J. Software Engineering & Applications, 2010, 3: 27-33
doi:10.4236/jsea.2010.31003 Published Online January 2010 (http://www.SciRP.org/journal/jsea)
27
Properties of Nash Equilibrium Retail Prices in Contract
Model with a Supplier, Multiple Retailers and
Price-Dependent Demand
Koichi NAKADE, Satoshi TSUBOUCHI, Ibtinen SEDIRI
Department of Industrial Engineering and Management, Nagoya Institute of Technology, Japan.
Received September 3rd, 2009; revised October 9th, 2009; accepted October 19th, 2009.
ABSTRACT
Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on
well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single
supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer
decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is
achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is
maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution
under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the
numerical values and proprieties on these retail prices, the number of products and their individual and overall profits.
In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of
Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we
compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’
profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the
demand distribution.
Keywords: Supply Chain Management, Nash Equilibrium, Stochastic Demand, Competing Retailers
1. Introduction
Recently, price contract models between suppliers and
retailers with stochastic demand have been analyzed
based on well-known newsvendor problems. Cachon [1]
has reviewed models with one supplier and one retailer
under several types of contracts. In a market, however,
many retailers exist and they compete in order to attract
the maximum number of consumers. In this context,
Yano and Gilbert [2] have been interesting in contracting
models in which the demand is stochastic and depends on
price. Wang et al. [3] and Petruzzi [4] have studied de-
centralized price setting newsvendor problems under
multiplicative retail demand functions. Song et al. [5]
have analyzed theoretically the optimal prices and the
fraction of a total profit under individual optimization to
that under supply chain optimization.
In Bernstein and Federgruen [6], they have analyzed a
contract model with single supplier and multiple retailers
and price dependent demand, where retailers compete on
retail prices. Each retailer decides a number of products
he procures from the supplier and his retail price to
maximize his own profit. This is achieved after giving
the wholesale and buy-back prices, which are determined
by the supplier as the supplier’s profit is maximized.
They have proved that the retail prices become a unique
Nash equilibrium solution under weak conditions on the
price dependent distribution of demand. They, however,
have not mentioned the numerical values and properties
on these retail prices, the number of products and their
individual and overall profits.
In this paper, we analyze the model numerically. We
first indicate some numerical problems with respect to
the theorem of Nash equilibrium solutions, which Bern-
stein and Federgruen [6] proved, and we show their
modified results. Then we present Nash equilibrium
prices, optimal wholesale and buy-back prices for the
supplier’s and retailers’ profits, and optimal retail prices
under supply chain optimization, analytically and nu-
merically. We also discuss the properties on a relation-
ship between these values and the demand distribution.
In the next section, we present the competing retailers

Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand
28
model introduced in [6], and we discuss the sufficient
conditions on the existence and the uniqueness of the
Nash solution. In Section 3 we investigate the model
with exponential and uniform distribution functions and
with linear and Logit demand functions. In Section 4, we
present numerical results and discuss the behavior of
Nash equilibrium solutions and properties of the profits
and prices.
2. Competing Retailers’ Model
The model of competing retailers for one supplier
and retailer introduced in [6], is shown
in Figure 1.
S
N,1
i
RiN
This model is set under wholesale and buyback pay-
ment scheme. The supplier incurs retailer S,1
i
R

a wholesale price for each product, com-
bined with an agreement to buyback unsold inventory at
. We assume that the supplier has ample capacity to
satisfy any retailer demand and produce products at a
constant production cost rate , which includes the
transportation cost to retailer . When and are
given, each retailer orders his quantity and cho-
oses his retail price . A salvage rate is
adopted in the supply chain. To avoid trivial setting, the
model parameters are chosen as and
iN
i
b
i
c
i
R
w
i
i
w
 
i
w
i
b
 
ii cv
i
R
i
p
i
y
i
v
ii
vb

for . 1iN
The demand is random and depends on the
price vector , with a cumulative dis-
tribution function . It is restrained to a
multiplicative form
i
pp,
G (x

D(p)

