American Journal of Industrial and Business Management, 2012, 2, 200-204
http://dx.doi.org/10.4236/ajibm.2012.24026 Published Online October 2012 (http://www.SciRP.org/journal/ajibm)
Modified Tournament Model Based on Perceived Wage
Qiang Guo1*, Chengyu Jiang2
1Management School, Northwest Polytechnical University, Shaanxi, China; 2Northwest Polytechnical University, Shaanxi, China.
Email: *guoqianghainan@gmail.com
Received July 26th, 2012; revised August 27th, 2012; accepted September 28th, 2012
ABSTRACT
In this paper, we combine the relative satisfaction and relative deprivation stemmed from wage comparison to form the
relative perception as the integrated influence factor on the individual’s utility function, which is the most different
point from the former tournament theory studies. We introduce the relative perception into the tournament model and
then analyze the Nash Equilibrium of the output competition game based on th is modified model. Consequently, some
new findings are obtained. Firstly, we find the relative perceptio n could affect the utility of workers as similar as what
the wage dispersion does. What’s more, the income-utility sensitivity can also affect the decision of workers to choose
the effort level. According to what is found in this study, the subjective perception should be paid enough attention to
since it could affect the worker both in utility and consequent action. Besides, the wage policy should design properly
and the differences in subjective sensitivity to relative perception or the proportion of income or perception among
workers should be taken in to account when the wage strategy is made.
Keywords: Relative Deprivation; Relative Satisfaction; Relative Perception; Tournament Model; Yitzhaki Index
1. Introduction
Undoubtedly, getting paid is an essential motivation for
people to work. Meanwhile, income is always regarded
as an indicator of their achievement and recognition
(Goodman, 1974) [1]. In the reign of neoclassical eco-
nomics, the equilibrium of free market achieves as wage
equals the value of marg inal produ ct ( Mank iw, 1995) [2].
However, Doeringer and Piore (1971) [3] argued that
there exist obvious differences between the external and
internal labor market. Jensen and Meckling (1976) [4]
further extended the con ception of wage allocation in the
internal labor market properly by defining the contract
between the principal and agent under the condition that
ownership and management are separated. To prevent the
moral hazard problems, by employing the conception of
relative performance evaluation, Lazear and Rosen (1981)
[5] proposed the tournament theory which presents a
hierarchical wage structure and a tournament-like way
for workers to achieve some preset wage level. The ra-
tional result obtained fro m this theory is that workers will
make more effort when they are faced with larger wage
dispersion.
No matter what differences exist among the tradition al
theories on pay-performance relationship, they are all
assumed that the only thing workers care is their own
wage. In other words, the subjective perception stemmed
from income comparison just like the air in the vacuum.
Unfortunately, more and more studies showed that peo-
ple actually care about others’ situation, which really
affect their own benefit. Crosby (1976, 1984) [6,7] and
Martin (1981) [8] stated that this kind of perception tends
to affect the workers’ productivity devastatingly by
causing stress symptoms, negative attitudes, undesirable
work behaviors and so on. Consistently, Sweeney et al.
(1990) [9] imposed th at this perception is associated with
pay dissatisfaction.
In this paper, we manage to incorporate the subjective
perception caused by the income comparison into the
original tournament model in order to show whether the
workers strategies and firm performance will be influ-
enced. The rest of the paper is organized as follows: Part
II is literature review, Part III is modeling, Part IV is
analysis and findings and Part V is implication and con-
clusion.
2. Literature Review
The term of relative deprivation originated in a post-war
psychological study on the US army (Stouffer et al.,
1949) [10] and the theory of relative deprivation was
articulated and formalized in Runciman (1966) [11]. In
