Linear, efficient and symmetric values for TU-games Abstract - CiteSeerX

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.pdf

1

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) X ' is e¢ cient if: 8v 2 ; 'i (v) = v(N ) i2N

' is symmetric if: 8v 2 ; 8 2

(N ); 8i 2 N; '

where v(T ) = v( (T )) 8T 2 2N ' is monotonic if : 8v; w 2 ; 8i 2 N;

(

v(S)

(i) (v)

w(S)

= 'i ( v)

if i 2 S

v(S) = w(S)

if i 2 =S

' 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

1

i

=) 'i (v)

'i (w)

' preserves additive games if : 8 2 Rn ; '( (:)) = X

where (:) is de…ned by : (S) =

l

l2S

' 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 A (v) i

8v 2 ; 8i 2 N;

=

, we de…ne2the value v(N ) n +

n X1 k=1

6 (n 6 4

Let us give two others expressions of Lemma 1 : 8v22 ; 8i 2 N; 1)

A (v) i

=

n X k=1

2)

A (v) i

6 (n 6 4

k)!(k 1)! A(k) n!

k=1

by :

k)!(k 1)! A(k) n!

X

(n k)!(k 1)! n!

(n k 1)!(k)! A(k) n!

v(S)

i2S jSj=k A

X

i2S = jSj=k

in the following lemma :

i2S = jSj=k

[A(k)v(S)

A(k

A

We can therefore deduce the following relation between

7 v(S)7 5

7 v(S)7 5

3

1)v(S

i2S jSj=k

3

3

X

(n k 1)!(k)! A(k) n!

v(S)

i2S jSj=k

2

n X 6X 6 = 4

X

A

7 fig)]7 5

and the Shapley value denoted

Shap. A (v)

Theorem 1 : 8v 2 ; 8i 2 N; 8A 2 ;

= Shap(v A )

where v A (S) = A(k)v(S) for each S such that jSj = k

Let us denoted by

3

A

the set of values

for all elements A on

.

Geometric properties of

It is easy to prove the following results : Proposition 1 : A = A1 + (1

A (v)

)A2 =)

Proposition 2 : vector of coordinates of k

is a convex set and furthermore:8A1 ; A2 2 ; 8 2 [0; 1] ; is a n 0

A1 (v)

=

+ (1

)

A2 (v)

8v 2

1 dimension a¢ ne set. Furthermore A = (A(k))k=1;::;n in the basis

is de…ned by : 8v 2 ; 8i 2 N

k (v) i

=

k

where

k=1;::;nX 1 (n k)!(k 1)! v(S) n! i2S jSj=k

2

0

1

is a

is the egalitarian value and X v(S)

(n k 1)!(k)! n!

i2S = jSj=k

4

Characterization of

and discriminate properties

The main result of this section is the following (see Hernandez-Lamoneda, Juarez and SanchezSanchez (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)

Shapley

1

1

1

Egalitarian

0

0

Solidarity

1 2 n 2

1 3 1 2

0

Consensus C.I.S or Equal surplus

n

1

:::

A(k)

:::

A(n

1)

A(n)

Authors

1

1

Shapley (1953)

0

0

1

Brink R. Van den (2007)

1 k+1 1 2

1 n 1 2

1

Nowak and Radzik (1994)

1

Yuan, Born and Ruys (2007)

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 n X1 ' = A with A(k) = n 1 k=1

One of the subset of linear, e¢ cient, symmetric values which8preserve additive games is the set > > 0 if k = 0 > > > < + (1 )(n 1) if k = 1 of consensus value whose vector A is de…ned by : A(k) = > 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. 3

Let 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 (v) i

We obtain the following result : Theorem 3 : ' is linear, e¢ cient, symmetric and satis…es the if and only if ' = (1

A dummy property

A

)Shap +

We can therefore deduce some well-known results such as : - the characterization of the Shapley value (Shapley, 1953) when - the theorem of Yang(1997) when - the characterization of the

A

=0

is the egalitarian value;

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) =

We obtain the following result : Theorem 4 : ' is linear, e¢ cient, symmetric and satis…es the if and only if ' = (1

)

0

+

A

where

0

We can deduce the characterization of egalitarian value when

A zero player property

is the egalitarian value. = 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. 4

A (v) i

5

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.

