In this paper, we study values for TU-games which satisfy three classical properties: Linearity, efficiency and symmetry. We give the general analytical form of these values and their relation with the Shapley value and the Egalitarian value.

Célestin Chameni Nembua

Nicolas Gabriel Andjiga

English

Research Papers in Economics

Linear, efficient and symmetric values for TU-games 
Célestin Chameni Nembua Nicolas Gabriel Andjiga
University of Yaoundé II (Cameroon) Ecole normale Supérieure (Cameroon)
Abstract
In this paper, we study values for TU-games which satisfy three classical properties:
Linearity, efficiency and symmetry. We give the general analytical form of these values and
their relation with the Shapley value and the Egalitarian value.
Citation: Chameni Nembua, Célestin and Nicolas Gabriel Andjiga, (2008) "Linear, efficient and symmetric values for
TU-games." Economics Bulletin, Vol. 3, No. 71 pp. 1-10
Submitted: October 2, 2008.  Accepted: November 2, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume3/EB-08C70057A.pdf1 Introduction
The main result on values for TU-games is due to Shapley (1953) who de￿nes a value which
satis￿es four properties: linearity, e¢ ciency, symmetry and the dummy axiom. The aim of this
paper is to study and characterize values which satisfy the ￿rst three classic properties. We
obtain almost the same results as Hernandez-Lamoneda, Juarez and Sanchez-Sanchez (2008)
which, independently, work in the same topic. We will focus our analysis on the discriminate
properties of values to obtain new axiomatization of some classic values.
2 Notations and preliminaries
Let N be a ￿nite set of n￿players, we denote by:
￿ P(N) the set of subsets of N and 2N the set of non empty subsets of N:
￿ R the set of real numbers and ￿(N) the set of automorphisms of N.
￿ An n-person TU-game on N is a pair (N;v) where v is a mapping from P(N) to R such
that v(;) = 0
￿ will denote the 2n ￿ 1 dimension linear space of all n￿person TU-game on N.
￿ A value ’ is a mapping from ￿ to Rn. ’ (v) = (’i(v))i2N is a vector of distribution of the
payo⁄s obtained in v.
Let us give some properties of values ;
￿ ’ is linear if : 8v;w 2 ￿; 8￿;￿ 2 R;’(￿v + ￿w) = ￿’(v) + ￿’(w)
￿ ’ is e¢ cient if: 8v 2 ￿;
X
i2N
’i(v) = v(N)
￿ ’ is symmetric if: 8v 2 ￿;8￿ 2 ￿(N);8i 2 N;’￿(i)(v) = ’i(￿v)
where ￿v(T) = v(￿(T)) 8T 2 2N
￿ ’ is monotonic if : 8v;w 2 ￿;8i 2 N;
(
v(S) ￿ w(S) if i 2 S
v(S) = w(S) if i = 2 S
=) ’i(v) ￿ ’i(w)
￿ ’ is covariant if : 8v 2 ￿;8￿ 2 R;8￿ 2 Rn;’(￿v + ￿) = ￿’(v) + ￿
where ￿v + ￿ is de￿ned by : (￿v + ￿)(S) = ￿v(S) +
X
i2S
￿i
1￿ ’ preserves additive games if : 8￿ 2 Rn;’(￿(:)) = ￿ where ￿(:) is de￿ned by : ￿(S) =
X
l2S
￿l
￿ ’ is non negative if : 8v 2 ￿;[8S ￿ N;v(S) ￿ 0] ) ’i(v) ￿ 0 8i 2 N
￿ ’ is marginalist if : 8v;w 2 ￿;8i 2 N;
[8S ￿ N;v(S) ￿ v(S ￿ i) = w(S) ￿ w(S ￿ i)] =) ’i(v) = ’i(w)
Let ￿ be the set of n + 1￿vectors of real numbers A = (A(k))k=0;::;n such that A(0) is a
￿xed real number and A(n) = 1:
For each element A of ￿, we de￿ne the value ￿A by :
8v 2 ￿;8i 2 N; ￿A
i (v) =
v(N)
n +
n￿1 X
k=1
2
6
6
4
(n￿k)!(k￿1)!
n! A(k)
X
i2S
jSj=k
v(S) ￿
