Application of Game Theory in Dynamic Competitive Pricing
With One Price Leader and n Dependent Followers
Monireh Sadat Mahmoudia, Ali Asghar Tofighb
a,b: Amirkabir University of Technology(Tehran Polytechnic),
Department of Industrial Engineering
[email protected]
[email protected]
Abstract
In this research UF cheese pricing is considered and Pegah, Pak, Kaleh, Rouzaneh and Mihan
firms’ data, as five main UF cheese competitive firms of Iran in breakfast cheese competitive
market, is used. By using these firm’s sales data, production data and price of each ton of UF
cheese in nineteen work-periods (each work-period is 6 months), their sales equations are
estimated for each work-period. With the objective of minimizing the difference between each
firm’s sales in each work-period and each firm’s target sales in the same work-period, optimal
prices are calculated in 4 states: static games with complete information, dynamic games with
complete information with one leader and four followers that decide simultaneously, dynamic
games with complete information with one leader and four followers that decide simultaneously
considering that Rouzaneh and Mihan firms collude in their product pricing, dynamic games
with complete information with one leader and four followers considering that followers collude
in their product pricing.
Keywords: Static Game with Complete Information, Dynamic Game with Complete Information,
Competitive Pricing, Stackelberg Game, Collusion
1. Introduction
In a statistical study in Japan, 70% of respondents stated that they would prefer rational prices
to high quality products [1]. Price is one of the marketing mixture factors which is affected
rapidly by the management decisions. MC KINSEY firm counselors believe that the quickest
and most efficient way for a firm to achieve the maximum benefit, is setting an appropriate
price for its products or services. These counselors also reported that in 2462 firms they have
studied, 1% improvement in price causes an average 11.1% increase in benefit [2].
Price leadership has long been recognized as an important and frequently occurring
phenomenon [3]. U.S. steel production firms maintained parallel price changes before foreign
competitors’ arrival [3]. General Motors acted as a price leader for many years and its prices
were followed by Chrysler and American Motors [4].
Effects of actions and marketing behavior of one brand can be distributed among its
competitors’ market shares in a complex manner. In 1988, Carpenter, Cooper, Hanssens, and
Midgley presented methods for modeling brands competition and brands strategies in markets
where competitive effects can be distributed differentially and asymmetrically. In that paper,
empirical studies were discussed, model parameters were estimated and competitive strategy
implications were explained in the proposed model. Price and advertisement competition
among brands of an Australian house appliances firm is used to illustrate application of these
procedures [4]. For example, Wall Street Journal has reported that Chrysler’s pricing strategy
seems to be following its competitor,Ford [5].
Example documented cases of price leaders enforcing companied to cooperate even through
price cuts, are Shell Oil in California, and Standard Oil in Ohio gasoline market [3]. Of other
markets with price leadership patterns, we can mention air travel agencies , turbo generators,
personal computers, and some packaged consumer goods such as cigarettes and breakfast
cereal [6].
2. Model Description
Presented model in this research, is obtained from presented model by Abhik Roy, Dominique
M. Hanssens, Jagmohan S.Raju in a paper named “Competitive Pricing by a Price Leader”.
Because of several reasons which are mentioned later in the article, the preceding paper’s
model is changed in some aspects and in other words the model has developed. In the
preceding paper it is assumed that the market consists of two firms, labeled as firm 1 and firm
2. This research focuses on the pricing problem of the leader (firm 1), although, the analysis
can also be used for optimal pricing rule of follower (firm 2). It is also assumed that pricing
decisions are made in discrete time periods. For example, forecasts of future demand are
obtained every month and prices are set based on these forecasts. Each period is presented by
subscript t, where t=1, 2, 3… T. Presented model in the preceding paper with the objective
function:
𝑇
min ∑ 𝜌𝑡−1 [(𝑞𝑡1 − 𝑞11∗ )2 + (𝑞𝑡2 − 𝑞12∗ )2 ]
(1)
𝑡=1
Subject to:
1
2
𝑞𝑡1 = 𝑎11 𝑞𝑡−1
+ 𝑎12 𝑞𝑡−1
− 𝑏11 𝑝𝑡1 + 𝑏12 𝑝𝑡2 + 𝑢𝑡1
(2)
1
2
𝑞𝑡2 = 𝑎21 𝑞𝑡−1
+ 𝑎22 𝑞𝑡−1
− 𝑏22 𝑝𝑡2 + 𝑏21 𝑝𝑡1 + 𝑢𝑡2
(3)
Where 𝑞ti is the sale of firm i in period t, and 𝑝ti is the price of firm i in period t. It’s assumed
that sales in period t depends on the sales in period (t-1), prices in period t, and other
exogenous factors that are not known completely at the beginning of period t. In these
equations, 𝑏11 reflects the effect of own price in period t on demand and its symbol is
consistent with the basic intuition that an increase in own price 𝑝𝑡1 , reduces demand in period
t. 𝑏12 captures the effect of the competitor’s price. An increase in the competitor’s price 𝑝𝑡2
increases demand for firm 1. Equation 2 also assumes that sales in period (t-1) affect sales in
period t, implying that there is a carryover effect from the past. This carryover effect may be
due to a problem in the distribution system, or due to the fact that consumer preferences do not
change instantaneously. Finally, demand in period t is dependent on exogenous factors like
changes in customer’s preferences or changes related to design or advertisement that is
determined by u1t [7].
