coalition of (N, v) if
S ∈ Argmax∅=R⊆N v(R) and there is no T ∈ Argmax∅=R⊆N v(R) such that
|R|
|R|
2

S ⊂ T . Given N , a set π = {P1 , . . . , Pm }, where Pi ⊆ N for all i ∈ {1, . . . , m},
with m ≤ |N |, is a partition of N if the following conditions hold: (i) Pi = ∅ for
all i ∈ {1, . . . , m}, (ii) ∪m Pi = N , and (iii) Pi ∩ Pj = ∅, for all i, j ∈ {1, . . . , m},
i=1
i = j.
The set of feasible payoff vectors of a game (N, v) is defined by X ∗ (N, v) :=
{x ∈ RN | x(N ) ≤ v(N )}. A solution on a class of games Γ ⊆ Γ is a mapping
σ which associates with each game (N, v) ∈ Γ a subset σ(N, v) of X ∗ (N, v).
Notice that σ is allowed to be empty. A solution on a class of games Γ ⊆ Γ is
said to be single-valued if |σ(N, v)| = 1 for all (N, v) ∈ Γ . Two games (N, v)
and (N, v ) are strategically equivalent if there is a vector (d1 , . . . , dn ) ∈ RN
and α > 0 such that for all coalitions S ⊆ N , v (S) = α v(S) + i∈S di . A
solution σ on Γ ⊆ Γ satisfies covariance if for all two strategically equivalent
games (N, v), (N, v ) ∈ Γ , σ(N, v ) = α σ(N, v) + i∈N di . The pre-imputation
set of (N, v) is defined by X(N, v) := {x ∈ RN | x(N ) = v(N )}, and the set of
imputations by I(N, v) := {x ∈ X(N, v) | xi ≥ v({i}), for all i ∈ N }. The core
of (N, v) is the set of those imputations where each coalition gets at least its worth,
that is, C(N, v) = {x ∈ X(N, v) | x(S) ≥ v(S) for all S ⊆ N }. A game (N, v)
is balanced if it has a non-empty core. By ΓBal we denote the class of balanced
games. A game (N, v) is superadditive if, for every S, T ⊆ N, S ∩ T = ∅,
v(S) + v(T ) ≤ v(S ∪ T ). A game (N, v) is convex if, for every S, T ⊆ N ,
v(S) + v(T ) ≤ v(S ∪ T ) + v(S ∩ T ). By ΓCon we denote the class of convex
games. Recall that ΓCon ⊂ ΓBal .
Given N ∈ N , and for any x ∈ RN , let us denote by x = (ˆ1 , . . . , xn ) the
ˆ
x
ˆ
vector obtained from x by rearranging its coordinates in a non-increasing order,
that is, x1 ≥ x2 ≥ . . . ≥ xn . In a similar way, for ∅ = T ⊆ N, x|T denotes
ˆ
ˆ
ˆ
the vector obtained from the restriction of x to T ordering its coordinates in
a non-increasing way: x|T 1 ≥ x|T 2 ≥ . . . ≥ x|T t , where t = |T |. For any two
vectors y, x ∈ RN with y(N ) = x(N ), we say that y Lorenz dominates x,
denoted by y L x, if k yj ≤ k xj , for all k ∈ {1, . . . , |N |}, with at least
j=1 ˆ
j=1 ˆ
one strict inequality. Given a coalition S ∈ N and a set A ⊆ RS , EA denotes
the set of allocations that are Lorenz undominated within A. That is, EA :=
{x ∈ A | y ∈ A such that y L x} . Given a game (N, v), the Lorenz core is
defined in a recursive way as follows. The Lorenz core of a singleton coalition
is L({i} , v) = {v({i})}. Now suppose that the Lorenz core for all coalitions of
cardinality k or less have been defined, where 1 < k < |N |. The Lorenz core of a
coalition S ⊂ N of size (k + 1) is defined by
L(S, v) = x ∈ RS | x(S) = v(S) and

T ⊂ S and y ∈ EL(T, v) such that y > x|T .

Note that, for all S ⊆ N, C(S, v) ⊆ L(S, v).
The weak constrained egalitarian solution (WCES) (Dutta and Ray,
1989), denoted by EL, selects the vectors that are Lorenz undominated within
the Lorenz core. For all (N, v) ∈ Γ, |EL(N, v)| ≤ 1 (Dutta and Ray,1989). The
3

constrained egalitarian solution, denoted by CE, is a single-valued solution
defined for two person games as follows: let (N, v) be a game with N = {i, j} and
suppose, without loss of generality, v(i) ≤ v(j), then CEj (N, v) = max v(N ) , v(j)
2
and CEi (N, v) = v(N ) − CEj (N, v).
The next two observations will be useful to prove our results.
Remark 1. (Hougaard et al. 2001 p. 153) Let N be a finite set of players, and let
S ⊆ N , S = ∅. If xS , yS ∈ RS , xS (S) = yS (S) and zN \S ∈ RN \S , then xS Lorenz
dominates yS if and only if xS , zN \S Lorenz dominates yS , zN \S .
Remark 2. Let N be a finite set of players, c ∈ R and (x1 , . . . , xn ) ∈ RN . It is
well-known that if i∈N xi = nc, then x is Lorenz dominated by (c, . . . , c) ∈ RN .
If i∈N xi > nc, let = i∈N xi − nc and define x = (x1 − n , . . . , xn − n ). Note
that x i = xi − n < xi , for all i ∈ N . Thus, x is Lorenz dominated by (c, . . . , c)
which implies, for all k = 1, . . . , n, x1 + . . . + xk > x 1 + . . . + x k ≥ kc.
ˆ
ˆ