(n￿k￿1)!(k)!
n! A(k)
X
i= 2S
jSj=k
v(S)
3
7
7
5
Let us give two others expressions of ￿A in the following lemma :
Lemma 1 : 8v 2 ￿;8i 2 N;
1) ￿A
i (v) =
n X
k=1
2
6 6
4
(n￿k)!(k￿1)!
n! A(k)
X
i2S
jSj=k
v(S) ￿
(n￿k￿1)!(k)!
n! A(k)
X
i= 2S
jSj=k
v(S)
3
7 7
5
2) ￿A
i (v) =
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)v(S) ￿ A(k ￿ 1)v(S ￿ fig)]
3
7 7
5
We can therefore deduce the following relation between ￿A and the Shapley value denoted
Shap.
Theorem 1 : 8v 2 ￿;8i 2 N; 8A 2 ￿;￿A(v) = Shap(vA)
where vA(S) = A(k)v(S) for each S such that jSj = k
Let us denoted by ￿ the set of values ￿A for all elements A on ￿.
3 Geometric properties of ￿
It is easy to prove the following results :
Proposition 1 : ￿ is a convex set and furthermore:8A1;A2 2 ￿;8￿ 2 [0;1];
A = ￿A1 + (1 ￿ ￿)A2 =) ￿A(v) = ￿￿A1(v) + (1 ￿ ￿)￿A2(v) 8v 2 ￿
Proposition 2 : ￿ is a n ￿ 1 dimension a¢ ne set. Furthermore A = (A(k))k=1;::;n￿1 is a
vector of coordinates of ￿￿￿0 in the basis
￿
￿k￿
k=1;::;n￿1 where ￿0 is the egalitarian value and
￿k is de￿ned by : 8v 2 ￿;8i 2 N ￿k
i (v) =
(n￿k)!(k￿1)!
n!
X
i2S
jSj=k
v(S) ￿
(n￿k￿1)!(k)!
n!
X
i= 2S
jSj=k
v(S)
24 Characterization of ￿ and discriminate properties
The main result of this section is the following (see Hernandez-Lamoneda, Juarez and Sanchez-
Sanchez (2008) for the proof) :
Theorem 2 :A value ’ is linear, e¢ cient and symmetric if and only if ’ is an element of ￿
To see the extend of our family of values, let us give some classical values which are elements
of ￿ and their corresponding elements of ￿:
Value A(1) A(2) ::: A(k) ::: A(n ￿ 1) A(n) Authors
Shapley 1 1 1 1 1 Shapley (1953)
Egalitarian 0 0 0 0 1 Brink R. Van den (2007)
Solidarity 1
2
1
3
1
k+1
1
n 1 Nowak and Radzik (1994)
Consensus n
2
1
2
1
2
1
2 1 Yuan, Born and Ruys (2007)
C.I.S or Equal surplus n ￿ 1 0 0 0 1 Driessen and Funaki (1991)
Let us give some results which are directly deduce from Theorem 1 :
Corollary 1 : A value ’ is linear, e¢ cient, symmetric and monotonic if and only if
’ = ￿A where A is positive (i.e. A(k) ￿ 0 8k = 1;::;n)
Corollary 2 : ’ is linear, e¢ cient, symmetric and preserves additive games if and only if
’ = ￿A with
n￿1 X
k=1
A(k) = n ￿ 1
One of the subset of linear, e¢ cient, symmetric values which preserve additive games is the set
of ￿￿consensus value whose vector A is de￿ned by : A(k) =
8
> > > > > <
> > > > > :
0 if k = 0
￿ + (1 ￿ ￿)(n ￿ 1) if k = 1
￿ if 2 ￿ k ￿ n ￿ 1
1 if k = n
The discrimination on values of ￿ can be done in two di⁄erent and equivalent ways :
- a particular vector A = (A(k))k=0;::;n of ￿ is given and therefore we obtain a speci￿c value
￿A:
- a property, called discriminate property, is added to linearity, e¢ ciency and symmetry to
obtain an unique element of ￿:
Let us introduce the non negativity and the marginalism as discriminate properties through
the following results :
Corollary 3 : ’ is linear, e¢ cient, symmetric and non negative if and only if
’ is the Egalitarian value.
Corollary 4 : ’ is linear, e¢ cient, symmetric and marginalist if and only if
’ is the Shapley value.
3Let us introduce another discriminate property used in the literature :