It’s also assumed that the leader’s objective is to keep unit sale as close as possible to a preset
target in each period. The follower is assumed to have a similar objective. It focuses on firm 1
here, because the process is identical for firm 2. Firm 1’s objective is to set price 𝑝𝑡1 at the
beginning of each period so that the actual sales are as close as possible to a preset target 𝑞11∗ .
Firm 1 also expects a certain level of 𝑞12∗ for firm 2’s sales and plans based on this
expectation. In other words, 𝑞11∗ is the target firm 1 wants to meet, while 𝑞12∗ is that part of
industry demand that firm 1 expects firm 2 to meet, therefore the objective is very similar to
achieving a pre-set market share of expected industry sales [7].
The target sales vectors of firm 1 and firm 2 are 𝑞1∗ = (q11∗ , q12∗ ) and 𝑞 2∗ = (q21∗ , q22∗ ),
respectively. If expectations and targets are in line, then 𝑞11∗ = 𝑞 21∗ and 𝑞12∗ = 𝑞 22∗ . In the
empirical illustration in this paper, it’s assumed that the target vectors of firms are in line.
More specifically, it assumes that each firm sets its prices so as to minimize the discounted
sum of squared deviations from the target vectors 𝑞1∗ and 𝑞 2∗ through periods 1 to T. It’s
assumed that both firms have the same discount rate,  [7].
Using this model for products and goods which are produced in our country is difficult and
sometimes impossible, because most firms in Iran set their product’s price under the
government order and government fixes the ceiling price for every product each year or every
six months so competitive pricing does not exist for many products and goods.
In this research, UF cheese pricing is considered and Pegah, Pak, Kaleh, Rouzaneh and Mihan
firms’ data, as five main UF cheese competitive firms of Iran in breakfast cheese competitive
market, is used. Since UF cheese is a product that should enter the market quickly and be
supplied among consumers, the problem of minimizing the gap between production in each
period and sale of the same period is counted as one of the most important objects of these
firms.
Since the maximum keeping time for these cheeses in fridge is 6 months and after that, they
will be inconsumable, there’s no need to consider sigma symbol in objective function for all
periods and it’s enough to study each period separately. Since  is constant for all five firms in
each period and the objective function is considered separately for each period,  is omitted in
the model. Also in presented model in the preceding paper, target sales which are equivalent to
production forecasts, are the same and equal to a specific quantity in all periods, but it’s not the
same for our model in this research. In this research the production forecast which is equivalent
to target sales, is estimated by OLS method in each period and the amount is different in each
period. Constant target sales assumption, as explained in the preceding paper, is one of the
defects of that model which is modified in model presented in this paper.
The problem is modeled in two types of objective functions for five mentioned firms. In type
one, the objective function for each firm is minimizing the difference between sales and target
sales each work-period. These objective functions are shown by (*). Objective functions for
five mentioned firms are:
min(𝑞𝑡1 − 𝑞𝑡11∗ )2
(4)
min(𝑞𝑡2 − 𝑞𝑡22∗ )2
(5)
min(𝑞𝑡3 − 𝑞𝑡33∗ )2
(6)
min(𝑞𝑡4 − 𝑞𝑡44∗ )2
(7)
min(𝑞𝑡5 − 𝑞𝑡55∗ )2
(8)
In type two, objective function for each firm is minimizing the summation of differences
between its sales and target sales in each period for two sequential periods. Since prices of
most products change annually, considering fixed prices assumption in two sequential periods
(each period is 6 months) in each year, objective functions for five mentioned firms are:
1
11∗ )2
min[(𝑞𝑡1 − 𝑞𝑡11∗ )2 + (𝑞𝑡+1
− 𝑞𝑡+1
]
(9)
2
22∗ )2
min[(𝑞𝑡2 − 𝑞𝑡22∗ )2 + (𝑞𝑡+1
− 𝑞𝑡+1
]
(10)
3
33∗ )2
min[(𝑞𝑡3 − 𝑞𝑡33∗ )2 + (𝑞𝑡+1
− 𝑞𝑡+1
]
(11)
4
44∗ )2
min[(𝑞𝑡4 − 𝑞𝑡44∗ )2 + (𝑞𝑡+1
− 𝑞𝑡+1
]
(12)
5
55∗ )2
min[(𝑞𝑡5 − 𝑞𝑡55∗ )2 + (𝑞𝑡+1
− 𝑞𝑡+1
]
(13)
These objective functions are shown by (**).
