Theoretical Economics Letters, 2012, 2, 369-372
http://dx.doi.org/10.4236/tel.2012.24068 Published Online October 2012 (http://www.SciRP.org/journal/tel)
Revenue Sharing in Hierarchica l Organiza tions: A New
Interpretation of the Generalized Banzhaf Value
Roland Pongou1, Bertrand Tchantcho2,3, Narcisse Tedjeugang2
1Department of Economics, University of Ottawa, Ottawa, Canada
2École Normale Supérieure, Yaoundé, Cameroon
3Université de Cergy Pontoise, Cergy-Pontoise, France
Email: rpongou@uottawa.ca, btchantcho@yahoo.fr, tedjeugang@yahoo.fr
Received February 24, 2012; revised March 23, 2012; accepted April 19, 2012
ABSTRACT
This paper examines the distribution of earnings in a new model of hierarchical multi-task organizations. Each such
organization is defined by a finite set of workers and tasks together with a production function that maps each allocation
of workers to the tasks into an aggregate output. Tasks are ordered in degree of importance so that aggregate output
increases when a worker climbs up the organization ladder. We show that the generalized Banzhaf value proposed by
Freixas [1] can be used as a theory of revenue sharing in such organizations, and provide a new interpretation and for-
mulation of this sharing ru le, proving th at a worker’s pay is proportional to the difference between his marginal produc-
tivity at the top level and at the bottom level of the hierarchy summed over all the possible configurations of the or-
ganization. This new formulation also facilitates computation.
Keywords: Hierarchical Organizations; Earnings; Marginal Productiv ity; Generalized Banzhaf Value
1. Introduction
This paper studies the distribution of earnings in a new
model of hierarchical organizations introduced in Pongou
et al. [2]. Each such organization is defined by a finite set
of workers and tasks, and a production function that
maps each allocation of workers to the tasks into an
aggregate output. Tasks are ordered in degree of im-
portance so that aggregate output increases when a
worker climbs up the organization ladder. Assuming that
the organization is competitive, output converts directly
into revenue.
We adapt the generalized Banzhaf value proposed by
Freixas [1] as a theory of revenue sharing in such organi-
zations. This notio n is an ex tension of the concep t known
as the Banzhaf-Coleman value. It was independently pro-
posed by Banzhaf [3] and Coleman [4] as a measure of
the ability of a voter to affect the outcome of a voting
game. An axiomatic study of this value was conducted
by Dubey and Shapley [5] and Laruelle and Valenciano
[6]. Important applications of the Banzhaf value to
corporations and corporate governance can be found in
Leech [7,8]. Feltkamp [9] generalizes the axiomatic
approach to any transferable utility cooperative game. In
the context of the firm, a cooperative game can be viewed
as a hierarchical organization that has only two levels. In
an attempt to generalize this notion, Freixas and Zwick er
[10] introduce the concept of simple games. A

