Download SOFTWARE IMPLEMENTATION OF THE MODEL OF GAME

SOFTWARE IMPLEMENTATION OF THE MODEL OF GAME THEORY
IN MARKETING DECISIONS
V. Stojanović1, M. Božinović2, N. Petković3
1
Faculty of Sciences and Mathematics, 2Faculty of Economics, Kosovska Mitrovica, Serbia,
3
Faculty of Management, Zaječar, Serbia.
Abstract: In this paper it considers an application of game theory, especially a model of antagonistic game,
which we applied in some cases of marketing decions, or, in general case, in conflicts that might arise in the
market. A mathematical model which we considered is closely related to model of linear programming, theory of
probability and statistics. In addition, for the model described in this paper we was done the original software’s
solution, i.e. the procedure implemented in statistical programming language „R”, that simulates given model.
1. INTRODUCTION
Mathematical modeling as a process is increasingly spreading to different areas, both
natural and social sciences, especially economics. The reason for this is certainly a very
turbulent movements, crises and conflicts that occur in the various forms of the local economy
at the global level. Contemporary problems in the financial market, especially the highintensity conflicts that are still very valid, are good motivation for an exact, mathematical way
to solve very difficult and intricate problems of the economy at the present, when the crisis
spread to most developed world economies. That is why modern marketing has developed
methods and models of advertising products, but they certainly can be improved by taking
into account first of all the possible reaction of market competition. It is well known that
consumers their decision to purchase a particular product often made under very aggressive,
sometimes irritating influence of various media, especially television, sometimes because of
their persistence and aggression consumers may be caused by a counter. Therefore, the
problem of marketing advertising management as well as advertising their products should
pay particular attention to the way they have been presented. Zaber products mainly optimizes
the cost of advertisement with the expected effects which should produce among consumers,
while, on the other hand, must be taken into all of the activities of competitors. Empirical fact
that the competition will certainly react as the advertising campaign of a product is affected
by the demand for the same or similar products with the competing companies, which,
ultimately, creates a conflict situation in the market. The final outcome of the conflict depends
primarily on the selection and combination of strategies that are positioned two antagonistic
sides chosen. The choice of inappropriate strategies and, accordingly, the wrong business
decisions in such circumstances certainly result in a decrease in demand, and on the other side
and loss of market position that a company has.
Game theory in operations research is widely used, especially in the optimization of
various conflict situations (see for example [3], [6]) and is thus closely related to decision
theory. Mathematical models created in game theory allows the analysis of different situations
in which the outcome of the game by two or more players do not depend only on one of them,
but the expected reactions of other participants in the game. So if we are to compare this with
the decision-making managers, it is clear that their decisions are clearly interdependent. From
the perspective of game theory goal of every player is a timely response to action opponents
in order to achieve a better result. Fundamental problem is finding the appropriate criteria for
the selection of the optimal strategy. In this case it is necessary to make a distinction between
pure and mixed strategies. Specifically, in a pure strategy players for the each situation of the
game G choose exactly one of the options that are available to them. However, in the mixed
strategy choice of players is not identical in every situation, which in turn means that the
mixed strategy consists of the shares of the different choices. Depending on the media in
which two competing firms advertise, they certainly have access to a number of strategies.
Payments in the model of matrix game G is created based on the estimated effects of all
possible combinations of strategies in relation to percentage change in the market share of one
of the players. The main feature of which will be further considered in the simulation consists
of a percentage of participation, where we assume that the random variables that they
represent a normal distribution whose parameters have to be estimated on the basis of relevant
data, while generating appropriate numerical values can be realized with appropriate software
package on the computer.
Since the matrix form of payment in the game G whose elements are simulated values of
percentage market share of each of them, the model can be transformed into a linear
programming model, and as such an appropriate method to solve linear programming (LP).
After the implementation of appropriate procedures, i.e. simulations and solving LP- problem,
the next step consists in forming the frequency distribution of appropriate strategies, and then
for each of the arithmetic mean of the distribution is calculated. Finally, in this way the output
from a share strategies that provide the greatest gains in the set game G. We will continue to
deal with these issues in order to develop an original model based on game theory that can be
applied to business decision-making in marketing and in this way, at least, and management.
For the application of the model developed the original software as logistical support to
operationalize and greatly facilitates its implementation.
2. MODEL CONSTRUCTION
Mathematical tools that we need to describe the decision problems in marketing, using
game theory applies primarily to concepts of non-coalitional, antagonistic, matrix game of a
zero sum. More precisely, these concepts we can describe formally as it is follows:
Definition 1. Non-coalitional game is the system G  I , Si iI ,  f i iI , where I = {1, 2,...,n}
represents the set of players, Si, i ∈ I are the sets of strategies of i-th player, and
f i :  Si  R
(1)
iI
are, so called, the gains functions, i.e. the limited functions which shows “the profit” of i-th
player in some situation x  iI S i .
Definition 2. Non-coalitional game G, defined in (1), is a finite game if all sets of strategies Si
are finite. Finite non-coalitional game is a bi-matrix game if I = {1, 2}, i.e. there are only two
players in the game G.
Definition 3. Non-coalitional game G, defined in (1), is a game of a zero sum if for any
situation x  iI S i it is valid
f
iI
i
 0.
Non-coalitional game with a zero sum and two players is called an antagonistic game.
Definition 4. Antagonistic bi-matrix game is called a matrix game.
Therefore, in a matrix game, two players in the conflict situation have access to a
finite number of different strategies whose choice of different events occurring in the game. In
addition, the gain of one player is equivalent to the loss of another, and collectively, the
"gain" of both players is zero. All strategies, as well as the individual gains of one, or the loss
of another player, enters the game matrix payments, which generally takes the form
 a11
a
A   21
 ...