3

Exact partition games

On the domain of convex games, Dutta and Ray (1989) show that the WCES picks
the payoff vector that is obtained by the following algorithm.
Let (N, v) be a convex game and EL(N, v) = {x}.
Step 1: Define v1 = v. Then find the unique coalition T1 ⊆ N such that for all
T ⊆ N , (i) v1 (T1 ) ≥ v1 (T ) , and (ii) if v1 (T1 ) = v1 (T ) and T = T1 , then |T1 | > |T |.
|T1 |
|T |
|T1 |
|T |
Uniqueness of such a coalition is guaranteed by convexity of (N, v). For all i ∈ T1 ,
xi =

v1 (T1 )
.
|T1 |

Step k: Suppose that T1 , . . . , Tk−1 have been defined.
Let Nk = N \ {T1 ∪ . . . ∪ Tk−1 } and let (Nk , vk ) be the marginal game defined as
follows:
vk (S) := v(T1 ∪ . . . ∪ Tk−1 ∪ S) − v(T1 ∪ . . . ∪ Tk−1 ),
(1)
for all S ⊆ Nk .
It can be shown that (Nk , vk ) is convex. Then find the unique coalition Tk ⊆ Nk
such that for all T ⊆ Nk , (i) vk (Tk ) ≥ vk (T ) , and (ii) if vk (Tk ) = vk (T ) and T = Tk ,
|Tk |
|T |
|Tk |
|T |
then |Tk | > |T |. For all i ∈ Tk ,
xi =

vk (Tk )
v(T1 ∪ . . . ∪ Tk ) − v(T1 ∪ . . . ∪ Tk−1 )
=
.
|Tk |
|Tk |

By construction, the WCES satisfies the following conditions: if π = (T1 , . . . , Tt )
is the ordered partition of N induced by EL(N, v) = {x}, then
4

• (C1): xi = xj for all i, j ∈ Tk and k = 1, . . . , t,
• (C2): x(T1 ∪ . . . ∪ Tk ) = v(T1 ∪ . . . ∪ Tk ), for all k = 1, . . . , t,
• (C3): xi > xj if i ∈ Tk , j ∈ Th , and k < h ≤ t.
The idea underlying this procedure is that agents in the unique maximal (w.r.t.
inclusion) coalition T1 maximizing the average worth v(T11|) share equally the amount
|T
v(T1 ) among them and leave the game. Then, the remaining agents N \ T1 play a
suitable reduced convex game where, again, agents in the unique maximal coalition
with highest average worth divide its worth equally among its members. The
process stops when all agents have been paid.
Theorem 2 in Dutta and Ray (1989) states that the output of this algorithm
is the WCES and that it belongs to the core. Theorem 3 in the same paper tells
us that, for convex games, the WCES Lorenz dominates every other core element.
Nevertheless, an analysis of the proofs of the aforementioned results reveals that
much weaker conditions than convexity are sufficient to guarantee the same results.
Definition 1. Let N = {1, . . . , n} be a finite set of players and x ∈ RN . We define
the ordered partition of N induced by x, π = (N1 , . . . , Nm ), as follows:
N1 = {i ∈ N | xi ≥ xk for all k ∈ N } ,
N2 = {i ∈ N \ N1 | xi ≥ xk for all k ∈ N \ N1 } ,
.
.
.
Nm = {i ∈ N \ N1 ∪ . . . ∪ Nm−1 | xi ≥ xk for all k ∈ N \ N1 ∪ . . . ∪ Nm−1 } .
Theorem 1. Let (N, v) be a balanced game, x ∈ C(N, v) and let π = (N1 , . . . , Nm )
be the ordered partition of N induced by x. If x (N1 ∪ . . . ∪ Nk ) = v (N1 ∪ . . . ∪ Nk ),
for all k = 1, . . . , m, then EL (N, v) = {x} and x L y, for all y ∈ C (N, v) \ {x}.
Proof. First we show that x L y, for all y ∈ C (N, v) \ {x}.
Assume, without loss of generality, that x1 ≥ x2 ≥ . . . ≥ xn . Then, the vector
obtained from x by rearranging its coordinates in non-increasing order is x = x.
Let us denote
ck =

 v(N )
1

 |N1 |


if k = 1

 v(N ∪...∪N ∪N )−v(N ∪...∪N )

1
1
k−1
k
k−1

|Nk |

if k > 1

for all k = 1, . . . , m, (m > 1).
Notice that xi = ck for all i ∈ Nk and k = 1, . . . , m. Let y ∈ C (N, v), y = x.
From Remark 1 we may suppose, without loss of generality, xi = yi for all i ∈ N .
Since y(N1 ) ≥ v(N1 ) = x(N1 ) = c1 |N1 |, and by Remark 2, we have that for all
t = 1, . . . , |N1 |,
tc1 ≤ y|N1 1 + . . . + y|N1 t ,
(2)
5

with at least one strict inequality.
Next we are going to prove that, for all t = 1, . . . , |N2 |,
x(N1 ) + tc2 ≤ y(N1 ) + y|N2 1 + . . . + y|N2 t .

(3)

If y(N2 ) ≥ x (N2 ) = |N2 |c2 , again by Remark 2, tc2 ≤ y|N2 1 + . . . + y|N2 t , for all
t = 1, . . . , |N2 |. This set of inequalities, together with (2), lead to expression (3).
If y(N2 ) < x(N2 ), let us denote ϕ1 = y(N1 ) − x(N1 ) ≥ 0 and β1 = x(N2 ) −
β
y(N2 ) > 0. Let z ∈ RN2 defined as zi = yi + |N1 | for all i ∈ N2 . Since x(N2 ) =
2
β
y(N2 ) + β1 = z(N2 ), by Remark 2 we have c2 ≤ z1 = y|N2 1 + |N1 | ≤ y|N2 1 + β1 ,
ˆ
2
which implies β1 ≥ c2 − y|N2 1 . This last inequality, together with y (N1 ∪ N2 ) ≥
v (N1 ∪ N2 ) = x (N1 ∪ N2 ) , lead to
ϕ1 = y(N1 ) − x(N1 ) ≥ x(N2 ) − y(N2 ) = β1 ≥ c2 − y|N2 1 .

(4)

Now from (4) it follows
x(N1 ) + c2 ≤ y(N1 ) + y|N2 1 .