Where Firm i=1: Pegah firm, firm i=2: Pak firm, firm i=3: Kaleh firm, firm i=4: Rouzaneh firm, firm
i=5: Mihan firm,
𝑝𝑡𝑖 : product price of firm i in period t.
𝑞𝑡𝑖 : firm i’s product sales in period t.
𝑖𝑗∗
𝑞𝑡 : Firm i anticipated production by firm j in period t.
3. Econometrics methodology
After defining and clarifying the econometrics model -which is achieved from economic
theories- the next step in econometrics investigation process is estimating parameters of the
model with available data. After estimating parameters of the model, econometrist should
examine appropriate criteria to assess the consistency of the estimated parameters with theory
expectations under test. If the selected model, verifies the hypothesis or theory under
investigation, the model can be used for prediction of future quantities of independent variable
based on expected future quantities or known quantities of descriptive variable. Eventually,
estimated equations of 𝑞𝑡1 ،𝑞𝑡2 ،𝑞𝑡3 ،𝑞𝑡4 and 𝑞𝑡5 by Eviews software are:
5
1
2
3
4
𝑞𝑡1 = 0.53𝑞𝑡−1
− 0.02431𝑞𝑡−1
− 0.0584𝑞𝑡−1
− 0.01788𝑞𝑡−1
−0.02916𝑞𝑡−1
−0.0029𝑝𝑡1 + 0.0016𝑝𝑡2 + 0.0019𝑝𝑡3
5
4
+ 0.0014𝑝𝑡 + 0.0023𝑝𝑡
(14)
5
1
2
3
4
𝑞𝑡2 = −0.0984𝑞𝑡−1
+0.5241𝑞𝑡−1
−0.0411𝑞𝑡−1
−0.0692𝑞𝑡−1
− 0.0529𝑞𝑡−1
+0.0028𝑝𝑡1 −0.0078𝑝𝑡2 + 0.0038𝑝𝑡3
5
4
+ 0.0034𝑝𝑡 + 0.003𝑝𝑡
(15)
5
1
2
3
4
𝑞𝑡3 = −0.0591𝑞𝑡−1
−0.0712𝑞𝑡−1
+0.48571𝑞𝑡−1
− 0.0847𝑞𝑡−1
−0.0986𝑞𝑡−1
+ 0.00338𝑝𝑡1
+ 0.0039𝑝𝑡2 −0.00869𝑝𝑡3 + 0.0036𝑝𝑡4 + 0.0027𝑝𝑡5
(16)
5
1
2
3
4
𝑞𝑡4 = −0.0641𝑞𝑡−1
−0.0617𝑞𝑡−1
−0.091𝑞𝑡−1
+0.5163𝑞𝑡−1
−0.0408𝑞𝑡−1
+0.0025𝑝𝑡1 + 0.0037𝑝𝑡2 + 0.0029𝑝𝑡3
5
4
− 0.0069𝑝𝑡 + 0.0017𝑝𝑡
(17)
5
1
2
3
4
𝑞𝑡5 = −0.0533𝑞𝑡−1
−0.0201𝑞𝑡−1
−0.0417𝑞𝑡−1
−0.0881𝑞𝑡−1
+0.27134𝑞𝑡−1
+0.0013𝑝𝑡1 + 0.0017𝑝𝑡2 + 0.0024𝑝𝑡3
5
4
+ 0.0021𝑝𝑡 −0.0045𝑝𝑡
(18)
4. Calculation of the game equilibrium point
After estimating 𝑞𝑡1 ،𝑞𝑡2 ،𝑞𝑡3 ،𝑞𝑡4 and 𝑞𝑡5 equations, we solve this problem in 4 following
states:
a. Static game with complete information
In this state, all firms set their price and present their product price to the market
simultaneously. This game is called static game with complete information in which
players move simultaneously. It is clear that in this game, competition is based on the
product pricing (UF cheese) in five firms: Pegah, Pak, Kaleh, Rouzaneh and Mihan, And
since price is a continuous quantity, this game is a static game with complete information
and continuous strategies.