12 N
p,,p
i1
|p,..., p
()
i
Dp d

i
N
)
()
i
p

, where i

is a
random variable with a cumulative distribution function
S
R
1
R
2
R
i
R
N
c
i
v
i
w
1
b
1
w
2
b
2
w
i
b
i
w
N
b
N
p
1
p
2
p
i
p
N
Figure 1. Competing retailers’ model
i
G(.) and a probability density function i
g
(.) . In addi-
tion i

is assumed to be positive only on
x

and independent of the price vector p. This
implies that
min
[,
i
xx
max ]
i
ii
i
x
G(x|p) Gd(p)



.
The demand function depends on the whole
price vector. It is supposed that decreases in pi-
and increases in pj for all j ≠ i, 1 ≤ I ≤ N, that is
()
i
dp
()
i
dp


0


i
i
p
pd and


0


j
i
p
pd .
Let


12
,,,
N
y
yy y denotes the order vector of
the model. The expected profit function for the retailer
is given by
i
R




(,)min, ()()
iiiiiii
pypEyD pbEyD pwy 
ii


 
,
where . It can be rewritten as


max 0,a


a


 )()()(),( pDyEbpywpyp iiiiiiii 
(1)
From (1), the retail prices p impact on the profits of all
retailers and his order quantity, however, affects only his
own profit. In addition the retailer wants to maximize his
order quantity. Then, the derivation of the retailer i’s
profit function given by Equation (1) on is equal to
zero
i
y
0


i
i
y
)
y
,p(

(2)
Therefore, the retailer i's optimal corresponding order
can be obtained from (1) and (2) by











ii
ii
iii bp
wp
G)p(d)p(y 1 (3)
This result reduces the no-cooperative game in the (p,
y)-space to a p-space game. In this space the retailers
compete only on prices (reduced retailer game). Then,
considering the Equations (1) and (3), we get the retailers
profits as a function of p only, as
1
1
ii
iiiii
ii
ii
ii ii
ii
pw
(p)d (p)(pw)Gpb
pw
(pb )EGpb















 

 

 








det (| )(())
iiiii
pwL fp   (4)
where is the profit function
with a deterministic demand ()p,
det (|) ()()
iiiii
pwpwd p  
ii
yd () () ()
ii
ii
ii
pw
fp pb



Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand29
is the critical fractile, and
i
We define and we apply
the logarithm
(S’), as
1(( 
))
11
1
()()()() /
iii
Gfp
iiiiiii i
Lf GffEGfuguduf 
 



 




1(( ))
ˆ() ()
iii
Gfp
ii i
Lp ugudu



to (4), we get for 1iN
log og
iii i
π(p)L (p)loglog l
ii
(pb)d (p)

 (5)
The supplier profit function is given by


()() ()
N
Miiiiiii
wcy bEyDp 
1i


 
. Using
Equation (3), it can be expressed as
 
1
M
1
ii ii
i
dp w cG


 



1
() .
N
ii
ii
ii
ii ii
ii
pw
pb
pw
bvEG pb 









 
 
 





(6)
Differentiating (5) on for
i
pNi 1
),p(U
p)p(dp ii
iii
)p(d)p(
~
log ii  

1 
with

1(
i i
G f


2
() )
1
() ()()
ii i
ii
ii iiii
wb p
Up pb pbLp


 (7)
Bernstein and Federgruen [6] have proved t
istence of a Nash solution for the reduced retailer
ga
the same reeach reta
price clos interval
o hold:
hat the ex-
*
p
me is assured by the following condition (A).
(A): For each {1,. .. ,}iNthe function log( )
i
dp is
increasing in (,)
ij
pp for all ij.
,
It is assumed in ference [6] that iler
i
Rchooses hisi p from aed
min max
,
ii
pp