(5)

|N2 |

If |N2 | ≥ 2 and

y|N2 i ≥ (|N2 | − 1) c2 , then from Remark 2, tc2 ≤ y|N2 2 + . . . +
i=2

y|N2 t+1 , for all t = 1, . . . , |N2 |−1, which leads, together with (5), to (3). Otherwise,
|N2 |

if |N2 | ≥ 2 and

y|N2 i < (|N2 | − 1) c2 , let us denote
i=2
|N2 |

ϕ2 = y(N1 ) + y|N2 1 − x(N1 ) − c2 and β2 = (|N2 | − 1) c2 −

y|N2 i > 0.
i=2

From (4) it follows ϕ2 ≥ β2 > 0. Next we show that β2 ≥ c2 − y|N2 2 . Choose
2
k ∈ N2 such that yk ≥ yi for all i ∈ N2 and define z ∈ RN2 \{k} as zi = yi + |Nβ|−1
2
for all i ∈ N2 \ {k}. Since z(N2 \ {k}) = y(N2 \ {k}) + β2 = x(N2 ) − c2 , by Remark
2
2 we have c2 ≤ z1 = y|N2 2 + |Nβ|−1 ≤ y|N2 2 + β2 , which implies β2 ≥ c2 − y|N2 2 .
ˆ
2
Since ϕ2 ≥ β2 , we obtain
ϕ2 ≥ c2 − y|N2 2 .
(6)
Now from (6) it can be checked that x(N1 ) + 2c2 ≤ y(N1 ) + y|N2 1 + y|N2 2 . Applying
the same reasoning for t = 3, . . . , |N2 | we obtain (3).
Following the same line of argument it can be proved that, for all k = 3, . . . , m
and all t = 1, . . . , |Nk |,
t

x (N1 ∪ . . . ∪ Nk−1 ) + tck ≤ y (N1 ∪ . . . ∪ Nk−1 ) +

y|Nk j .
j=1

6

(7)

Finally, combining (2), (3) and (7) we get
x1 = c1 ≤ y|N1 1 ≤ y1
x1 + x2 = 2c1 ≤ y|N1 1 + y|N1 2 ≤ y1 + y2
.
.
.
x1 + . . . + x|N1 | = x(N1 ) ≤ y(N1 ) ≤ y1 + . . . y|N1 |
x1 + . . . + x|N1 |+1 = x(N1 ) + c2 ≤ y(N1 ) + y|N2 1 ≤ y1 + . . . y|N1 |+1
.
.
.
x1 + . . . + x|N1 |+|N2 | = x (N1 ∪ N2 ) ≤ y (N1 ∪ N2 ) ≤ y1 + . . . y|N1 |+|N2 |
.
.
.
x1 + . . . + xn = x (N1 ∪ . . . ∪ Nm ) = y (N1 ∪ . . . ∪ Nm ) = y1 + . . . + yn ,
with at least one strict inequality,1 which means that x L y.
To see that EL(N, v) = {x}, we replicate the induction argument used by
Dutta and Ray (1989) to prove their Theorem 2 (step 2).2
Note first that EL (N1 , v) = x|N1 . Next we see that for all t = 1, . . . , m−1, if
EL (N1 ∪ . . . ∪ Nt , v) = x|N1 ∪...∪Nt , then EL (N1 ∪ . . . ∪ Nt+1 , v) = x|N1 ∪...∪Nt+1 .
Suppose that EL (N1 ∪ . . . ∪ Nt , v) = x|N1 ∪...∪Nt but EL (N1 ∪ . . . ∪ Nt+1 , v) =
x|N1 ∪...∪Nt+1 , for some t. Since x (N1 ∪ . . . ∪ Nt+1 ) = v (N1 ∪ . . . ∪ Nt+1 ) and
x ∈ C (N, v), we have
x|N1 ∪...∪Nt+1 ∈ C N1 ∪ . . . ∪ Nt+1 , v|N1 ∪...∪Nt+1 ⊆ L (N1 ∪ . . . ∪ Nt+1 , v) ,
and thus there exists y ∈ L (N1 ∪ . . . ∪ Nt+1 , v) with y
y1
ˆ
y1 + y2
ˆ
ˆ

L

x|N1 ∪...∪Nt+1 . Then,

≤ x1
≤ x1 + x2
.
.
.

(8)

y1 + . . . + y|N1 ∪...∪Nt+1 | = x1 + . . . + x|N1 ∪...∪Nt+1 |
ˆ
ˆ
with at least one strict inequality.
Since y (N1 ∪ . . . ∪ Nt+1 ) = x (N1 ∪ . . . ∪ Nt+1 ), if yj ≥ xj for all j ∈ N1 ∪ . . . ∪
Nt+1 then we would have y = x|N1 ∪...∪Nt+1 , in contradiction with y L x|N1 ∪...∪Nt+1 .
As a consequence, the set J := {j ∈ N1 ∪ . . . ∪ Nt+1 | yj < xj } must be non-empty.
Take then q ∗ = min {k ∈ {1, . . . , t + 1} |J ∩ Nk = ∅}. We claim that,
yi ≤ xi for all i ∈ Nq∗ .
1
2

This strict inequality follows from expression (2).
We describe in detail the induction argument for the convenience of the reader.

7

Indeed, if q ∗ = 1, for all i ∈ N1 it follows from (8) that yi ≤ y1 ≤ x1 = xi . If q ∗ > 1,
ˆ
from yi ≥ xi for all i ∈ N1 and expression (8) we have yi = xi for all i ∈ N1 .Then,
again from (8), we obtain y|N1 |+1 ≤ x|N1 |+1 . The repetition of the same argument
ˆ
leads to yi = xi for all i ∈ N1 ∪ . . . ∪ Nq∗ −1 . Now, taking into account (8) and the
definition of π we obtain, for all i ∈ Nq∗ ,
yi ≤ y|N1 ∪...∪Nq∗ −1 |+1 ≤ x|N1 ∪...∪Nq∗ −1 |+1 = xi .
ˆ
Note that q ∗ ≤ t, since otherwise y (N1 ∪ . . . ∪ Nt+1 ) < x (N1 ∪ . . . ∪ Nt+1 ).
So, denote T = N1 ∪ . . . ∪ Nq∗ . By hypothesis, EL(T, v) = x|T . But then,
since yi ≤ xi for all i ∈ T and there exists j ∗ ∈ Nq∗ such that yj ∗ < xj ∗ , we
conclude that y ∈ L (N1 ∪ . . . ∪ Nt+1 , v), getting a contradiction. This means that
EL(N, v) = {x}.
Remark 3. Under some conditions of positivity, a similar result was stated by
Sánchez-Soriano et al. (2014). In that paper, Proposition 2 says the following:
The vector a = (1n1 a1 , 1n2 a2 . . . , 1nt at ) such that a1 ≥ a2 ≥ . . . ≥ at > 0 and
t
ni
for all i = 1, . . . , t, Lorenz dominates
i=1 ni = n, where 1ni = (1, . . . , 1) ∈ R
n1
n
each other element x ∈ R satisfying i=1 xi ≥ n1 a1 , n1 +n2 xi ≥ 2 ni ai , . . .,
i=1
i=1
n−nt
t
n
t−1
i=1 xi ≥
i=1 ni ai .
i=1 xi =
i=1 ni ai , and
In our context, this implies v(N1 ∪ . . . ∪ Ni ) > 0, for all i = 1, . . . , m, being
(N1 , . . . , Nm ) a partition of N as described in Definition 1. At this point, it is
important to pointed out that the WCES fails to satisfies covariance (see Dutta
and Ray, 1989) and so the problem of existence of the WCES and the properties
of Lorenz domination cannot be solved just by looking at positive games.
Let us show an example to illustrate this point. Let (N, v) be a game with N =
{1, 2, 3} and v({1}) = 0.8, v({2}) = −1, v({3}) = −2, v({12}) = −0.1, v({13}) =
−0.8, v({23}) = −3.5 and v({123}) = −1.5. Let x = (0.8, −0.9, −1.4) ∈ C(N, v).
Then, the ordered partition of N induced by x is π = ({1} , {2} , {3}), with x1 =
v({1}) > 0, x1 + x2 = v({1} ∪ {2}) < 0 and x1 + x2 + x3 = v({1} ∪ {2} ∪ {3}) <
0. From Theorem 1, EL(N, v) = {x} and x Lorenz dominates every other core
element. However, this last assertion can not be derived from Proposition 2 in
Sánchez-Soriano et al. (2014).
Theorem 1 generalizes both Theorem 2 and Theorem 3 in Dutta and Ray
(1989), and it can be useful to check that a core element is the WCES.
Let us introduce the class of games that satisfies the conditions stated in Theorem 1.
Definition 2. A game (N, v) is an exact partition game if there exists a core
element x such that the ordered partition of N induced by x, π = (N1 , . . . , Nm ),
satisfies x(N1 ∪ . . . ∪ Nk ) = v(N1 ∪ . . . ∪ Nk ), for all k = 1, . . . , m.