In these types of games, it is enough to derive each firm’s objective function from its own
price and the final equation is equaled to zero in order to achieve critical points of the
objective function. By doing this, five equations are achieved according to 𝑃𝑡1 ،𝑃𝑡2 ،𝑃𝑡3 ،
𝑃𝑡4 and𝑃𝑡5 . Since in this paper, 18 work- periods is considered, each of 𝑃𝑡1 ،𝑃𝑡2 ،𝑃𝑡3 ،𝑃𝑡4
and 𝑃𝑡5 are an 18×1 matrix. Because of so much calculation, MATLAB software is used
for calculating the game equilibrium points and we calculate the equilibrium quantities in
18 work-periods which are stated as follow. If this equations system doesn’t have any
single solution or the equilibrium prices are less than zero, it’s stated that this game
doesn’t have any equilibrium point.
b. Dynamic game with complete information with one price leader and four followers
that decide simultaneously (Stackelberg Game).
In this state information these firms provided this research, at the beginning of each
period, leader firm (Pegah) presents its product price to the market, after that, followers
which are Pak, Kaleh, Rouzaneh and Mihan firms, price their cheese products, after
observing the leader firm’s price. In other words, followers start a static game with
complete information among themselves. Each game in each period is a dynamic game
with complete information and continuous strategies, which is solved by backward
method. Problem solving process is: since each period is considered separately, at first, in
each period we derive followers’ objective functions from their prices, and then final
equation is equaled to zero to find critical points. By doing this, four equations are
achieved. Using these equations, 𝑃𝑡2 ،𝑃𝑡3 ،𝑃𝑡4 and 𝑃𝑡5 are calculated according to 𝑃𝑡1 :
𝐴4×4 𝑃4×18 + 𝐶4×1 𝑃′1×18 + 𝐵4×18 = 0
(24)
3
5
2
4
Where P: a 4×18 matrix, rows are 𝑃𝑡 ،𝑃𝑡 ،𝑃𝑡 and 𝑃𝑡 and number of columns is the
number of work-periods.
𝑃′ : a 1×18 matrix, this matrix is 𝑃𝑡1 and number of columns is the number of workperiods.
𝐴𝑃 = −(𝐶𝑃′ + 𝐵)
−1 (𝐶𝑃′
−1
′
−1
𝑃 = −𝐴
+ 𝐵) = −𝐴 𝐶𝑃 − 𝐴 𝐵
(25)
3
5
2
4
Finally, by using this equation, 𝑃𝑡 ،𝑃𝑡 ،𝑃𝑡 and 𝑃𝑡 are calculated according to 𝑃𝑡1 .
Calculated variables of 𝑃𝑡2 ،𝑃𝑡3 ،𝑃𝑡4 and 𝑃𝑡5 according to 𝑃𝑡1 are placed in equation (9).
Then, we derive firm 1’s objective function from𝑃𝑡1 , and final equation is then equaled to
zero to find 𝑃𝑡1 . Calculated 𝑃𝑡1 is placed in equation (25) and p matrix, which consists of
𝑃𝑡2 ،𝑃𝑡3 ،𝑃𝑡4 and𝑃𝑡5 , will be calculated.
c. Dynamic game with complete information with one price leader and four followers
that decide simultaneously considering that Rouzaneh and Mihan firms collude in
their pricing.
Again in this state, leader firm (Pegah) presents its product price to the market at the
beginning of each period, after that, followers price their products, after observing the
leader firm’s price. This state is similar to state 2 in pricing being done among
followers simultaneously, but we should enter the concept of collusion between
Rouzaneh and Mihan firms in this state:
For entering collusion concept to the model, we act as follow:
Like what is considered in theoretical discussions about players’ collusion concept,
we assume players, which collude together, as a single player and solve the problem
by this assumption. In this research if we assume Rouzaneh and Mihan firms have
colluded in their product pricing, we replace these two firms with a firm labeled
“Both”, which sale is the sum of Rouzaneh and Mihan frims’ sales and the target
sales or production anticipate is the sum of target sales of these two firms, and the
product’s actual price in Both firm is the average of products prices of Rouzaneh and
Mihan firms. In this state, leader firm (Pegah) presents its product price to the market,
after that, Pak, Kaleh and Both firms price their products after observing the leader
firm’s price. By using the Eviews software, 𝑞𝑡1 ،𝑞𝑡2 ،𝑞𝑡3 and 𝑞𝑡𝐵𝑜𝑡ℎ equations are
estimated:
1
3
4
𝑞𝑡1 = 0.537854𝑞𝑡−1
− 0.061102𝑞𝑡−1
−0.043991𝑞𝑡−1
− 0.002794𝑝𝑡1 + 0.001373𝑝𝑡2 + 0.001857𝑝𝑡3
4
+ 0.003781𝑝𝑡
(26)
1
2
3
𝑞𝑡2 = −0.101504𝑞𝑡−1
+ 0.511151𝑞𝑡−1
− 0.036003𝑞𝑡−1
− 0.053794+0.002759𝑝𝑡1 − 0.007649𝑝𝑡2
3
4
(27)
+ 0.003804𝑝𝑡 + 0.006324𝑝𝑡
1
2
3
4
𝑞𝑡3 = −0.067847𝑞𝑡−1
−0.090239𝑞𝑡−1
+0.479661𝑞𝑡−1
−0.067391𝑞𝑡−1
+ 0.003259𝑝𝑡1
3
+ 0.004016𝑝𝑡2 −0.008617𝑝𝑡 + 0.006312𝑝𝑡4
(28)
1
3
4
𝑞𝑡𝐵𝑜𝑡ℎ = −0.105538𝑞𝑡−1
−0.197690𝑞𝑡−1
+ 0.315380𝑞𝑡−1
+ 0.003952𝑝𝑡1 + 0.004155𝑝𝑡2
+ 0.005462𝑝𝑡3 −0.006849𝑝𝑡4
(29)
The continuation of solving this state’s problem is similar to state 2.