this seminal work, an individual’s feeling of deprivation
was illustrated by an example of promotion and four
preconditions are defined for an individual to obtain this
sort of feeling, which are 1) he does not have X, 2) he
*Corresponding a uthor.
Copyright © 2012 SciRes. AJIBM
Modified Tournament Model Based on Perceived Wage 201
sees some other person or persons as having X, 3) he
wants X, and 4) he sees it as feasible that he should
have X. In another nobly work, five preconditions are
noted (Crosby, 1976) [6], which are 1) they want some
object X, 2) they feel entitled to X, 3) they perceive that
someone else possesses X, 4) they think it feasible to
attain X, and 5) they refuse personal responsibility for
their cur rent failure to posses s X themselve s. After s im-
plified in Crosby (1984) [7], two foundational precon-
ditions are suggested, n amely, wanting X and deserving
X. Since then, this theory providing insights into the
subjective well-being has attracted much attention in
many fields.
Beyond psychology view, Sen (1976) [12] firstly in-
troduced the income distribution into the study on rela-
tive deprivation. What’s more specifically, one salient
index in the context of relative deprivation depicting the
relative-weighted income from all the persons in the b et-
ter-off condition came forth in Yitzhaki (1979) [13 ]. This
Yitzhaki index defines the relative deprivation as the sum
of the differentials between some individual income and
all upper-ranked incomes, which is usually utilized as the
measurement for relative deprivation. After that, Martin
(1981) [8] proposed that the relative deprivation stems
from a comparison on rewards among persons or groups
and finally introduced the conception of relative depriva-
tion into the work and organization study.
Relative deprivation always focuses on the negative
cognition referring to the perception of disparity. How-
ever, another sort of feeling can also be obtained from
the income comparison when the individual has a better
salary. Contrast to the relative deprivatio n, this feeling is
positive and consequently can be defined as relative sat-
isfaction. Such as what Moyes (2007) [14] pointed out,
an individual could find some comfort when the poorer
persons are chosen as the reference group. By recon-
structing it, Hey and Lambert (1980) [15] divided the
Yitzhaki index into the mean income and the satisfaction
index. Stark and Yitzhaki (1988) [16] claimed that the
satisfaction originates from having X and the depriv ation
from having no more than X and assumed that the satis-
faction and the deprivation functions are complement
each other and the sum of them is mean income. Chak-
ravarty (1997) [17] represented the notion of satisfaction
as the complement of the deprivation to the mean inco me,
which is in lines with Hey and Lambert (1980) [15] and
Stark and Yitzhaki (1988) [16]. In addition, Ebert and
Moyes (2000) [18 ] found a dual r elationship betw een the
index of deprivation and the one reflecting satisfaction.
All in all, people always obtain the perception from the
income comparison, positive, negativ e or both. Generally,
we could get an aggregated perception (defined as rela-
tive perception in this paper) by choosing the different
referent groups at the same time when we are among the
richer ones and poorer ones.
It is obvious that the subjective perceptio n does matter
to the well-beings of people according to the theory of
relative deprivation. However, the original tournament
model doesn’t take it into account and assumes the indi-
vidual’s utility can be affected only by her own income.
In this paper, what we are interested in is whether the
wage dispersion can also work well as an incentive
mechanism to improve wo rkers’ output when the relative
perception exists. In addition, how much relative percep-
tion affects the workers’ effort is also an issue we would
like to explore.
3. Modeling
Following Lazear and Rosen (1981)[5], we assume two
identical workers and are employed in one repre-
sentative firm ij
X
and all of them are risk-neutral. The
output of worker is
k
,
kkkkk
eqe
, for ,kij
,
where k and k
e
denote the effort and random influ-
ence on the output of worker . The firm’s output is the
sum of workers’ outputs which can be calculated by the
equation
k