8

Let ΓEP denote the class of exact partition games. This class is large enough
to include convex games and dominant diagonal assignment games,3 but also nonsuperadditive games.
Example 1. Let (N, v) be a balanced game with set of players N = {1, 2, 3} and
characteristic function:
S
{1}
{2}
{3}

v(S)
S
1
{12}
1
{13}
1
{23}

v(S)
S
0
{123}
7
0

v(S)
9

This games is not supperadditive since v({12}) < v({1}) + v({2}), but (N, v) ∈
ΓEP . Indeed, take x = (3.5, 2, 3.5) ∈ C(N, v). The ordered partition of N induced
by x, π = ({13} , {2}), satisfies x1 + x3 = v({12}) and x(N ) = v(N ). Hence,
EL(N, v) = {x} and (N, v) ∈ ΓEP .
In Section 4, we will axiomatize the WCES on ΓEP .

4

Axiomatic characterizations

The main concern of this section is to characterize the WCES over the domain
of exact partition games, ΓEP . As particular cases, we obtain new axiomatic
characterizations over the class of convex games.
On the domain of convex games, the first characterization was provided by
Dutta (1990) by means of constrained egalitarianism and consistency with respect
to both the max reduced game (Davis and Maschler, 1965) and the self reduced
game (Hart and Mas-Colell,1989). Constrained egalitarianism is a prescriptive
property that imposes to select, for two person games, the Lorenz maximal allocation within the core. Consistency is a sort of internal stability requirement that
relates the solution of a game to the solution of the game when some players leave
the game.
A solution σ on Γ ⊆ Γ satisfies
• Constrained egalitarianism if for all N ∈ N with |N | = 2, and all (N, v) ∈
Γ , it holds σ(N, v) = CE(N, v).
Note that any two person exact partition game is convex. Thus, the WCES
satisfies constrained egalitarianism on ΓEP .
To define consistency, we need to introduce the notion of reduced game.
3

Using different arguments, Llerena (2012) shows that on the class of dominant diagonal
assignment games, the τ -value (Tijs, 1981) satisfies the requirements of Theorem 1.

9

Definition 3. (Davis and Maschler, 1965) Let (N, v) be a game, ∅ = N ⊂ N and
x ∈ RK where N \ N ⊆ K ⊆ N . The max reduced game relative to N at x is the
N
game N , rM,x (v) defined by

N
rM,x (v)(S)






=

0

if S = ∅,
max {v(S ∪ Q) − x(Q)} if ∅ = S ⊂ N ,

 Q⊆N \N


v(N ) − x(N \ N )

(9)

if S = N .

Remark 4. The max reduction operation is transitive (See, for instance, Chang
N
N
N
and Hu, 2007). That is, rM,x|N rM,x (v) = rM,x (v), for all N ∈ N , all (N, v) ∈ Γ,
all coalitions ∅ = N ⊂ N ⊂ N and all payoff vector x ∈ RK with N \ N ⊆ K ⊆
N.
In the max reduced game (relative to N at x), the worth of a coalition S ⊂ N
is determined under the assumption that S can choose the best partners in N \ N ,
provided they are paid according to x. Max consistency says that in this max
reduced game, the original agreement should be confirmed.
A solution σ on Γ ⊆ Γ satisfies
• Max consistency if for all N ∈ N , all (N, v) ∈ Γ , all N ⊂ N, N = ∅, and
N
N
all x ∈ σ(N, v), then N , rM,x (v) ∈ Γ and x|N ∈ σ N , rM,x (v) .
• Weak max consistency if for all N ∈ N , all (N, v) ∈ Γ , all N ⊂ N
N
with 1 ≤ |N | ≤ 2 and all x ∈ σ(N, v), then N , rM,x (v) ∈ Γ and x|N ∈
N
σ N , rM,x (v) .