d. Dynamic game with complete information with one price leader and four followers
who collude in their product pricing.
Again in this state, leader firm (Pegah) presents its product price to the market at the
beginning of each period, after that, followers price their products, after observing the
leader firm’s price. This state is so similar to state 3, the only difference is that in this
state, all followers collude in their pricing.
For entering collusion concept to the model in this state, we act like state 3:
We assume players, which collude together, as a single player and solve the problem by
this assumption. In this research, if we assume that Pak, Kaleh, Rouzaneh and Mihan
firms have colluded in their cheese product pricing, we replace these four firms with a
firm called “All”, which sale is the sum of Pak, Kaleh, Rouzaneh and Mihan firms’ sales
and All’s anticipated production is the sum of Pak, Kaleh, Rouzaneh and Mihan firms’
anticipated productions and the target production or anticipated production is the sum of
target productions of Pak,Kaleh ,Rouzaneh and Mihan firms and the actual price of All’s
product is the average of cheese product prices of Pak,Kaleh,Rouzaneh and Mihan firms.
In this state, leader firm (Pegah) presents its product price to the market first, after that,
“All” sets its product price simultaneously after observing leader firm’s price. By using
the Eviews software, 𝑞𝑡1 and 𝑞𝑡𝐴𝑙𝑙 equations are estimated:
1
2
𝑞𝑡1 = 0.537830𝑞𝑡−1
−0.031071𝑞𝑡−1
− 0.002766𝑝𝑡1 + 0.006999𝑝𝑡2
(30)
1
2
𝑞𝑡𝐴𝑙𝑙 = −0.270770𝑞𝑡−1
+ 0.284101𝑞𝑡−1
+0.010162𝑝𝑡1 + 0.00711𝑝𝑡2
(31)
Price in tomans
The continuation of solving this state’s problem is similar to state 2.
5. Results of solving the model:
In figure 1, actual prices for UF cheeses of Pegah, Pak, Kaleh, Rouzaneh and Mihan
firms are compared in 17 work-periods.
5000000
4500000
4000000
3500000
3000000
2500000
2000000
1500000
1000000
500000
0
Pegah Actual Price
Pak Actual Price
Kaleh Actual Price
Rouzaneh Actual Price
Mihan Actual Price
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
time(semi-annual)
Fig 1. Comparison of the actual prices of five firms’ UF cheeses (Pegah, Pak, Kaleh, Rouzaneh and Mihan) in 17 workperiods.
Optimal (equilibrium) prices of each firm’s UF cheese are illustrated separately in figures
2,3,4,5,6 in a model with objective function (*) and its actual price, in four mentioned states.
Optimal (equilibrium) prices of each firm’s UF cheese are illustrated separately in figures 7,8,9,10,11 in a
model with objective function (**) and its actual price, in four mentioned states.
4500000
Pegah Optimal Price in Independent
Followers State
Pegah Optimal Price in Rouzaneh &
Mihan Collusion State
Pegah Optimal Price in a Nash Game
Price in tomans
4000000
3500000
3000000
2500000
Pegah Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
Pegah Actual Price
2000000
1500000
1000000
500000
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 2. Optimal (equilibrium) prices of Pegah firm in four mentioned states and comparing them with actual
prices of objective function (*).