. The authors proved the uniqueness of the


max
Na

sh solution in the price space

min max
, 2,
ii
i
i
wbp. This has provided the
following conditions (D) and (S) t
(D):

p


2det 2det
2
log( |
iii
ji
i
pw b
p




 
log( |))
iii
ij
pw b
pp
  ,
(S):  







 x
i
i
i
i
ix)x(Gdu)u(ug
)x(g
)x(G
x)x( 02 2 ψ,
for aanll is the medi
of theon under the
{1,..., }iN
distribution G
onditions ma
i
, where i mx 
i). However, th
n the b
(i m
e soluti
ndabove cy exist oouary of the area
 

i
ii pbwp maxmin ,2,max . In this case, it does not
satisfy log()0.
ip
i
p


 the condition (S) is modified to
(S’):

2
() ()()0
()
ix
i
xuguduGxx

() 2
ii
i
G
xx
gx





] Then, the following theorem can
ained.
S’) hold, th
um prices on
 
 
ψ
for all min max [,
ii
xxx.
be obt
Theenorem : If conditions (A), (D) and (
there is a unique set of Nash equilibri
[,)
i
i
w

wtisfy hich salog( )0
ip
 

for all
i
{1,..., }iN
p

.
Proof: In the same way as in Bnd Federgruen
s a unique Nash s[,)
i
w
ernstein a
[6], there iolution in
*
p
i


which satisfies , because
*
ii
pwfor each {1,..., }iN

,
() 0
ip


when ii
p
w

whereas () 0
ip

 when
ii
pw. This ims that plielog( )
i
p
p
0
i




when
*
pp

for all iN{1,. . . ,}

.
following, the retailerst these
um prices, whereas the sthe be-
retailers and de
k prices t
In the sell products a
equilibri upplier knows
havior of thetermines the wholesale and
buybaco maximize his own profit. This system
is
faces a random de-
called “individual optimization”. On the other hand,
the problem of determining retail prices and quantities of
products to maximize the entire profits of supply chain is
called “supply chain optimization”
3. Determination of the Nash Equilibrium
As shown in Figure 2, we study a two competing retail-
ers’ model. Each retailer {1, 2}i
mand (),
i
Dp where 12 (, )ppp

. We assume two
types of cumulative distribution functions of demand.
We consider first the exponential case and then the uni-
form one.
3.1 Exponential Case
The cumulative distribution function in the exponential
S
R
1
c
i
w
1
1
R
2
b
1
w
2
b
2
p
p
2
v
i
Figure 2. Two competing retailers’ model

Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand
30
() 1
x
i
Gxe

 for all 0x, case is given by
where i
E

is set as
rse function of
01y. Wit
one without loss
()
i
Gxis given by
h
of generality. The
1()log(1 )
i
Gy y
 inve
for all ()
()
ii
ii
, we get ()
ii
fp  
pw
pb












)
bp
bw
()bw(wp
bp
)p(L
iiii
ii
iiiiii log
1

Then, using (7), we obtain


 
log
ii
ii
ii
ii ii
i
pw
U(p )
w
pb wb






i i
i
b
p wb
p




 


1) Linear demand function
The linear demand is given for by
j
,, {1,2}jiij
()
iiiiij
ji
dpp p
 

 
with 0,,0 ijii



.(8)
we obtain the system of
equations
With this demand function,
11
11
1111122
2
log( )p 


2
22
211
log( )()0,
()0.
pUp
ppp
Up
p

 









It can be rewritten as
22 22
pp 



2222 2222
1
() ()
,
()
Up pUp
pUp
 





21 2 2
1 1111111
2
12 11
() ()
.
()
Up pUp
pUp
 





(9)
The optimal order quantities and canbe
evaluated to
1
y2 y


11
111
()yp 

1122
11
22
2222211
log ,
() log
pb
p pwb
pb
ypp pw

 
 





 


Since
22
.
b


1log ,
iiii i
i
pwpb wp
EG p




  






i
i
iiiiii
bwb pb



 

from (6), the supplier profit function can be expressed as
 
2
M1
log iii i
ii iiii
iii ii
pb pw
dpw c bb
wb pb 



 



 
 
 
 
11
11 11
11 11
22
22 22
22 22
log ,
log
pb
pdpbwpw
wb
pb
pdpbw pw
wb






 










 













2) Logit demand function
Now, the problem is studied with a logistic demand
function, expressed by
2
()
i
p
i
i
1
j
p
ij
j
Ck e
 


ke
dp
 
 for 0and 
iik,C,

. (10)
With this demand function we obtain the system of
equations
2
12
1
12
112
11
1112
221
22
221 2
log( )()() 0,
log( )()()0.
p
pp
p
pp
pCke
Up
pCkeke
pCke
Up
pCke ke























Then, we have


2
2222 222
1
() () 1log ,
p
C CUpkeUp
p








1
12 2
111 11
2
211
()
() 1log ()
p
kU p
C CUpkeUp
pkUp










11
() .