• Rich player max consistency if for all N ∈ N , all (N, v) ∈ Γ and all
x ∈ σ(N, v), if N1 ⊆ N, N1 = N, is the set of players with highest payoff
N \N
N \N
(w.r.t. x), then N \ N1 , rM,x 1 (v) ∈ Γ and x|N \N1 ∈ σ N \ N1 , rM,x 1 (v) .
Weak max consistency applies the condition of max consistency to reduced
games with at most two players. Rich player max consistency weakens max consistency just requiring this condition when rich players leave the game. Clearly,
max consistency implies both weak and rich player max consistency.
Proposition 1. The WCES satisfies max consistency on ΓEP .
Proof. For two person games, max consistency clearly holds. Let (N, v) ∈ ΓEP
and x = EL(N, v) with |N | > 2. Since the max reduction operation is transitive
N \{i}
(see Remark 4), it is enough to see that, for all i ∈ N , N \ {i}, rM,x (v) ∈ ΓEP
N \{i}

and x|N \{i} = EL N \ {i}, rM,x (v) .
Let π = (N1 , . . . , Nm ) be the ordered partition of N induced by x. We distinguish two cases:
10

1) If m = 1, then x =

v(N )
, . . . , v(N|)
|N |
|N

∈ C(N, v). Let i ∈ N . By max
N \{i}

consistency of the core (Peleg, 1986), x|N \{i} ∈ C N \ {i}, rM,x (v) . Hence,
N \{i}

N \{i}

N \ {i}, rM,x (v) ∈ ΓEP and x|N \{i} = EL N \ {i}, rM,x (v) .
2) If m > 1, take k ∈ {1, . . . , m} and i ∈ Nk . The ordered partition of N \ {i}
induced by x|N \{i} is either π = (N1 , . . . , Nk−1 , Nk \ {i}, Nk+1 , . . . , Nm ), if
|Nk | > 1, or π = (N1 , . . . , Nk−1 , Nk+1 , . . . , Nm ), otherwise.
From the max consistency of the core, the definition of max reduced game
and the fact that x(N1 ∪ . . . ∪ Nk ) = v(N1 ∪ . . . ∪ Nk ) for all k ∈ {1, . . . , m},
we have
• For h ∈ {1, . . . , k − 1},
N \{i}

x(N1 ∪ . . . ∪ Nh ) ≥ rM,x (v)(N1 ∪ . . . ∪ Nh )
≥ v(N1 ∪ . . . ∪ Nh )
= x(N1 ∪ . . . ∪ Nh ),
which means that
N \{i}

x(N1 ∪ . . . ∪ Nh ) = rM,x (v)(N1 ∪ . . . ∪ Nh ).

(10)

• For h ∈ {k, . . . , m},
x(N1 ∪ . . . ∪ Nk \ {i} ∪ . . . ∪ Nh ) ≥
≥
=
=

N \{i}

rM,x (v)(N1 ∪ . . . ∪ Nk \ {i} ∪ . . . ∪ Nh )
v(N1 ∪ . . . ∪ Nk ∪ . . . ∪ Nh ) − xi
x(N1 ∪ . . . ∪ Nk ∪ . . . ∪ Nh ) − xi
x(N1 ∪ . . . ∪ Nk \ {i} ∪ . . . ∪ Nh ),

which means that
N \{i}

x(N1 ∪ . . . ∪ Nk \ {i} ∪ . . . ∪ Nh ) = rM,x (v)(N1 ∪ . . . ∪ Nk \ {i} ∪ . . . ∪ Nh ).
(11)
From (10) and (11) it follows that x|N \{i} satisfies the conditions stated in
N \{i}
Theorem 1 (w.r.t. π ). Hence, we conclude that N \ {i}, rM,x (v) ∈ ΓEP
N \{i}

and x|N \{i} = EL N \ {i}, rM,x (v) .
To prove that max consistency together with constrained egalitarianism characterize the WCES over the class of convex games, Dutta (1990) invokes converse
max consistency, which is the dual property of max consistency. This property is
crucial in his proof of uniqueness.
A solution σ on Γ ⊆ Γ satisfies
11

• Converse max consistency if for all N ∈ N with |N | ≥ 3, all (N, v) ∈ Γ
and all x ∈ RN with x(N ) = v(N ), if for all N ⊂ N with |N | = 2,
N
N
N , rM,x (v) ∈ Γ and x|N ∈ σ N , rM,x (v) , then x ∈ σ(N, v).
Converse max consistency says that if the projection of an efficient allocation
x is chosen for every two player max reduced game, then x should be chosen for
the original game.
Unfortunatly, Example 2 bellow reveals that the WCES is in conflict with
converse max consistency on ΓEP .
Example 2. (Arín and Iñarra, 2001) Let (N, v) be a balanced game with set of
players N = {1, 2, 3, 4} and characteristic function:
S
{1}
{2}
{3}
{4}

v(S)
0
0
0
0

S
v(S)
S
{12}
0
{123}
{13}
2
{124}
{14}
2
{134}
{23}
2
{234}
{24}
2
{34}
0

v(S)
S
v(S)
0
{1234}
4
0
0
0

Take x = (1, 1, 1, 1) ∈ C(N, v). The ordered partition of N induced by x is π =
({N }) and x(N ) = v(N ). Hence, EL(N, v) = {x} and (N, v) ∈ ΓEP . Now choose
N
y = (2, 2, 0, 0) ∈ C(N, v). Below, we describe the max reduced games N , rM,y
relative to N ⊂ N at y with |N | = 2,
{12}

S
{12}

rM,y (v)
4

{14}

S
{14}

rM,y (v)
2

{24}

S
{24}

rM,y (v)
2

S
{1}
{2}

rM,y (v)
2
2

S
{1}
{4}

rM,y (v)
2
0

S
{2}
{4}

rM,y (v)
2
0

{13}

rM,y (v)
2

{23}

rM,y (v)
2

{34}

rM,y (v)
0

{12}

S
{1}
{3}

rM,y (v)
S
2
{13}
0

{14}

S
{2}
{3}

rM,y (v)
S
2
{23}
0

{24}

S
{3}
{4}

rM,y (v)
S
0
{34}
0

{13}

{23}

{34}

The corresponding constrained egalitarian solution are:
{12}

CE {13}, rM,y (v) = (2, 0) = y|{13} ,

{13}

{14}

CE {23}, rM,y (v) = (2, 0) = y|{23} ,

{24}

CE {34}, rM,y (v) = (0, 0) = y|{34} .