5

6

Appendix

Lemma 1 : 1) is obvious A (v) i

2)

=

2

n X 6 (n 6 4

k)!(k 1)! A(k) n!

n X (n

k)!(k 1)! A(k) n!

n X (n

k)!(k 1)! A(k) n!

X

v(S)

n X1

n X X

(n k)!(k 1)! n!

i2S = jSj=k

n X (n

v(S

i)

i2S jSj=k+1 p)!(p 1)! A(p n!

1)

p=2

[A(k)v(S)

7 v(S)7 5

X

(n k 1)!(k)! A(k) n!

k=1

i2S jSj=k

k=1

=

v(S)

i2S jSj=k

k=1

=

X

X

(n k 1)!(k)! A(k) n!

v(S)

i2S jSj=k

k=1

=

X

3

A(k

X

v(S

i)

i2S jSj=p

1)v(S

i)]

k=1 i2S jSj=k

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 A(k)v(S) A(k)w(S) 8S 3 i; jSj = k ) for all k = 1; 2; :::n ) v(S) = w(S) if i2S = A(k 1)v(S) = A(k 1)w(S) if i2S; = jSj = k X (n k)!(k 1)! X (n k)!(k 1)! [A(k)v(S) A(k 1)v(S i)] [A(k)w(S) A(k 1)w(S i)] n! n!

i2S jSj=k

i2S jSj=k

2

n X 6X 6 ) 4 k=1

i2S jSj=k

A (v) i

)

(n k)!(k 1)! n!

[A(k)v(S)

A(k

1)v(S

A (w): i

Conversely, suppose

A

3

2

n X 6X 6 4

7 i)]7 5

k=1

(n k)!(k 1)! n!

[A(k)w(S)

i2S jSj=k

is monotonic;

for any i 2 N , for all k = 1; 2; :::; n, there exists(at least one coalition Si N(such that : 1 1 if S = Si 2 if S = Si Si 3 i and jSi j = k:Consider the games v(S) = and w(S) = 0 if S 6= Si 0 if S 6= Si ) v(S) w(S) 8S 3 i It is obvious that : As A is monotonic,this implies v(S) = w(S) if i 2 =S A (w) , (n k)!(k 1)! A(k) i n! A(k) ) A(k) 0: 2

A (v) i

, A(k)

(n k)!(k 1)! A(k) 2n!

Corollary 2 : .' linear, e¢ cient, symmetric and preserves additive games ,'= A

A

by Theorem 2 and

A

preserves additive games

preserves additive games , 8 2 Rn ; '( (:)) = 2

n X 6X 6 , 4 k=1

(n k)!(k 1)! n!

[A(k) (S)

A(k

1) (S

i2S jSj=k

6

, 3

A( i

7 i)]7 5=

i

(:)) =

i

8~i2 N and 8 2Rn

8i 2 N and 8 2 Rn

A(k

1)w(S

i)]

,

,

, , , ,

2

n X 6X 6 4

(n k)!(k 1)! n!

[A(k) ( (S

i) +

i)

A(k

(n k)!(k 1)! n!

jSj=k

> > > > > > > > > > :

2

n X 6X 6 4 k=1

n X (n

k)!(k 1)! n!

A(k

P

1)]

j2S i

3

i2S jSj=k

[A(k)

[A(k)

3

7 (n k)!(k 1)! 7 A(k) n! 5

A(k

1)] (k

7 7=0

j5

(k

(n 2)! 2)!(n k)! j

(k

8i 2 N; 8 2 Rn

n X

= 0 and

A(k) = n

8 j 2R

k=1

1) [A(k)

A(k

1)]

j

1) [A(k)

A(k

n X1

= 0 and

k=1

k=1 n X1

8i 2 N; 8 2 Rn

i

=1

k=1

n X

7 i)]7 5=

1) (S

i2S

k=1

2 8 jSj=k > > n > X 6X > > 6 > > 4 > > > < k=1 i2S

n X

3

1)] = 0 and

A(k) = n

k=1 n X1

A(k) = n

1

8 j 2R

1

k=1

A(k) = n

1

k=1

Corollary 3 : Suppose ' is2linear, e¢ cient, symmetric and non negative; 8v 2 ; 8i32 N;

'i (v) =

A (v) i

=

v(N ) n

+

n X1 k=1

Suppose ' 6=

0,

6 (n 6 4

k)!(k 1)! A(k) n!