,jk
,jk simple game can be interpreted as a hierarchical
organization that has levels of hierarchy, and is
equipped with a real-valued production function whose
range has cardinality . In our model of a hierarchical
organization, is not fixed, but is endogenously
determined by the production function. Freixas [1] ge-
neralizes the relative Banzhaf value to this class of games.
Diffo Lambo and Moulen [11] and Freixas et al. [12]
study this value in relation to other theories within the
class of (2, 2) simple games, and Tchantcho et al. [13]
and Pongou et al. [14] extend this an alysis to the class of
(3, 2) simple games.
j
k
k
A new formulation and interpretation of the genera-
lized Banzhaf value is provided. It proves that in a hie-
rarchical organization, the Banzhaf value of a worker is
proportional to the difference between his marginal pro-
ductivity at the top level and at the bottom level of the
hierarchy summed over all the possible configurations of
the organization. This new formulation also makes this
notion very easy to compute.
The remainder of the paper is organized as follows.
Section 2 defines hierarchical organizations. In Section 3,
a new formulation of the generalized Banzhaf value is
provided. Section 4 conc ludes.
2. Hierarchical Organiz ati ons
An organization is a list
,,NTV where is a non- N
C
opyright © 2012 SciRes. TEL
R. PONGOU ET AL.
370
empty finite set of workers,
1
=,,
j
TT T a finite col-
lection of finite sets of tasks with cardinility =2Tj,
and a production function that maps each allocation
of workers to the tasks into a real number which
measures the aggregate output or productivity of the
organization under that particular configuration (Pongou
et al. [2]). For every , i is a set of identical
tasks. Denote by i a representative task of i. An
allocation of workers to the tasks is a j-partition
1
V
=,
=1,ijT
t
T
,
j
A
AA of , where i
N
A
is the set of w ork e rs
assigned to type tasks (or simply to task i). Under
an allocation 1j
i
tt
=,
,
A
AA, i
A
may be empty for
any , which would mean that no worker has
been assigned to task . The set of all possible
allocations is denoted
=1,ij
i
t
j
.
In a hierarchical organization, tasks are ordered in de-
gree of importance: 1 is the most important task, 2
the second most important task, and so on. A typical ex-
ample is a military organization where each higher-
ranking member is more competent and has more im-
portant tasks than his subordinates. A hierarchical organi-
zation is said to be monotonic if aggregate output in-
creases as a worker moves from a less important task to a
more important one. The concept of monotonicity is for-
malized below.
t
t
Let
1
=,,
j
A
A
1,2,l
A
j
be an ordered j-partition of .
For any , denote by 1l
N
,=
l
A
A
=1,
l
A
j
the set of workers executing a task at least as important
as task . Let 1 be an ordered j-parti-
tion of such that for any.
That is, is obtained from by moving some
workers to more important tasks. This is denoted by
l
tN
B
=,BB
l
A
,
jB
l
B
A
j
B. A multi-task organization is said to
be monotonic if for any ordered j-parti tions
,,NTV
,
j
AB
such that j
A
B,

VAVB1.
3. The Generalized Banzhaf Value and a
New Formulation
The generalized Banzhaf value developed by Freixas [1]
is adapted to our framework as follows. Let
,,VNT
be a monotonic hierarchical organization, and
the range of Without loss of
generality, assume 12
vv Let
12
,..vv
,., k
v.V
>.
k
v>>
A
be an al-
location of workers and a worker. Denote by
the task assigned to in the allocation
pp
,pA
A
,
and by
p
A
the allocation obtained from
A
by moving
ne level down the organization ladder :
p o

,=
p
pA

,1,pA


,=,
p
qA qA

. ,qNq p
Let
,1,2,,lm k be such that .
is said to be 1<
lmk
p
,lm-critical in
A
if:


=> =
lm
p
vVAVA v
.
Denote by
,lm
pV
the number of allocations in
which p is
, -critical,lm and by

,lm
ip V
num-
ber of allocations the
,...,j1
=
A
AA in which is p
,lm-critical and p
i
A
(1 follows that: 1). Itij
 
1
,,
=1
=.
j
lm lm
pip
i
VV

The Banzhaf value is defined below.
Definition 1. Let
,,NTV be a mon
dNp otonic hierarchi-
cal organizatio n, an
a worker.
1) The Banzhaf scor is: e of
p
V
2) The normalized Banzhaf index of is:
p

=km
V v



1,
=2 =1
lm
pl
m
ml
v
p
 

=,
pV
V
j
pi
i
V
The Banzhaf index of a worker measures his earnings
in terms of the proportion of the organization’s revenue
he is expected to obtain. Our goal is to give a new
interpretation of this sharing rule. Let
A
be an al-
location of workers and pN a worker. Denote by
1
p
A
the allocation obtained by moving p from his
position in
A
up to the top level of the organization.
Similarly, denote by
j
p
A
the allocation obtained by
moving p from his position in
A
down to the bottom
level of the organization. We show below that the
Banzhaf score of p is proportional to the difference
between his marginal productivity at the top level of the
organization (
p
VA VA) and his marginal pro-
ductivity at the bottom level (


j
p
VA VA) summed
over all the possible worker allocations.
Theorem 1. Let
1
,,NTV be a mono hierarchi-
cal organization, and pN tonic
a worker. The Banzhaf score
of p is given by:

 
1
1
=.
j
j
p
V
pp
A
VAVA
j
The definition below will be needed in the proof of
Theorem 1.
Definition 2. Let
A
and B be two j-partitions of N
and p a worker.
B is said to be p-euivalenqt to
A
if ,qN q,p
,= ,.q qA