a m1
a12
a 22
...
am2
a1n 
... a 2 n 
.
... ... 

... a mn 
...
Here, each horizontal line represents a choice of one of the m possible strategies of the first
player, and the vertical line to select one of n strategies of the other player. Finally, our
payment matrix elements aij, i = 1, . . . ,m; j = 1, . . . , n, are the value of the gains of the first
player, or losses of the other player in the selection of appropriate strategies, i.e. in certain
game situations. In the following, we will notice the matrix game G with matrix payments A
as GA. The basic principle of optimization in matrix games, as well as no matter where
antagonistic game, is the principle that consists in the realization of the so-called, situation of
equilibrium. This principle requires that in the game GA choose it i * row and j * column of
matrix A  aij mn so it is valid
 
aij  ai  j   ai j .
This, in fact, means that the first player, by choosing the optimal strategy i* provides certain,
guaranteed gain ai  j  , no matter which strategy j choice his opponent. Similarly, the second
player, assuming the rational behavior, choosing an optimal strategy j* whose selection
provides maximum value of loss ai  j  . In this way, i* and j* are the optimal pure strategy in
the game GA, while ai  j  is the saddle point, i.e. the optimal value of the game GA, denoted as
    G A   ai j .
 
In a game where there is a unique optimal strategies both of players can not of improve their
score by changing strategies, and such a game in a state of equilibrium. For these games then
we say than have a clean (optimal) strategies, and thus, matrix game completely resolved.
2.1 Matrix games with mixed strategies
If the game matrix A there is no saddle point, players have not the choice of optimal
(pure) strategies that enable a single set limit profit or loss. Determining the optimal strategies
of players then are based on introduction the random elements in the game, which is reflected
in the formation of a series of probability with which each of them chooses a particular
strategy. So, they do not choose just one strategy, but for more of them, with a certain
probability, and such choice of two or more pure strategies is called a mixed strategy.
Suppose that the first player, who can have m strategies, each of which can be selected with
probabilities p1, . . . , pm, and we will denote the appropriate probabilities of selection one of n
strategies of other player with q1, . . . , qn. These values, of course, satisfy the non-negativity
conditions pi ≥ 0, qj ≥ 0, as well as the conditions
m
n
 p  q
i 1
i
j 1
j
 1,
because both of players will most sure choose at least one of the offered strategies.
Probabilities of all possible choices of the strategies above are representing the mixed
strategies of given matrix game. The optimal mixed strategies in zero-sum games between
two players can be determined in several ways. However, one of the lengthiest ways of
solving these games was to express them as a linear programming problem, which will also
create the opportunity for a software implementation of the model. For this purpose, we
introduce the following terms:
Definition 5. Matrix games GA and GB, with matrices A and B of the same size m×n, called
affine-equivalent if there exists k > 0 and τ ∈ R, such that
B = k·A + τ·1 m×n,
where 1 m×n is matrix of the same size as the matrices A and B, such that all theirs elements are
equal to one.
It is easily to verifying that affine-equivalent games have the same set of optimal
strategies, where
υ(GA) = k·υ(GB) + τ.
In the special case when k = 1, the games GA and GB we will call an equivalent games. Then,
the general procedure to finding optimal mixed strategy is based on the direct application of
the LP principles. In addition, based on the above concepts, the arbitrary matrix game GA,
specifics with a game matrix A = [aij ]m×n, we can assume that min aij   0 . Otherwise, if the
condition of non-negativity is not fulfilled, we will solve equivalent game with matrix
A′ = A + τ ·1 m×n,
where τ =|min(aij)|.
Now, we can formulate a LP model for the first player, where the coordinates of the vector
p   p1 ,, pm  are probabilities that the first player chooses its strategies i=1,...,m. Let us
T
introduce the vector
x  x1 ,, x m  
T
1
 p,
 p 
where