X

v(S)

(n k 1)!(k)! A(k) n!

i2S jSj=k

X

i2S = jSj=k

7 v(S)7 5

then there exist at least one k0 2 f1; 2; :::; ( n 1g such that A(k0 ) 6= 0: 1 if i 2 = S and jSj = k0 If A(k0 ) > 0; for i0 2 N; consider the game v0 (S) = 0 elsewhere It is straightfoward that 'i0 (v0 ) < 0 ( 1 if i 2 S and jSj = k0 If A(k0 ) < 0; f or i0 2 N; consider the game w0 (S) = 0 elsewhere It is straightfoward that 'i0 (w0 ) < 0

Corollary 4 : any games 8v; w 2

Suppose ' =

A

is linear, e¢ cient, symmetric and marginalist, consider

such that; 8i 2 N; [8S 2 N; v(S)

As ' is marginalist, 'i (v) = 'i (w),

n X 6X 6 4 k=1

2

n X 6X 6 4 k=1

(n k)!(k 1)! n!

[A(k)w(S)

A(k

(n k)!(k 1)! n!

i2S jSj=k

1)w(S

i2S jSj=k

7

v(S

3

7 i)]7 5

i) = w(S)

w(S

[A(k)v(S)

A(k

i)] 1)v(S

3

7 i)]7 5=

,

,

2

n X 6X 6 4

(n k)!(k 1)! n!

[A(k)(v(S)

w(S)) + A(k

1)(w(S

i)

v(S

i2S

k=1

2jSj=k

n X 6X 6 4

(n k)!(k 1)! n!

[(A(k)

A(k

1)) (v(S

i)

w(S

i2S jSj=k

k=1

Since this condition must be satis…ed for any games v; w 2 8i 2 N; [8S

) A(k)

N; v(S)

A(k

v(S

i) = w(S)

w(S

3

7 i))]7 5=0

3

7 i))]7 5=0

such that;

i)]

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) = X (n k)!(k 1)! [A0 (k)v(S i) A0 (k 1)v(S i)] , 'i (v) = n! =

X

i2S jSj=k (n k)!(k 1)! n!

i2S jSj=k

X

, 'i (v) = = ,

X

, A0 (k) ,

A(k

1)v(S

i)]

[A0 (k)

A0 (k

1)] v(S

i)

A(k

1)] v(S

[A(k)

i)

(A(k)

A(k

1)) for all k = 2; 3; :::; n and A(n) = A0 (n) = 1

A0 (k

1) =

(A(k)

A(k

1)) for all k = 2; 3; :::; n

1) = (1

= (1

) + A(n

1

1)

) + A(k) for all k = 1; 2; :::; n

= (1

A

)Shap +

Theorem 4 : ' is linear, e¢ cient, symmetric and satis…es the if and only if for any 2 v2 n X 6X 6 , 'i (v) = 4 k=1

=

2

n X 6X 6 4 k=1

,

n X k=1

2

A (v) i

1) =

A0

,'=

i)

=

A0 (k

and A0 (n A0 (k)

(n k)!(k 1)! n!

i2S jSj=k (n k)!(k 1)! n!

i2S jSj=k A0 (k)

[A(k)v(S

A0 (v) i

and for any v zero player (n k)!(k 1)! n!

[A0 (k)v(S)

i2S jSj=k

A0 (k

1)v(S

i2S jSj=k

(n k)!(k 1)! n!

[A(k)v(S)

i2S jSj=k

6X 6 4

i 2 N; 'i (v) 3=

(n k)!(k 1)! 0 A (k n!

1)v(S

A(k

3

1)v(S

3

7 i)7 5= 8

n X k=1

2

7 i)]7 5

6X 6 4

i2S jSj=k

A zero player property A0 (v) i

=

A (v) i

7 i)]7 5

(n k)!(k 1)! A(k n!

1)v(S

3

7 i)7 5

, A0 (k

1) for all k = 2; 3; :::; n and A0 (n) = A(n) = 1

1) = A(k

, A0 (k) = A(k) for all k = 1; 2; :::; n

,'=

A0

= (1

)

0

+

1

A

9