Let (N;v) a TU-game and i 2 N;
￿ i is v-dummy player if : 8S ￿ N ￿ fig;v(S) = v(S + i)
Let ’ be a value, A = (A(k))k=0;::;n a vector in ￿, and ￿ a real number :
￿ ’ satis￿es the dummy property if : 8v 2 ￿;8i 2 N;[i is v-dummy player ) ’i(v) = 0]
￿ ’ satis￿es the ￿￿A￿dummy property if : 8v 2 ￿;8i 2 N;
￿
i is v-dummy player ) ’i(v) = ￿￿A
i (v)
￿
We obtain the following result :
Theorem 3 : ’ is linear, e¢ cient, symmetric and satis￿es the ￿ ￿ A￿dummy property
if and only if ’ = (1 ￿ ￿)Shap + ￿￿A
We can therefore deduce some well-known results such as :
- the characterization of the Shapley value (Shapley, 1953) when ￿ = 0
- the theorem of Yang(1997) when ￿A is the egalitarian value;
- the characterization of the ￿￿consensus value (when ￿A is the equal surplus value) due to
Yuan, Born and Ruys (2007)
Another discriminate property used in this paper is obtained by replacing the dummy player
by the zero player as follows :
Let (N;v) a TU-game and i 2 N;
￿ i is v-zero player if : 8S ￿ N;i 2 S ) v(S) = 0
Let ’ be a value, A = (A(k))k=0;::;n a vector in ￿, ￿ a real number :
￿ ’ satis￿es the zero player property if : 8v 2 ￿;8i 2 N;[i is v-zero player ) ’i(v) = 0]
￿ ’ satis￿es the ￿￿A￿zero player property if : 8v 2 ￿;8i 2 N;
￿
i is v-zero player ) ’i(v) = ￿￿A
i (v)
￿
We obtain the following result :
Theorem 4 : ’ is linear, e¢ cient, symmetric and satis￿es the ￿ ￿ A￿zero player property
if and only if ’ = (1 ￿ ￿)￿0 + ￿￿A where ￿0 is the egalitarian value.
We can deduce the characterization of egalitarian value when ￿ = 0, result due to Van den
Brink (2007).
The last discriminate property in this paper can be seen as the dual of the previous one and
is de￿ned by :
Let (N;v) a TU-game and i 2 N;
￿ i is v-winning player if : 8S ￿ N;v(S ￿ fig) = 0
One can extend this property as the previous ones and obtain new values.
45 Conclusion
The paper has outlined the properties of the class of values which are linear, e¢ cient and
symmetric. In particular, it has been proved that the class is an a¢ ne linear space for which a
"canonical" basis has been identi￿ed. On the one hand, this has been helpful in bringing out
some common analytic expression of all the values in the class. We have obtained that every
value in the class can be put in the random order form, similar to the well-known expression of
the Shapley value. On the other hand, we have explained conditions for a given value which is
presumed as member of the class to satisfy some basic properties as well as recasting in simplify
and extensive terms some well-known results.
References
[1] Brink, R. Van Den (2007), Null or Nullifying players: The di⁄erence between Shapley
value and the equal division solutions. Journal of Economic Theory , 136: 767-775.
[2] Driessen TSH, Funaki Y.(1991) Coincidence of and Collinearity between game theoretic
solutions. OR Spektrum 13: 15-30.
[3] Hermandez-Lamoneda, Luis, Ruben Juarez, and Francisco Sanchez-Sanchez (2008)
￿solution Without dummy axiom for TU cooperative games￿Economics Bulletin,
Vol.3, No.1 pp 1-9
[4] Nowak A.S. and Radzik T.(1994) A solidarity Value for n-person Transferable Utility