m
m
m

i 1
i 1

 p   min   pi ai1 ,  pi ai 2 ,,  pi ain  .
 i 1
It can be shown (see, for instance [1], [2], [5], [7]) that the optimal value of the game is
   max  p , and that is to find the optimal mixed strategy p    p1 ,, p m  equivalent to
T
solving the following LP problem:
Find the optimal, minimum value of the function
m
min F1 x    xi
i 1

1

,
as per system of constrains AT  x  1 n1 .
Similarly, we can construct an appropriate LP model of the other player. To this end,
T
denote with q  q1 ,, qn  the probabilities of choice some of its strategies j=1,...,n, and
y   y1 ,, y n  
T
1
 q,
 q 
where

n
n
n


j 1
j 1
j 1

 q   max   q j a1 j ,  q j a2 j ,,  q j amj  .
Based on such a set of conditions, we have the problem of the maximum:
Find the optimum, the maximum value of the function
n
max F2 y    y j
j 1

1

,
as per system of constrains A  y  1nm1 .
Obviously, this is a LP problem which is dual to the problem of minimum described
above. After determine the system of vector (x∗, y∗), that they represent the required optimum
solutions of the LP problem, the optimal mixed strategy for the first and second players are,
respectively,
p∗ = υ∗ · x∗, q∗ = υ∗ · y∗,
where
 
1
1

.

F1 x
F2 y 
 
 
On this way, the problem of finding optimal mixed strategy of an arbitrary matrix game is
entirely resolved.
2.2 The parameters of market share
Immediately prior to application of the model of the game theory in marketing
decision, it is necessary to designate the percentage value of the market share, which
represents the elements of the matrix payment of the game GA. These values, in general,
cannot be safely and reliably. For this reason, they are usually defined as random variables
that are at least approximately normally distributed. Density function in this case is given by
f x  
1
 2
e

 x   2
2 2
, xR ,
where   R i   0 are the parameters of normal distribution which need to be determined.
Specifically, management assesses their values, thus narrowing the choice of some strategy
which is not optimal. In order to achieve this, it is important to take into account all the
necessary information on which to assess changes in the market share of the company. In this
way, they get the rated values of the parameters  ij  R and  ij  0 of normally distributed
random variables a ij . They suit to each particular situation in matrix game GA with the matrix
of payment A = [aij ]m×n, and we can write