CE {12}, rM,y (v) = (2, 2) = y|{12}

{23}

CE {14}, rM,y (v) = (2, 0) = y|{14}

{34}

CE {24}, rM,y (v) = (2, 0) = y|{24}
12

However, y = EL(N, v).
To be precise, Dutta (1990) only uses bilateral max consistency, that is, max
consistency for only two person games, together with constrained egalitarianism,
to characterize the WCES on ΓCon . Let us see that on ΓEP , these two properties
do not characterize the WCES. To do this, we introduce the egalitarian core (Arin
and Iñarra, 2001).
Definition 4. The egalitarian core of a balanced game (N, v), denoted by Eg C,
is the set Eg C(N, v) = {x ∈ C(N, v) | xi > xj ⇒ Sij (x) = 0}, where Sij (x) =
max{v(S) − x(S) | i ∈ S, j ∈ S, S ⊂ N }.
Arín and Iñarra (2001) show that the egalitarian core satisfies max consistency and constrained egalitarianism on ΓBal . Note that a two person balanced
games is an exact partition game since the constrained egalitarian solution is a
core element satisfying the conditions stated in Theorem 1. Thus, the egalitarian
core satisfies bilateral max consistency and constrained egalitarianism on ΓEP . In
Example 2, EL(N, v) = {(1, 1, 1, 1)} and (2, 2, 0, 0) ∈ Eg C(N, v), which means
that EL(N, v) = Eg C(N, v). The same example also illustrates that the egalitarian core is not max consistent on ΓEP . Indeed, consider the max reduced
N \{4}
game N \ {4}, rM,y (v) with y = (2, 2, 0, 0). As the reader can easily check,
N \{4}

N \{4}

Eg C N \ {4}, rM,y (v) = {(2, 2, 0} and N \ {4}, rM,y (v) ∈ ΓEP .
The second characterization of the WCES provided by Dutta (1990) uses self
consistency (Hart and Mas-Collel, 1989). This property is defined for single-valued
solutions.
A single-valued solution σ on Γ ⊆ Γ satisfies
• Self consistency if for all N ∈ N , all (N, v) ∈ Γ and all N ⊂ N, N = ∅,
N
N
then N , rS,σ (v) ∈ Γ and, for all i ∈ N , σi (N, v) = σi N , rS,σ (v) ,
N
where N , rS,σ (v) is the self reduced game of (N, v) relative to N and
σ definded as follows:

N
rS,σ (v)(R) =





0
 v(R ∪ (N \ N )) −


σi R ∪ (N \ N ), v|R∪(N \N )

if R = ∅,
if ∅ = R ⊆ N .

i∈N \N

(12)
In the self reduced game (relative to N at σ), the worth of a coalition R ⊆ N
is the worth of R ∪ (N \ N ) in the original game minus the sum of the payoffs that
the solution assigns the members of N \ N for the subgame faced by the group
R ∪ (N \ N ). Self consistency states that in this self reduced game, the original
agreement should be accepted. The next example shows that the WCES fails to
satisfies self consistency on ΓEP .
13

Example 3. Let (N, v) be a balanced game with set of players N = {1, 2, 3} and
characteristic function:
S
{1}
{2}
{3}

v(S)
S
2
{12}
1
{13}
0
{23}

v(S)
S
4
{123}
2
1.5

v(S)
4

Take x = (2, 2, 0) ∈ C(N, v). The ordered partition of N induced by x, π =
({12} , {3}), satisfies x1 + x2 = v({12}) and x(N ) = v(N ). Hence, from Theorem
1 we have that EL(N, v) = {x} and (N, v) ∈ ΓEP .
Let N = {13}. Then,
N
rS,EL (v)({1})

= v({12}) − EL2 {12}, v|{12} = 4 − 2 = 2,

N
rS,EL (v)({3}) = v({23}) − EL2 {23}, v|{23} = 1.5 − 1 = 0.5 and
N
rS,EL (v)({13}) = v(N ) − EL2 (N, v) = 4 − 2 = 2.

(13)

N
N
Note that N , rS,EL (v) has no imputations. Thus, N , rS,EL (v) ∈ ΓEP and the
WCES is not defined.

In order to characterize the WCES within the domain of exact partition games
we will make use, together with consistency, the following properties.
A solution σ on Γ ⊆ Γ satisfies
• Nonemptiness if for all N ∈ N and all (N, v) ∈ Γ , it holds σ(N, v) = ∅.
• Efficiency if for all N ∈ N , all (N, v) ∈ Γ and all x ∈ σ(N, v), then
x(N ) = v(N ).
• Individual rationality if for all N ∈ N , all (N, v) ∈ Γ , all x ∈ σ(N, v)
and all i ∈ N , then xi ≥ v({i}).
• Core selection if for all N ∈ N , all (N, v) ∈ Γ , all x ∈ σ(N, v) and all
S ⊆ N , then x(S) ≥ v(S).
• Rich player feasibility if for all N ∈ N , all (N, v) ∈ Γ and all x ∈ σ(N, v),
it holds x(N1 ) ≤ v(N1 ), where N1 denotes the set of players with highest
payoff (w.r.t. x).
• Internal Lorenz stability if for all N ∈ N with |N | ≥ 2, all (N, v) ∈ Γ
and all x, y ∈ σ(N, v), neither x L y nor y L x.
• External Lorenz stability (over the core) if for all N ∈ N with |N | ≥ 2
and all (N, v) ∈ Γ , if x ∈ C(N, v) \ σ(N, v), then there is y ∈ σ(N, v) such
that y L x.
14

Efficiency says that all the gains from cooperation should be shared among
the players. Individual rationality means that the proposed solution can not be
improved upon by a single player, while core selection extends this impossibility
to any coalition. Note that core selection, together with the feasibility assumption
of a solution, imply efficiency. Rich player feasibility states that the total amount
received by players with the highest payoff can not exceed what they can get for
themselves. Internal Lorenz stability is a natural requirement in an egalitarian
framework. External Lorenz stability (over the core) gives priority to the social
goal of equality in front of particular interests, in the sense that if a core element
is not an outcome of the solution is because there is an allocation in the solution
which is more egalitarian (w.r.t. the Lorenz criterion).
Next, we state our first characterization result.
Theorem 2. The WCES is the unique solution on ΓEP that satisfies weak max
consistency, individual rationality, internal Lorenz stability and external Lorenz
stability (over the core).
Proof. Proposition 1 implies weak max consistency and individual rationality
comes from the fact that the WCES selects a core element. Internal Lorenz stability
is because the WCES is single-valued, and external Lorenz stability (over the core)
follows from Theorem 1.
In order to show uniqueness, suppose there is a solution σ = EL satisfying
the above four properties. Let (N, v) ∈ ΓEP . Note that external Lorenz stability
(over the core) implies nonemptiness. If |N | = 1, by nonemptiness and individual
rationality (and feasibility) σ(N, v) = EL(N, v). Suppose |N | ≥ 2. We first show
that σ(N, v) ⊆ C(N, v). Let x ∈ σ(N, v) and i ∈ N . Then, weak max consistency
{i}
and efficiency for one person game imply xi = rM,x (v)({i}) = v(N ) − j∈N \{i} xj ,
which proves efficiency. To check coalitional rationality, let ∅ = S ⊂ N and i ∈
{ik}
N \ S. Chose k ∈ S and consider the max reduced game {ik}, rM,x (v) . By weak
{ik}