games. Int Jour of Game Theory , 23: 43-48.
[5] Shapley L.S (1953), A value for n-person games. In: Kuhn H, Tucker AW (eds)
Contributions to the theory of games, vol. II. Princeton University Press,
Princeton,NJ.
[6] Yang C.H (1997), A Family of Values for n-person Cooperative Transferable Utility
Games. MS. Thesis. University at Bu⁄alo.
[7] Young. H.P (1985). Monotonic Solutions of Cooperative games. Int. J. Game Theory
14: 65-72.
[8] Yuan Ju, P. Born and P. Ruys (2007),The Concensus Value: a new solution concept for
cooperative games. Soc. Choice Welfare , 28 685-703.
56 Appendix
Lemma 1 :
1) is obvious
2) ￿A
i (v) =
n X
k=1
2
6
6
4
(n￿k)!(k￿1)!
n! A(k)
X
i2S
jSj=k
v(S) ￿
(n￿k￿1)!(k)!
n! A(k)
X
i= 2S
jSj=k
v(S)
3
7
7
5
=
n X
k=1
(n￿k)!(k￿1)!
n! A(k)
X
i2S
jSj=k
v(S) ￿
n￿1 X
k=1
(n￿k￿1)!(k)!
n! A(k)
X
i2S
jSj=k+1
v(S ￿ i)
=
n X
k=1
(n￿k)!(k￿1)!
n! A(k)
X
i2S
jSj=k
v(S) ￿
n X
p=2
(n￿p)!(p￿1)!
n! A(p ￿ 1)
X
i2S
jSj=p
v(S ￿ i)
=
n X
k=1
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)v(S) ￿ A(k ￿ 1)v(S ￿ i)] ￿
Corollary 1 : Suppose A(k) ￿ 0 8k = 1;2;:::n; for any i2N; and for any v;w 2 ￿;
v(S) ￿ w(S) 8S 3 i
v(S) = w(S) if i= 2S
)
) for all k = 1;2;:::n
A(k)v(S) ￿ A(k)w(S) 8S 3 i;jSj = k
￿A(k ￿ 1)v(S) = ￿A(k ￿ 1)w(S) if i= 2S;jSj = k
)
)
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)v(S) ￿ A(k ￿ 1)v(S ￿ i)] ￿
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)w(S) ￿ A(k ￿ 1)w(S ￿ i)]
)
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)v(S) ￿ A(k ￿ 1)v(S ￿ i)]
3
7 7
5 ￿
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)w(S) ￿ A(k ￿ 1)w(S ￿ i)]
3
7 7
5
) ￿A
i (v) ￿ ￿A
i (w):
Conversely, suppose ￿A is monotonic;
for any i 2 N, for all k = 1;2;:::;n, there exists at least one coalition Si ￿ N such that :
Si 3 i and jSij = k:Consider the games v(S) =
(
1 if S = Si
0 if S 6= Si
and w(S) =
(
1
2 if S = Si
0 if S 6= Si
It is obvious that
v(S) ￿ w(S) 8S 3 i
v(S) = w(S) if i = 2 S
)
: As ￿A is monotonic,this implies
￿A
i (v) ￿ ￿A
i (w) ,
(n￿k)!(k￿1)!
n! A(k) ￿
(n￿k)!(k￿1)!
2n! A(k)
, A(k) ￿
A(k)
2 ) A(k) ￿ 0: ￿
Corollary 2 : .’ linear, e¢ cient, symmetric and preserves additive games
, ’ = ￿A by Theorem 2 and ￿A preserves additive games
￿A preserves additive games , 8￿ 2 Rn;’(￿(:)) = ￿ , ￿A
i (￿(:)) = ￿i 8~ i2 N and 8￿ 2Rn
,
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)￿(S) ￿ A(k ￿ 1)￿(S ￿ i)]
3
7 7
5 = ￿i 8i 2 N and 8￿ 2 Rn
6,
n X
k=1
2
6
6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)(￿(S ￿ i) + ￿i) ￿ A(k ￿ 1)￿(S ￿ i)]
3
7
7
5 = ￿i 8i 2 N; 8￿ 2 Rn
,
8
> > > > > > > > > > <
> > > > > > > > > > :
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k) ￿ A(k ￿ 1)]
P
j2S￿i
￿j
3
7
7
5 = 0
n X
k=1
2
6
6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! A(k)
3
7
7
5 = 1
8i 2 N; 8￿ 2 Rn
,
n X
k=1
(n￿k)!(k￿1)!
n! [A(k) ￿ A(k ￿ 1)]
(n￿2)!
(k￿2)!(n￿k)!￿j = 0 and
n X
k=1
A(k) = n 8￿j2R
,
n X
k=1
(k ￿ 1)[A(k) ￿ A(k ￿ 1)]￿j = 0 and
n￿1 X
k=1
A(k) = n ￿ 1 8￿j2R
,
n X
k=1
(k ￿ 1)[A(k) ￿ A(k ￿ 1)] = 0 and
n￿1 X
k=1
A(k) = n ￿ 1
,
n￿1 X
k=1
A(k) = n ￿ 1 ￿
Corollary 3 : Suppose ’ is linear, e¢ cient, symmetric and non negative;8v 2 ￿;8i 2 N;