4500000
Pak Optimal Price in Independent
Followers State
Pak Optimal Price in Rouzaneh &
Mihan Collusion State
Pak Optimal Price in a Nash Game
4000000
Price in tomans
3500000
3000000
2500000
Pak Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
Pak Actual Price
2000000
1500000
1000000
500000
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Price in tomans
Fig 3. Optimal (equilibrium) prices of Pak firm in four mentioned states and comparing them with actual prices
of objective function (*).
4500000
4000000
3500000
3000000
2500000
2000000
1500000
1000000
500000
0
Kaleh Optimal Price in Independent
Followers State
Kaleh Optimal Price in Rouzaneh & Mihan
Collusion State
Kaleh Optimal Price in a Nash Game
Kaleh Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
Kaleh Actual Price
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 4. Optimal (equilibrium) prices of Kaleh firm in four mentioned states and comparing them with actual
prices of objective function (*).
Price in tomans
5000000
Rouzaneh Optimal Price in Independent
Followers State
4000000
Rouzaneh Optimal Price in Rouzaneh &
Mihan Collusion State
3000000
Rouzaneh Optimal Price in a Nash Game
2000000
Rouzaneh Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
1000000
Rouzaneh Actual Price
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 5. Optimal (equilibrium) prices of Rouzaneh firm in four mentioned states and comparing them with actual
prices of objective function (*).
4500000
Mihan Optimal Price in Independent
Followers State
Mihan Optimal Price in Rouzaneh &
Mihan Collusion State
Mihan Optimal Price in a Nash Game
3500000
3000000
2500000
Mihan Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
Mihan Actual Price
2000000
1500000
1000000
500000
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Price in tomans
Fig 6. Optimal (equilibrium) prices of Mihan firm in four mentioned states and comparing them with actual
prices of objective function (*).
4500000
4000000
3500000
3000000
2500000
2000000
1500000
1000000
500000
0
Pegah Optimal Price in Independent
Followers State
Pegah Optimal Price in Rouzaneh &
Mihan Collusion State
Pegah Optimal Price in a Nash Game
Pegah Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
Pegah Actual Price
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 7. Optimal (equilibrium) prices of Pegah firm in four mentioned states and comparing them with actual
prices of objective function (**).
Price in tomans
Price in tomans
4000000
Pak Optimal Price in Independent
Followers State
Pak Optimal Price in Rouzaneh &
Mihan Collusion State
Pak Optimal Price in a Nash Game
4500000
4000000
3500000
3000000
2500000
2000000
1500000
1000000
500000
0
Pak Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
Pak Actual Price
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 8. Optimal (equilibrium) prices of Pak firm in four mentioned states and comparing them with actual prices
of objective function (**).
4500000
Kaleh Optimal Price in Independent
Followers State
Price in tomans
4000000
3500000
3000000
Kaleh Optimal Price in Rouzaneh &
Mihan Collusion State
2500000
Kaleh Optimal Price in a Nash Game
2000000
Kaleh Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
1500000
1000000
Kaleh Actual Price
500000
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 9. Optimal (equilibrium) prices of Kaleh firm in four mentioned states and comparing them with actual
prices of objective function (**).
5000000
Rouzaneh Optimal Price in
Independent Followers State
4500000
Price in tomans
4000000
Rouzaneh Optimal Price in
Rouzaneh & Mihan Collusion State
3500000
3000000
Rouzaneh Optimal Price in a Nash
Game
2500000
2000000
Rouzaneh Optimal Price in Pak,
Kaleh, Rouzaneh & Mihan Collusion
State
1500000
1000000
500000
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 10. Optimal (equilibrium) prices of Rouzaneh firm in four mentioned states and comparing them with actual
prices of objective function (**).
4500000
Mihan Optimal Price in Independent
Followers State
Mihan Optimal Price in Rouzaneh &
Mihan Collusion State
Mihan Optimal Price in a Nash Game
Price in tomans
4000000
3500000
3000000
2500000
Mihan Optimal Price in Pak, Kaleh,
Rouzaneh & Mihan Collusion State
Mihan Actual Price
2000000
1500000
1000000
500000
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time(semi-annual)
Fig 11. Optimal (equilibrium) prices of Mihan firm in four mentioned states and comparing them with actual
prices of objective function (**).
As shown in figures 2 to 11, optimal (equilibrium) prices of firms in each period are so alike in four
mentioned states. Also, all these prices are less than their actual prices, except for Pegah firm’s diagrams
that have a reverse trend from period 14. This issue is shown in figures 2 and 7.