,,e
iiij jj
Qqe q
 . Given the hierar-
chical wage structure formed by high-level
H
W and
low-level
W is exogenous, we assume the winner in
the output competition obtains the higher wage
H
W and
the loser gets the lower one
Wi
. Based on the definition,
i, the probability for worker to achieve the higher
wage, is
P







prob prob
prob
prob d
ij
iij iijj
ij ji
ee
ij ij
Pqqe e
ee
eeg Gee




 

 
where
j
i

and

g
.

G
is the cumu-
lative distribution function of
and .

0E
In line with Clark and Oswald (1998) [19], we assume
individual’s utility is impacted by three items, namely,
the effort to work, income for the work and perception
originated from comparison of the income with the rele-
vant group. Following the most literature, we consider
the effort to work as a sort of disutility which means the
effort will be negatively related to the utility. We denote
this kind influence on worker as and as-
sume and kk
ue . Contrarily, income
is always positive to individual’s utility,
k0

kk
ue

0
kk
ue


kk
uW
k
is
denoted to present this kind influence on worker . Be-
sides, the comparison of income will affect the individ-
ual’s utility when the subjective perception is taken into
account, which is denoted as
kk k in this paper.
As mentioned above, two components devote to contrib-
uting the perception from comparison, the relative depri-
vation component discussed in many former studies and
vW W
Copyright © 2012 SciRes. AJIBM
Modified Tournament Model Based on Perceived Wage
202
the relative satisfaction component discussed in Spilim-
bergo and Ubeda (2004) [20] where the social satisfac-
tion is incorporated into the utility function. Because the
perception obtained from different relevant groups is
obviously different,
kk k
vW W
should be the aggre-
gated effect on the utility of worker . So we define
k
 
max0,+ min0,
kkk k
kkk
vW WW WW W

 ,
where
is the sensitivity of relative satisfaction to the
wage dispersion and
is the sensitivity of relative
deprivation to the wage dispersion. Similar with Carlsson
and Qin (2010) [21], we adopt an additive comparison
utility function, where the relative deprivation is depicted
as the difference of the wage gap between the individual
and the richer ones. Enlightened by Johansson-Stenman
et al. (2002) [22], we denote the income-utility sensitiv-
ity as
and the perception-utility sensitivity as 1
consequently. And the utility function of worker
could be expressed as follows. k
 


1
kkk kkkk
k
Uue uWvWW

i


 .
Since the probability for worker to win the higher
wage is i, the expected utility function in Von Neu-
mann-Morgenstern form could be employed to describe
the decision preference of workers (Neumann and
Morgenstern, 1944) [23] and the function is as follows.
P



 

 
 




()1
1
11
1( )1
()
iiiii ii
i
H
iii XHL
iiiLLH
i
HL Lii
EUEueuWv WW
Pue WWW
Pue WWW
P
WW Wue

 


 


 


 

 


 
Rationally and strategically, workers choose the best
effort level to maximize their own expected utility. The
first and the second order condition for this effort-mak-
ing decision problem are as follows.
F.O.C.
 

1
HL 0
i iii
WWPuee

 

i
e
S.O.C.
222
1
0
2
H
Lii ii
WWPeuee

 

i
Given the competitor’s action, the best response func-
tion for worker is
i
 
1
H
Lii ii
WWPeuee
 
 
 i
Since
 
ii ij, the equa-
tion above can be rewritten as
 


1
H
Lij
ii i
WWgee
ue e

 


 
Given the competitor’s action, the best response func-
tion for worker is
j
 

1
H
Ljj
jj j
WW Pe
ue e

 

 
Since

j
jjijj
PeGeee gee
i
,
the equation above can be rewritten as
 


1
H
Lj
jj j
WWgee
ue e

i
 


 
Because the best functions of worker and are
symmetric and these two workers are identical as as-
sumed, the best solutions for them should be same such
as j
ij
*
i
ee
*
. As a result,
 

10
HL
kk k
WWg
uee

 

 
should be the Nash Equilibrium for the worker , where
k
,kij
.
4. Analysis and Findings
As Lazear and Rosen stated in the seminal work of tour-
nament theory, a hierarchical wage structure with two
different wage levels is preset and two workers compete
for the higher wage via output. The main conclusion ar-
gues that wage dispersion and environment noise could
affect the effort decision of workers. More effort will be
exerted with the wage dispersion or the noise level.
Comparing the original tournament model, we take the
workers’ subjective perception on the wage dispersion
into account, which makes the model in this paper is dif-
ferent from the original one and help us to extend the
existing knowledge in this field via the comparative
static method. We find the effort level which the workers
choose is affected by both the wage dispersion and the
subjective perception in term of the income-utility sensi-
tivity and the sensitivity of relative deprivation (satisfac-
tion) to the wage dispersion. Some new findings in
detail are followed.
Findings A. Given other conditions fixed, workers
tend to make more effort with the wage dispersion (noise
level) increasing, which is in line with the result obtained
from the original tournament model.
Proof A. Given other condition fixed,

kk
ue
will
increase with wage dispersion

H
L
WW (or noise
level
0g) since
i ij
PeGeee gee 
Copyright © 2012 SciRes. AJIBM
Modified Tournament Model Based on Perceived Wage 203
 


1
.
HL
kk
WWg
ue
 
 


0
What’s more, effort will increase with

kk
ue
be-
cause . Combining the deduction above, the
conclusion can be obtained that effort will increase with
wage dispersion (noise level).