’i(v) = ￿A
i (v) =
v(N)
n +
n￿1 X
k=1
2
6 6
4
(n￿k)!(k￿1)!
n! A(k)
X
i2S
jSj=k
v(S) ￿
(n￿k￿1)!(k)!
n! A(k)
X
i= 2S
jSj=k
v(S)
3
7 7
5
Suppose ’ 6= ￿0, then there exist at least one k0 2 f1;2;:::;n ￿ 1g such that A(k0) 6= 0:
If A(k0) > 0; for i0 2 N; consider the game v0(S) =
(
1 if i = 2 S and jSj = k0
0 elsewhere
It is straightfoward that ’i0(v0) < 0
If A(k0) < 0; for i0 2 N; consider the game w0(S) =
(
1 if i 2 S and jSj = k0
0 elsewhere
It is straightfoward that ’i0(w0) < 0 ￿
Corollary 4 : Suppose ’ = ￿A is linear, e¢ cient, symmetric and marginalist, consider
any games 8v;w 2 ￿ such that;8i 2 N;[8S ￿ N;v(S) ￿ v(S ￿ i) = w(S) ￿ w(S ￿ i)]
As ’ is marginalist, ’i(v) = ’i(w),
n X
k=1
2
6
6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)v(S) ￿ A(k ￿ 1)v(S ￿ i)]
3
7
7
5 =
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)w(S) ￿ A(k ￿ 1)w(S ￿ i)]
3
7 7
5
7,
n X
k=1
2
6
6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)(v(S) ￿ w(S)) + A(k ￿ 1)(w(S ￿ i) ￿ v(S ￿ i))]
3
7
7
5 = 0
,
n X
k=1
2
6
6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [(A(k) ￿ A(k ￿ 1))(v(S ￿ i) ￿ w(S ￿ i))]
3
7
7
5 = 0
Since this condition must be satis￿ed for any games v;w 2 ￿ such that;
8i 2 N;[8S ￿ N;v(S) ￿ v(S ￿ i) = w(S) ￿ w(S ￿ i)]
) A(k) ￿ A(k ￿ 1) = 0 for all k = 2;3;:::;n and A(n) = 1
) A(k) = 1 for all k = 1;2;:::;n
) ’ = Shap: ￿
Theorem 3 : ’ is linear, e¢ cient, symmetric and satis￿es the ￿ ￿ A￿dummy property if
and only if , for any v 2 ￿ and for any v￿dummy player i 2 N;’i(v) = ￿A0
i (v) = ￿￿A
i (v)
, ’i(v) =
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A0(k)v(S ￿ i) ￿ A0(k ￿ 1)v(S ￿ i)]
= ￿
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)v(S ￿ i) ￿ A(k ￿ 1)v(S ￿ i)]
, ’i(v) =
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A0(k) ￿ A0(k ￿ 1)]v(S ￿ i)
= ￿
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k) ￿ A(k ￿ 1)]v(S ￿ i)
, A0(k) ￿ A0(k ￿ 1) = ￿(A(k) ￿ A(k ￿ 1)) for all k = 2;3;:::;n and A(n) = A0(n) = 1
, A0(k) ￿ A0(k ￿ 1) = ￿(A(k) ￿ A(k ￿ 1)) for all k = 2;3;:::;n ￿ 1
and A0(n ￿ 1) = (1 ￿ ￿) + ￿A(n ￿ 1)
, A0(k) = (1 ￿ ￿) + ￿A(k) for all k = 1;2;:::;n
, ’ = ￿A0
= (1 ￿ ￿)Shap + ￿￿A ￿
Theorem 4 : ’ is linear, e¢ cient, symmetric and satis￿es the ￿￿A￿zero player property
if and only if for any v 2 ￿ and for any v￿zero player i 2 N;’i(v) = ￿A0
i (v) = ￿￿A
i (v)
, ’i(v) =
n X
k=1
2
6
6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A0(k)v(S) ￿ A0(k ￿ 1)v(S ￿ i)]
3
7
7
5
= ￿
n X
k=1
2
6
6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! [A(k)v(S) ￿ A(k ￿ 1)v(S ￿ i)]
3
7
7
5
,
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! A0(k ￿ 1)v(S ￿ i)
3
7 7
5 = ￿
n X
k=1
2
6 6
4
X
i2S
jSj=k
(n￿k)!(k￿1)!
n! A(k ￿ 1)v(S ￿ i)
3
7 7
5
8, A0(k ￿ 1) = ￿A(k ￿ 1) for all k = 2;3;:::;n and A0(n) = A(n) = 1
, A0(k) = ￿A(k) for all k = 1;2;:::;n ￿ 1
, ’ = ￿A0
= (1 ￿ ￿)￿0 + ￿￿A ￿
9

Linear, efficient and symmetric values for TU-games

http://www.accessecon.com/pubs/EB/2008/Volume3/EB-08C70057A.pdf

Linear, efficient and symmetric values for TU-games

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