Objective
Function(Ton^2)
6. Comparison of objective function quantities for four mentioned States:
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Objective Function of Pegah
in Nash State
Objective Function of Pak in
Nash State
Objective Function of Kaleh
in Nash State
Objective Function of
Rouzaneh in Nash State
Objective Function of Mihan
in Nash State
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Objective Function(Ton^2)
Fig 12. Comparison of objective function(*) quantities for five firms (Pegah, Pak, Kaleh, Rouzaneh and Mihan)
in state 1.
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
Objective Function of Pegah in
Independent Followers State
Objective Function of Pak in
Independent Followers State
Objective Function of Kaleh in
Independent Followers State
Objective Function of Rouzaneh in
Independent followers State
Objective Function of Mihan in
Independent Followers State
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Objective Function(Ton^2)
Fig 13. Comparison of objective function(*) quantities for five firms (Pegah, Pak, Kaleh, Rouzaneh and Mihan)
in state 2.
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Objective Function of Pegah in
Rouzaneh & Mihan Collusion State
Objective Function of Pak in
Rouzaneh & Mihan Collusion State
Objective Function of Kaleh in
Rouzaneh & Mihan Collusion State
Objective Function of Both in
Rouzaneh & Mihan Collusion State
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Fig 14. Comparison of objective function (*) quantities for Pegah, Pak , Kaleh and Both firms in state 3.
Objective Function(Ton^2)
0.45
Objective Function of Pegah
in All followers Collusion
State
0.4
0.35
Objective Function of All in All
followers Collusion State
0.3
0.25
0.2
0.15
0.1
0.05
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Fig 15. Comparison of objective function (*) quantities for Pegah and All firms in state 4.
Objective Function(Ton^2)
1,600,000
Objective Function of Pegah
in Nash State
Objective Function of Pak in
Nash State
Objective Function of Kaleh in
Nash State
Objective Function of
Rouzaneh in Nash State
Objective Function of Mihan
in Nash State
1,400,000
1,200,000
1,000,000
800,000
600,000
400,000
200,000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Fig 16. Comparison of objective function (**) quantities for five firms (Pegah, Pak, Kaleh, Rouzaneh and
Mihan) in state 1.
Objective Function(Ton^2)
1,600,000
Objective Function of Pegah in
Independent Followers State
Objective Function of Pak in
Independent Followers State
Objective Function of Kaleh in
Independent Followers State
Objective Function of Rouzaneh in
Independent followers State
Objective Function of Mihan in
Independent Followers State
1,400,000
1,200,000
1,000,000
800,000
600,000
400,000
200,000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Fig 17. Comparison of objective function (**) quantities for five firms (Pegah, Pak, Kaleh, Rouzaneh and
Mihan) in state 2.
Objective Function(Ton^2)
3,500,000
Objective Function of Pegah in
Rouzaneh & Mihan Collusion State
Objective Function of Pak in
Rouzaneh & Mihan Collusion State
Objective Function of Kaleh in
Rouzaneh & Mihan Collusion State
Objective Function of Both in
Rouzaneh & Mihan Collusion State
3,000,000
2,500,000
2,000,000
1,500,000
1,000,000
500,000
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Fig 18. Comparison of objective function (**) quantities for Pegah, Pak , Kaleh and Both firms in state 3.
Objective Function(Ton^2)
14,000,000
Objective Function of Pegah in All
followers Collusion State
Objective Function of All in All followers
Collusion State
12,000,000
10,000,000
8,000,000
6,000,000
4,000,000
2,000,000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Time(semi-annual)
Fig 19. Comparison of objective function (**) quantities for Pegah and All firms in state 4.
By comparing these diagrams, we realize that considering objective function (*) and pricing strategy in
state (1) or (2) gives the best result. It means that if players(production firms) price their products every
six months and set their pricing strategy in a manner that the leader firm (Pegah) presents its product
price to the market at the beginning of each period, after that, followers which are Pak, Kaleh, Rouzaneh
and Mihan firms either price their cheese products independently or all firms specify their prices and
present their production price to the market simultaneously after observing the leader firm’s price, the
difference between each firm’s sales in each work-period and each firm’s target sales in the same workperiod will be least. Diagrams show that the objective function (*) is much better than the objective
function (**). Therefore, pricing in each six-month period is more suitable than annual pricing where
price is assumed to be constant in each year.
Also, in both objective functions (*) and (**), objective function’s quantities in state (1) and (2) are less
than others; since the objective function is minimization, states (1) and (2) are the best choices as pricing
strategies for both objective functions (*) and (**) for all firms.
The differences between actual prices and calculates prices by Game theories, are reasonable due to these
reasons:
1. Considering disutility function of “minimizing the difference between each firm’s sales in each
period and the target sales of the same period” as the objective function of the model.