B
obvious that the relation defined ave is
er o
It isbo an equi-
1A monotonic hierarchical organization as defined here can be viewed
as a reinterpretation of the notion of (j, k) simple games (Freixas and
Zwicker [10]). Note, however, that not all hierarchical organizations
are monotonic (Pongou et al. [2]). valencerelation. The numbf elements in each equi-
Copyright © 2012 SciRes. TEL
R. PONGOU ET AL. 371
valence class is
j
, and since the number of j-partitions
is n
j, there are exactly 1n
j equivalence classes. We
denote th ese equi vale nce classe s by

j
pt,
1
=1,2,, n
tj
.
Theorem 1 is proved below.
Proof. It is the case that the Banzhaf score of a worker
s in which is
p is the number of j-partition p
,-lm
cr
For any , the cardinality of is , and
where
itical, so that:
 

=
pVVAVA




1
()
=1
1
=1 ()
=
=.
jp
A
p
n
jj
At
p
t
n
j
p
j
tAt
p
VA VA
VA VA

t

j
ptj



=
p
j
At
p
A VAVVt





1j
pp
VAt A

A
t
. Thusis any representative j-partition in the class
,
A
On the other hand,
p
It follows that:

j
pt
1n
j
pV
 





=1 ()
1
1
=1
=
=
p
j
tAt
p
n
jj
pp
t
VAV
VAt VAt

 
jpp
A
VA VA

 









 

1
1
1
=1
1
1
=1
1
1
=1
=
=
=
=from the expression of above.
j
j
pp
n
jj
At
p
t
n
jj
pp
t
n
jj
pp
t
p
VA VA
jV AtV At
jVAt VAt
jV V


 
1
1
=j
j
p
pp
A
VAVA
j
A simple application is provided below.
Example 1. A monotonic hierarchical organization
and four
effort). Ars
V
N
tas
,,TV involves two workers p and q
ks (the tasks may also be viewed as different levels of
ll the possible allocations of worke are given
in the first column of the table below. Note that we write
,,,pq, for instance, for
 

,,,qq , which
denotes a configuration in which p and q are as-
sks 1
t and 4
t, respe worker
is assigned to task 2
t or task 3
t. T prodution fun-
ction of the organization is defined in the second column.
The difference betwen the marginal productivity at the
top level and at the bottom level of the hierarchy is given
in the third column for worker p and in the fourth
column for worker q.
4-Partition (A) V(A
signed to tactively, and no
he c
e
) V(1
p
A
V(
j
p
A
) V(1
q
A
) - V(
j
q
A
)) -
(Ø,Ø,Ø,pq) 0 0 0
(Ø,Ø,p,q) 0 0 1
(
(
(
(
(
Ø, p ,Ø ,q)0 0 1
(p,Ø,Ø, q) 0 0 2
(Ø,Ø,q,p) 0 1 0
Ø ,q, Ø, p)0 1 0
(q,Ø,Ø, p) 0 2 0
Ø,Ø,pq,Ø) 0 1 1
(Ø,p,q,Ø) 1 1 1
(p, Ø,q,Ø) 1 1 2
(Ø, q,p, Ø) 0 1 1
(q,Ø, p,Ø) 1 2 1
Ø,pq, Ø,Ø)1 1 1
(p,q, Ø,Ø) 1 1 2
(q, p,Ø,Ø) 1 2 1
pq, Ø,Ø,Ø)2 2 2
f of each workere-
re
The Banzhascore of the two rs is the
fo 164=4, which implies that normBanzhaf
va the alized
lue is 12 for each. So each is expected to have half
of theut or revenue.
4. Conclusion
total outp
distribution rning a newly
hical ornizationsngs are
We have studied t
defined model of hh
ie
erarc of
ga eas in
. Earni
measured by the generalized Banzhaf value proposed by
Freixas [1]. A new interpretation of this sharing rule has
been given. More precisely, in any hierarchical organi-
zation, the normalized Banzhaf value of a worker is pro-
portional to the difference between his marginal pro-
ductivity at the top level and at the bottom level of the
hierarchy summed over all the possible configurations of
the organization. This new formulation is also easy to
compute.
Copyright © 2012 SciRes. TEL
R. PONGOU ET AL.
Copyright © 2012 SciRes. TEL
372
ixas, “The Banzhaf Measures for Games with Sev-
eral Levels of Approval in the Input and Output,” Annals
of Operations 1, 2005, pp. 45-66.
doi:10.1007/s1
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