0
kk
ue
Findings B. Given other condition s fixed, the sensitiv-
ity of relative satisfaction and deprivation will be posi-
tively related to the effort level o f workers, which means
increasing the sensitivity of relative satisfaction or dep-
rivation will enhance the effort level. In other words,
wage dispersion should be decreased to maintain the ef-
fort level when the sensitivity of relative satisfaction or
deprivation increases, vice verse. This result shows that
the relative satisfaction or deprivation could motivate the
workers as similar as what the wage dispersion does.
Proof B: Given other conditions fixed,
 
1101
 
 

because
,
which implicates will increase with

kk
ue
ac-
cording to the equation
 

10
H
Lk
WWg ue

k
 

 .
Since , effort will increase with

0
kk
ue

kk
ue
and consequently increase with the sensitivity of relative
satisfaction. The effort level will increase with the sensi-
tivity of relative de privation could be proved in the si mi-
lar way.
Findings C. In case of 1
, the subject percep-
tion will not affect the effort level, which is a very spe-
cial case. Under this situation, all conclusions are the
same with the ones from the original model.
Proof C: The Nash Equilibrium is simplified to


0
H
Lkkk
which is the same with
the result obtained from the original model when
WWgue e

1
 and then .
 
11
 

Findings D. Given other conditions fixed, the in-
come-utility sensitivity will be helpful to increase the
effort level when 1
. In other words, the effort
level will increase with the income if the subjective per-
ception of the wage disparity is less obvious. Under this
situation, decreasing the wage disparity properly will
maintain the effort level. Vice verse, the income-utility
sensitivity will be negative to increase the effort level
when 1
 and other conditions are fixed, which
means the effort level will decrease with the income in-
creasing if the subjective perception of the wage dispar-
ity is more obvious. Under this situation, enlarging the
wage disparity is the way to maintain the effort level of
workers.
Proof D: If the worker’s effort increases with
, it
implies
 
1

is monotone increasing
function. So,
10
 



. In other
words, 1
. If the worker’s effort increases with
decreasing,
1

should be mono-
tone decreasing, whic h means
10


1

. In other words,
.
5. Conclusions and Implication
Considering the relative perception combined by the
relative satisfaction and relative deprivation as one in-
fluence factor on the individual’s utility function is the
most different point from the former tournament theory
studies. After we analyzing the Nash Equilibrium of this
output competition game and implementing the co mpara-
tive static studies, some new interesting findings are
shown in front of us. First of all, the relative perception
could affect the utility of workers and motivate the
workers as similar as what the wage disparity does unless
the sensitivity of relative satisfaction to the wage dispar-
ity and the sensitivity of relative deprivation to the wage
disparity are complementary to 1. What’s more, the in-
come-utility sensitivity being related to worker’s utility
implies the degree of importance of income for workers
will influence the workers’ effort-making decision. In
detail, the income-utility sensitivity will be helpful to
increase the effort level when the sum of the sensitivity
of relative satisfaction and deprivation to the wage dis-
parity is less than 1, otherwise, the income-utility sensi-
tivity will be negative to increase the effort level when
the sum of the sensitivity of relativ e satisfaction and dep-
rivation to the wage disparity is more than 1. Last but not
least, the enlarged wage gap can motivate workers to
make more effort and the bigger noise from outside
needs more compensation gap is also found in this study,
which is in line with the results originated from the tradi-
tional tournament model.
According to what is found in this study, the subjec-
tive perception should be paid enough attention to since
it could affect the worker both in utility and consequent
action. Whether the influence is positive or negative is
depend. More detailed, increasing the proportion of in-
come influence will stimulate worker to make more ef-
fort when the workers is less sensitive to the perception
of relative satisfaction or deprivation, vice verse. This
finding implies that the wage policy should design care-
fully and the differences in subjective sensitivity to rela-
tive perception or the proportion of income and percep-
tion among workers should be taken into account when a
proper wage strategy is made.
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Modified Tournament Model Based on Perceived Wage
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