Since one of the most important goals of the firms in their products pricing, is increasing their
income and maximizing the firm’s benefit, considered firms in this research may have prices their
products based on the same objective function. Therefore, objective function in pricing models of
these firms may be utility function of “maximizing the firm’s benefit” instead of disutility
function of “minimizing the difference between each firm’s sales in each period and the target
sales of the same period”
2. Insensitivity of objective function to the sign of (qit – qii*t).
Production firms prefer their sales in period t , (qit) , to be bigger than their target sales in the
same period ,(qii*t), it means that : (qit – qii*t) > 0 ; Because firm i has sold its product more than
its expectation (target sales). But if (qit – qii*t) is less than zero, firm i prefers this quantity to be
minimized. Therefore, it’s necessary to use objective function min { (0, - (qit – qii*t) )} instead of
min(qit – qii*t) in order to enter the sign of (qit – qii*t) to the model’s objective function.
3. Inability in using Game rules in UF cheese competitive market.
Assumptions and rules of the Games theory may not be completely applicable to UF cheese
competitive market in practice and several factors of the strategic environment may substantially
affect the UF cheese price. For example, firms may not have freedom enough in pricing their
product and Games theories may not be completely applicable to their product pricing.
4. Linear sales equations assumption.
The assumption of linear sales equations may also cause differences between calculated prices by
Game theories and the actual prices.
5. Method for target sales calculation.
As explained in this article, target sales in each period are estimated by OLs method. Target sales
anticipation in each period by using time series in MINITAB software, is also another method for
estimating the target sale for each firm. Using time series and setting the most suitable model for
data for target sales anticipation may cause better results and closer estimated quantities to actual
ones.
7. Results
 Optimal prices of Pegah firm in models with objective function (*) or (**) are higher than other






firms, and in fact it hasn’t happened because of the government’s involvement in Pegah firm’s
pricing.
Optimal (equilibrium) prices of Pegah, Pak, Kaleh, Rouzaneh and Mihan firms are nearly the
same in states 1, 2, 3 and 4 in each period.
Actual prices of the followers (Pak, Kaleh , Rouzaneh and Mihan) are more than optimal
(equilibrium) prices in each period for all the states.
Actual prices of the leader firm (Pegah) are more than optimal (equilibrium) prices in each period
for all the states, and this trend is reversed from period 14; i.e: from period 14, actual prices are
less than optimal (equilibrium) prices in each period.
Pricing in each six-month period is more suitable than annual pricing, where price is assumed to
be constant in each year.
States 1 and 2 are the best choices as pricing strategies for both objective functions (*) and (**)
for all firms.
If players (production firms) price their products each six months and either follow the state 1
pricing strategy or all five firms present their products’ prices to the market simultaneously.
In this research, a model with objective function of minimizing the difference between the firm’s sale in
each period and their production in the same period was presented. Since one of the important goals of
production firms is maximizing their income level, it’s recommended to consider this model with two
objective functions: 1. minimizing the difference between the firms’ sales in each period and their
production in the same period, 2.maximizing the income level; solving the model and finding the optimal
(equilibrium) points are considered in 4 states. Also, it’s recommended to consider the model in the
following states too:




We face multi-leadership in production of a specific product or service.
There’s no constant leader in all periods, it means that the leader firm (the player that moves first)
may change in some periods.
Followers don’t make decision simultaneously, it means that the leader firm may move first,
follower 1 will decide and move after observing the leader’s move, then follower 2 will decide
and mover after observing follower 1’s move and …
In stackelberg game, model is designed in a way that two or more follower firms discuss and
collude randomly.
Resources
Carol Howard, Paul Herbig, “Japanese Pricing Strategy”, Journal of Consumer Marketing,
Vol 1, No 34, (1996), pp. 5-17.
Charles R.Duck, “Mathing Appropriate Pricing Strategy”, Journal of Product Management,
Vol 3, No 2, (1994), pp. 15-27.
Nagle, T. T., the Strategy and Tactics of Pricing, Prentice-Hall, Englewood Cliffs, NJ, 1987.
Carpenter, G. S., L. G. Cooper, D. M. Hanssens and D. F. Midgley, “Modeling Asymmetric
Competition," Marketing Sci., 7(1988), 393-412.
Wall Street Journal, "Chrysler Plans To Raise Prices By 3% to 5%,"July 28, 1992, A4.
Rabah Amir, Anna Stepanova, “Second-mover advantage and price leadership in Bertrand
duopoly”, Games and Economic Behavior 55 (2006), 121-141.
Abhik Roy, Dominique M. Hanssens, Jagmohan S. Raju, “Competitive Pricing by a Price
Leader,” Management Science, Vol. 40, No. 7, (Jul., 1994), pp. 809-823.