max consistency, x|{ik} ∈ σ {ik}, rM,x (v) and, by individual rationality, xk ≥
{ik}

rM,x (v)({k}) ≥ v(S) − x(S \ {k}), which implies x(S) ≥ v(S). Hence, x ∈ C(N, v).
Let us denote x∗ = EL(N, v). If x∗ ∈ σ(N, v), by external Lorenz domination
(over the core) there is y ∈ σ(N, v) such that y L x∗ , a contradiction. Hence,
x∗ ∈ σ(N, v). Finally, by internal Lorenz stability we conclude that σ(N, v) =
EL(N, v).
To see that the properties in Theorem 2 are independent we introduce the
following solutions:
• Let σ1 defined as follows: σ1 (N, v) = ∅, for each (N, v) ∈ ΓEP . Then, σ1 satisfies weak max consistency, individual rationality, internal Lorenz stability,
but no external Lorenz stability (over the core).
• Let σ2 defined as follows: σ2 (N, v) = C(N, v), for each (N, v) ∈ ΓEP . Then,
15

σ2 satisfies weak max consistency, individual rationality, external Lorenz stability (over the core), but not internal Lorenz stability.
• Let σ3 defined as follows: σ3 (N, v) = EI(N, v), for each (N, v) ∈ ΓEP .
That is, σ3 chooses the Lorenz maximal allocations in the imputation set.
Llerena and Mauri (2015) show that this solution is single-valued and Lorenz
dominates all core elements. Then, σ3 satisfies individual rationality, internal
Lorenz stability, external Lorenz stability (over the core), but not weak max
consistency.
• Let σ4 defined as follows: σ4 (N, v) = EL(N, v) if |N | ≥ 2, and σ4 (N, v) =
X ∗ ({i}, v) if N = {i}, for each (N, v) ∈ ΓEP . Then, σ4 satisfies weak max
consistency, internal Lorenz stability, external Lorenz stability (over the core),
but not individual rationality.
It is well-known that the max reduced game of a convex game relative to a core
element is also convex (see, for instance, Hokari, 2002). Moreover, on this domain
the WCES selects the unique Lorenz maximal allocation within the core (Dutta
and Ray, 1989). Thus, Theorem 2 holds on the domain of convex games.
Theorem 3. The WCES is the unique solution on ΓCon that satisfies weak max
consistency, individual rationality, internal Lorenz stability and external Lorenz
stability (over the core).
Defined on the domain of convex games, σ1 , σ2 , σ3 and σ4 show the independence of the properties in Theorem 3.
Although the WCES satisfies nice properties on the domain of convex games,
and some of them are inherited on the domain of exact partition games, its existence is not linked to the nonemptiness of the core. On the domain of balanced
games, an alternative route, already suggested by Dutta and Ray (1989) and latter
adopted by Arin and Iñarra (2001) and Hougaard et al. (2001), is to focus on the
Lorenz maximal allocations within the core.
Definition 5. The Lorenz maximal core of a balanced game (N, v), denoted by
EC(N, v), is the set EC(N, v) = {x ∈ C(N, v) | y ∈ C(N, v) such that y L x} .
By definition, the Lorenz maximal core satisfies individual rationality and internal Lorenz stability. External Lorenz stability (over the core) follows by compactness of the core. Arin and Iñarra (2001) and also Hougaard et al. (2001), show
that the Lorenz maximal core satisfies max consistency. Since weak max consistency and individual rationality imply core selection, uniqueness follows directly
from internal Lorenz stability and external Lorenz stability (over the core). Thus,
properties in Theorem 2 also characterize the Lorenz maximal core on the domain
of balanced games.
16

Theorem 4. The Lorenz maximal core is the unique solution on ΓBal that satisfies
weak max consistency, individual rationality, internal Lorenz stability and external
Lorenz stability (over the core).
Solution σ1 , σ2 and σ3 defined on ΓBal , together with solution σ5 defined bellow,
show the independence of the properties in Theorem 4.
• Let σ5 defined as follows: σ5 (N, v) = EC(N, v) if |N | ≥ 2, and σ5 (N, v) =
X ∗ ({i}, v) if N = {i}, for each (N, v) ∈ ΓBal . Then, σ4 satisfies weak max
consistency, internal Lorenz stability, external Lorenz stability (over the core),
but not individual rationality.
Our second characterization is by means of nonemptiness, rich player max
consistency, core selection and rich payer feasibility.
Theorem 5. The WCES is the unique solution on ΓEP that satisfies nonemptiness,
rich player max consistency, core selection, and rich player feasibility.
Proof. Proposition 1 implies rich player max consistency, nonemptiness and core
selection follow from the fact that the WCES selects a core element, rich player
feasibility comes from the structure of the WCES on ΓEP .
In order to show uniqueness, suppose there is a solution σ = EL satisfying the above four properties. Let (N, v) ∈ ΓEP , EL(N, v) = {x} and π =
(N1 , N2 , . . . , Nm ) be the ordered partition of N induced by x. First, we will see that
N1 is the unique maximal equity coalition of (N, v). Let R ⊆ N be an equity coalition. Recall that xk = v(N11|) , for all k ∈ N1 . Since x ∈ C(N, v), there exists i ∈ R
|N
v(R)
. Thus, for each k ∈ N1 , it holds xk = v(N11|) ≥ xi ≥ v(R) ≥ v(N11|) ,
|R|
|N
|R|
|N
v(R)
v(N1 )
that |R| = |N1 | . Hence, N1 is an equity coalition. Suppose that

such that xi ≥

which means
R \ N1 = ∅. Then,
x(R) =

xi = |N1 ∩ R|

xi +
i∈N1 ∩R

< |N1 ∩ R|

i∈R\N1

v(N1 )
+
xi
|N1 |
i∈R\N1

v(N1 )
v(N1 )
v(N1 )
+ |R \ N1 |
=
|R| = v(R),
|N1 |
|N1 |
|N1 |

contradicting x ∈ C(N, v). Hence, R ⊆ N1 .
By nonemptiness, σ(N, v) = ∅. Let y ∈ σ(N, v) and π = (R1 , R2 , . . . , Rk )
be the ordered partition of N induced by y. By core selection and rich player
feasibility, yi = v(R11|) for all i ∈ R1 . If R1 = N , by core selection y = x and
|R
thus σ(N, v) = EL(N, v). Otherwise, since x L y, x1 ≤ y1 which means that
ˆ
ˆ
v(R1 )
v(N1 )
v(N1 )
yi ≥ |N1 | for all i ∈ R1 . Hence, |R1 | ≥ |N1 | . This, together with the fact that