The order quantities are given by
1
12
2
12
11
1
11
11 2
22
2
22
21 2
1
2
)log ,
()log .
pp
p
pp
kep b
wb
Cke ke
kep b
yp wb
Cke ke






 




 
 

 



 


The supplier profit function and retailers’ profit func-
tio in the same way as for the linear de-
ma
3.2 Uniform Case
The cumulative distribution function in the uniformase
is given by
(
p
yp
 


ns are obtained
nd function.
c
(1 )
() ,
2
i
i
i
Gx a
11, 01,1, 2,
ii i
xa
axaai



 

where



1.
i
E


1()1Gy 
(
i
The inverse function of is gven
by i for . With
()
i
Gx
01y
i





.
The retailers’ profit functions are given by
2
ii
a ay 
())/( )
ii iii
f
ppwpb

, we get
() 1
ii ii
iiii
ii ii
pw pw
Lpa a
pb pb

 



 


 



Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand 31
r {1, 2}i Then by (7) fo
12
1
() 1
1
ii
ii
iii iii
ii
pw
pb pwpw
aapb



 








1) Linear demand function
With the linear demand given by (8) and , we
ii
ii
ii
aa
pb
wb
Up









ii

)p(U ii
obtain
2 222222
(
)
Up
2
1
21 2 2
1111 1111
2
12 11
() )
,
(
() ()
.
()
Up p
pUp
Up pUp
pUp
 

 









r quantities are given by The optimal orde


11
111112211
11
22
222221122
22
()1 2,
()1 2
.
pw
ypppaa pb
pw
ypppaapb
 
 



 









 







equal to The supplier profit function is
 
2
M
p



1
2
12
ii
iiiiii
iii
ii
ii
ii
w
dpw caapb
pw
ab pb




  










 







.
The retailers’ profit functions are given by
 


 


11
111 1
11
11 2
11
11 1
11
22
2
22
22 2
22
22 2
22
12
,
.
pwa apb
pdp
pw
ap bpb
pw
pb
pdp
pw
ap bpb





  









 






 













 









2) Logit demand function
With the Logit function given by (10), we obtain
and as
22 2
12pw aa

  



pw





1
p2 p


2
1
2222 222
1
122
1111111
2
211
1log ,
1log .
λp
λCCU(p)keU


λp
(p)
pλkλU(p)
λCCU(p )keU(p )
pλkλU(p)








The optimal order quantities are given by
1
12
2
12
11
11
11 11 2
22
22
22
21 2
()1 2,
()1 2.
p
pp
p
pp
kepw
ypaa pb
Cke ke
kep w
ypaa pb
Cke ke















 









  


1
1
2
2
The supplier profit function and retailers’ profit func-
tions are obtained in the same way as for the linear de-
mand function.
3.zation
W
anordantities to maximize the overall profit of
the supply chain, the wholesale and buyback prices are
meaningless because they are payments between the
supplier and the retailers. As the whole of the supply
chain is equivalent to a single retailer with who
price and buyback
3 Supply Chain Optimi
hen the supplier and the retailers determine the prices
d the er qu
lesale
12
(, )cc 12 (, )

 , and by using (3),
the optimal order quantity (the amount of products) is
given by 1 () ()
Iii
iii
pc
yp dpG
ii
p



 . By u


sing (4), the
overall expected profit of the supply chain is
2
1
()() ()
Iii
iii i
iii
pc
ppcdpL
pv


 
 