17

N1 is the unique maximal equity coalition of (N, v), leads to R1 ⊆ N1 . Suppose
that |R1 | < |N1 |. Then,
x1
ˆ
x1 + x2
ˆ
ˆ

= y1
ˆ
= y1 + y2
ˆ
ˆ
.
.
.

x1 + . . . + x|R1 |
ˆ
ˆ
= y1 + . . . + y|R1 |
ˆ
ˆ
x1 + . . . + x|R1 | + x|R1 |+1 > y1 + . . . + y|R1 | + y|R1 |+1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
in contradiction with x L y. Thus, R1 = N1 and xi = yi for all i ∈ N1 ,
N \N
N \R
which imply N \ N1 , rM,x 1 (v) = N \ R1 , rM,y 1 (v) . By rich player max consisN \N

N \N

tency, y|N \N1 ∈ σ N \ N1 , rM,x 1 (v) and x|N \N1 = EL N \ N1 , rM,x 1 (v) , with
N \N

N \ N1 , rM,x 1 (v) ∈ ΓEP . Applying the same arguments as before, it can be
checked that N2 = R2 and xi = yi for all i ∈ N2 . Following this reasoning step by
step we reach x = y, which means that σ = EL.
Solution σ1 defined on ΓEP , together with the following σ6 , σ7 and σ8 show the
independence of the properties in Theorem 5.
• Let σ6 defined as follows: σ6 (N, v) = {x ∈ C(N, v) | x(N1 ) = v(N1 )}, for
each (N, v) ∈ ΓEP , where N1 denotes the set of players with highest payoff
(w.r.t. x). Then, σ6 satisfies nonemptiness, core selection and rich player
feasibility, but not rich player max consistency.
v(N )
, . . . v(N|) , for each (N, v) ∈
• Let σ7 defined as follows: σ7 (N, v) =
|N |
|N
ΓEP . Then, σ7 satisfies nonemptiness, rich player max consistency and rich
player feasibility, but not core selection.

• Let σ8 defined as follows: σ8 (N, v) = EL(N, v) if |N | ≥ 3, and σ8 (N, v) =
C(N, v) if |N | ≤ 2, for each (N, v) ∈ ΓEP . Then, σ8 satisfies nonemptiness,
rich player max consistency and core selection, but not rich player feasibility.
Theorem 5 also holds on the domain of convex games.
Theorem 6. The WCES is the unique solution on ΓCon that satisfies nonemptiness, rich player max consistency, core selection and rich player feasibility.
Definded on the domain of convex games, σ1 , σ6 , σ7 and σ8 show the independence of the properties in Theorem 6.
Finally, let us pointed out that on the domain of balanced games, the properties
stated in Theorem 5 do not characterize the Lorenz maximal core since it fails to
satisfy rich player feasibility.

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5

Final remarks

We have introduced a subclass of balanced games, called exact partition games
ΓEP . This class is large enough to include convex games and dominant diagonal
assignment games, but also nonsuperadditive games. On ΓEP , we have shown
that the WCES behaves as in convex games, that is, it exists, belongs to the core
and Lorenz dominates every other core element. Moreover, we have provided two
axiomatic characterizations by means of consistency, rationality, and two properties
of fairness based on the Lorenz criterion. Interestingly, both characterizations hold
over the domain of convex games. Additionally, one of them could be extended to
balanced games characterizing the Lorenz maximal core on this domain. Finally,
for future research it could be interesting to study whether the characterizations
of the WCES given by Klijn et al. (2000), Hougaard et al. (2001) and Arin et al.
(2003) over the domain of convex games can be extended to ΓEP .

Acknowledgments
We are very grateful to J.M. Izquierdo for his valuable suggestions. The first
author acknowledges the support from research grants ECO2014-52340 (Ministerio
de Economía y Competitividad) and 2014SGR631 (Generalitat de Catalunya).

References
[1] Arín, J. and E. Iñarra (2001) Egalitarian solutions in the core. International
Journal of Game Theory, 30: 187-193.
[2] Arín, J., J. Kuipers and D. Vermeulen (2003) Some characterizations of the
egalitarian solutions on classes of TU-games. Mathematical Social Sciences, 46:
327-345.
[3] Davis, M. and M. Maschler (1965) The kernel of a cooperative game. Naval
Research Logistics Quarterly, 12: 223-259.
[4] Dutta, B. (1990) The egalitarian solution and reduced game properties in convex games. International Journal of Game Theory, 19: 153-169.
[5] Dutta, B. and D. Ray (1989) A concept of egalitarianism under participation
constraints. Econometrica, 57: 615-635.
[6] Hokari, T. (2002) Monotone-path Dutta-Ray solutions on convex games. Social
Choice and Welfare, 19: 825-844.

19

[7] Hougaard, J.L., B. Peleg and L. Thorlund- Petersen (2001) On the set of
Lorenz-maximal imputations in the core of a balanced game. International journal of Game Theory, 30: 147-165.
[8] Klijn, F., M. Slikker, S. Tijs and J. Zarzuelo (2000) The egalitarian solution
for convex games: some characterizations. Mathematical Social Sciences, 40:
111-121.
[9] Llerena, F. (2012) The pairwise egalitarian solution for the assignment game.
Operations Reseach Letters, 40: 84–88.
[10] Llerena, F. and Ll. Mauri (2015) On the Lorenz-maximal allocations in the
imputation set. Economics Bulletin, 4: 2475-2481.
[11] Sánchez-Soriano J., R. Branzei, N. Llorca and S.H. Tijs (2014) On Lorenz
dominance and the Dutta-Ray algorithm. International Journal of Mathematics,
Game Theory and Algebra, 23: 21-29.
[12] Shapley, L.S. (1971) Cores of convex games. International Journal of Game
Theory, 1: 11-16.
[13] Solymosi, T. and T.E.S. Raghavan (2001) Assignment games with stable core,
International Journal of Game Theory, 30: 177185.
[14] Tijs, SH. (1981) Bounds for the core and the t-value. In Game Theory and
Mathematical Economics, O. Moeschlin and D. Pallaschke, eds. North Holland
Publishing Company 123132.

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