, (11)
wh ere, the retail prices 12 (, )pp are given. The optimal
retail prices 12 (,)
I
I
pp in the integrated supply chain
maximize the profit function given by (11).
ote th
4. Numerical Examples
4.1 Geometric Analysis of the Nash Solution
The system of equations on 12 (, )pp that solves the
profit functions for the two retailers is obtained in Section
3. In the case of exponential demand and linear functions,
we dene right hand sides of two equations in (9) by
22
()
f
p and 11 ()
f
p, respectively. Then the equations
(9e ) becom12
()pf2 p

and Note that in
s sa
21
().pfp
tisfied
1
other caseby (, )pp form
122
()pfp
the equations 12

and 211 ()pfp

similarly. Geom
he
etrically,
Nash to analyze the
ution, we
behavior
ot the f
of th
ncti
e system around t
ons ()
ii
sol plu
f
p fo
le solutions fo
solution
r 1 pand 2 p
r the equa-in Figure 3. There are multip
e Nashtions, but there is a uniqu12 (,)pp with
(1,2)
ii
pwi, which has been proved in the theorem
of Section 2.

Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand 32
P
1
P
2
w
2
w
1
12
(,)pp
Figure 3. Nash solution and system of equations
Given wholesale and buyback prices, we derive these
Nash retail prices, and profits of the supplier and two
retailers. We compute them for all combinations of
wholesale and buyback prices, which are integers and
satisfy and i
U
iii
cww ii vbw

, where is
set as the upper bound for the optimal wholesale price for
the supplier, and derive optimal wholesale and buyback
prices for the supplier. We also compute the overall prof-
its and retail prices under the supply chain optimization,
and compare them with the ones under individual i-
mization.
.2 Numerical Results
U
i
w
opt
4
In numerical examples we set parameters as shown in the
following:
12
(, ) (0,0)

 ,
12
( ,))(100,100

 ，12 (, )(1,1)

，
12 21
(,)(0.3,0.3)

 (linear function ) ,
0.03

，12 ( ,)(0.005,0.005)CC ，
12
(, ) (1,1)kk , (Logit function).
The program is coded by C and the computations are
mpiler on PC. In Table 1, we
d Logit functions, wher-
cost parameter settings are considered:
etric) and
done by using Fujitsu C co
assume exponential demand an
eas in Table 2 the linear function is assumed. In these
tables two
12
( ,)(30,30)cc  (symm12 (,)(30, 20)cc

(anti-symmetiric).
The values in tables are the optimal pfor supplier,
the profit for each retaim of
supplier’s and retailers’ profits), optimal whole-sale and
buyback prices for the supplier, Nashilibrium retail
prices and orderes in paranthesis ()
are th
function
i
rofit
ler; entire expected profit (su
equ
quantities. The valu
e total profit, optimal retail prices and order quanti-
ties for retailers under the supply chain optimization.
In the cases of Tables 1 and 2, optimal whole sale
prices and buybacks determined by the supplier give
more profits to the supplier than retailers. In the symme-
Table 1. Exponential demand and logit
1 2 1 2
Ci 30 30 30 20
()
M
p

32.195 35.792
(,)
iii
py

10.227 10.227 7 3.843
Entire
expected
profits
52.649
(62.430)
58.552
(7
8.91 1
0.153)
i
w 98 98 100 88
i
b 47 47 47 47
175.420
(172.428)
175.420
(172.428)
175.376
(182.095)
168.444
(161.07)
0.311 0.311 0.276 0.418
i
p
i
y (0.606) (0.606) (0.444) (0.965)
ential demand and linear function Table 2. Expon
i
a 0.1 0.3 0.5 0.7
()
M
p
 2531.42 2352.36 2176.38 2003.38
(,)
iii
py 
513.03 481.51 450.56 414.12
Entire
expected
profits
3557.49
(4303.71)
3315.38
(3999.12)
3077.51
(3700.00)
2832.62
(3407.00)
12
()ww87 87 87 87
12
()bb75 75 75 74
1
()
2
p
p110.31 110.97 111.69 112.55
) (87.08) (88.46) (89.96) (91.56
2
(4)
2
(4)
2
(4)
2
(4)
12
()yy3.51
0.26
4.55
1.75
5.59
3.20
6.05
4.57
tric cost casesptimil prf two retailers
bec same. red to supply ptimiza-
tioail pricgher and the s of or-
ders are smaller in the individual optimal case. It is be-
cause uer then optiation mamountf de-
mand a satisf decsing re prices in-
creasing ordeiess idivti-
mal case ther o ofit,
which leads toer wle prnd asult
retail prices bighemmost
ase, the optimal wholesale price to the retailer with the
smalther
retailer, which leads to mrofits he fo-
tailer. The reaso that thailer wmall w
price setshe il d mntr-
der, which imat mouur
in td the supplier can sell more products to cus-
tompa wit he
demepethri ly,
and the wholespricestail prind th
quantities changore.
In both cases entirepected profits in indi-
vidual optimalis a0 toof er
supply chain on tn cof
, the oal retaices o
ome the
n, the ret
Compa chain o
es are hiquantitie
nd chaimizore s o
reied byreatail and
r quantit, wherean the inidual op
suppliewants to btain itswn pro
highholesaices as a re
ecome hher. In t anti-syetric c
c
ler production cost is smaller than that to ano
ore p
e ret
for t
ith s
rmer re
holesale n is
tless retaprice anore quaities of o
plies thore amnts of demand occ
otal an
ers. In
and d
rticular,ith Logdemand function t
nds on e retail pces more intensive
ale , reces ae order
e m
the exthe
cases bout 8 85 % that und
optimizatin. Whehe chaionsists

Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand 33
Table 3. Uniform deand unct
i
mand linear fion
1 2 1 2
Ci30 30 30 20
()
M
p
 1200.548 1473.307
(,)
iii
p
y  242.306 242.306 228.119 380.888
Entire
expected
profits
1685.160
(2041.22)
2082.314
(2515.01)
i
w 89 89 89 82
i
b 77 77 77 73
i
p
116.154
(96.902)
116.154
(96.902)
115.532
(97.788)
112.445
(90.259)
i
y 22.105
(37.717)
22.105
(37.717)
21.233
(34.608)
32.826
(58.887)
one supplier and one retailer, it is shown in Song et al.
(2008) that the fraction is 3/4(in linear case) or
2 /0.736e (in Logit case). The competition among
retailers makes retail prices lower, which makes the frac-
tion higher.
In Table 3, the uniform distribution of demand is
assumed with the symmetric production costs
((12 , )(30,30)cc ), and the i a, which corresponds to
the width of the uniform distribution, is changed from 0.1
to 0.7. It implies that large i a means the high variance
of dems are
higher, d profitsf the sulier andtailers drease.
This is because when the vace inces, theantity
of order mt be into apply tation of
demand, whereas thelso incd
to obtain proitsle
Wchanges the optimal wholesale prices an
buybes for tr are almoe. Note
that even it is cared w result the exponential
case shown in Fige 2, whi h has mvariance
these unif disttions, difference on the
is very small. ohand
buyback pricese tnce
of the demand d
tail prices. In numerical ex-
am
rant-in-Aid for Scientific
t with revenue sharing,” Man-
hain modeling implications and in-
sights,” Workusiness, University
of Illinois, Urb4.
5. Concluding Remarks
In this paper, we first show the sufficient condition that
unique Nash equilibrium retail prices exist and they are
greater than wholesale prices. We then give the equations
whose solutions are those re
ples we compute these equilibrium prices, optimal
wholesale and buy-back prices for the supplier and sup-
ply chain optimal retailers’ prices, and discuss properties
on these values. In future research, a two-supplier prob-
lem and other types of problems will be modeled and the
properties will be discussed analytically and numerically.
6. Acknowledgments
This work was supported by G
Research (C) 19510145.
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rianreas qu
uscreased he fluctu
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of retai
price m
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ust be arease
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ack pric
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he suppliest the sam
ifompiths in
ur
ribu
c
the
ore than
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orm se prices
It means that the ptimal wolesale
for the supplier arrobust inhe varia
istribution.