aij ~ N ij ,  ij2

 aij ~ ij   ij .
In the next step, using a software application that simulates the decision-making
model, we can access to LP program, which is implemented as a series of T successive
simulations, where, for example T = 100, 150, 200, ... Each simulation individually, i.e. all
realized values of the variables aij represents the probability of changes in market share
expressed as a percentage, which appears as a last combination of appropriate strategies first
and second players in the game GA. By solving this model, we come to the optimal strategies,
determined on the basis of a given set of simulations. Obviously, obtained strategies are
somewhat various and that enabling the formation of their frequency distribution. After that,
the optimal strategies of the appropriate model are determined as the means of the established
distribution.
3. SOFTWARE IMPLEMENTATION AND APPLICATION OF THE MODEL
As an illustration, consider two companies, for example RODA and IDEA, which
share a particular segment of the consumer goods market. To increase market share, we
assume that the management of both companies launch campaign to advertising theirs
products. Given that the total sales of limited purchasing power of consumers, it is the
demand of one of the company can only increase the downloading of the market share held by
the competitor, so it is essential to choose the right advertising strategy. It can further be
assumed that the managements of both companies start advertising campaigns in each of the
three possible media: newspapers, radio and television. In this way, we can form a matrix
payment format 3 x 3, which includes the estimated value of all the effects, ie. all possible
combinations of strategies with percentage change in the market share of the company, as
well as the estimated value of the standard deviation of market effects. Suppose that the
matrix payment company RODA as the first, the underlying IDEA as the other player,
outlining the estimated value of the expectation and standard deviation of the percentage
change in its market share, as follows:
2,5  0,3  1,5  0,2
 0,5  0,1

A ~  1,5  0,2  0,5  0,1 0,5  0,1 .
 0,5  0,1 0,5  0,1 1,5  0,2 
Thus, the elements of the payment matrix A are obtained in a random way, as the
realization of a normally distributed random variables, described above. In addition, business
strategies of company RODA are arranged as a rows, and a strategies of other company IDEA
by columns of matrix A. For both companies the appropriate strategies are their decision to
advertise their products, respectively, in the press (the first rows and columns), followed by
radio (the other rows and columns), and at the end of the TV (the third rows and columns).
That, precisely, it means that the value a11 ~ 0,5% ± 0,1% is the percentage of increase in the
market share of the first player, i.e. RODA company, in relation to the firm IDEA, as other
player, if both choose their strategies as advertising in newspapers. Similarly, the next value
a12 ~ 2,5% ± 0,3% shows the percentage of the market share of company RODA if it chooses
advertising in newspapers, and competitive company advertising on the radio, etc.
In this way, we have created the necessary conditions for the application of the above
procedure to find the optimal strategies of the two opposing companies. Formally, this
procedure can be shown by the algorithm which consists of the following iterative steps:
Step 1: For k = 1,2,...,T repeat the following steps.
 
Step 2: Compute the k-th realization of the game’s matrix A  aij




aij ~ N ij ,  ij2 , and denote such realization as A[k]  aij k  mn .
Step 3: If   min aij k   0 , then A[k]← A[k] + |τ|.
mn
, where
i, j
Step 4: Find the minimum of objective function F1 x under linear constrains
A[k ]  x  1 n1 . Denote the solution of this minimization problem as x  [k ] and obtained
T
 
minimum as F1 x   F1 k .
Step 5: Similarly as above, solve a dual problem, i.e. find the maximum of function
F2 y  under constrains A[k ]  y  1m1 and denote this solution as y  [k ] .
Step 6: Compute p∗[k] = υ∗[k] · x∗[k] and q∗[k] = υ∗[k] · y∗[k], where   k   F1 k  .
1
Step 7: Next k.
Step 8: Find the empirical distributions of vector sequences p∗[k], q∗[k], and numeric
sequence υ∗[k]. Take theirs means as estimates of optimal strategies of the both players and
the game's value, respectively.
Software’s implementation of the algorithm above is realized by the original authors’
code, i.e. the procedure written in statistical programming language „R“. For this purpose, we
used a random numbers generators algorithm to generate the members of matrix sequence
A[k], as well as the Nelder-Mead’s method of constrained optimization, also implemented in
„R“. Using these procedures in our model, after T = 100 simulations of normal distributed
random variables aij, we obtained the estimated values of optimal strategies p* and q*, as well
as estimates of the game’s value υ∗.
Figure 1. Empirical densities of the optimal strategies.
In the next step, we examined their stochastic properties, i.e. we funded the empirical
distributions of realizations of p* and q*, interpreted as realizations of some random variables
with the appropriate (unknown) distribution. The empirical distributions (histograms with the
appropriate empirical densities functions) of the obtained estimates are shown in Fig.1. As we
can see, the estimates of the first player’s strategies (company RODA) show some variability,
in dependence on the realizated values of normally distributed random variables aij in matrix
A[k]. On the other hand, it is obviously that estimated values of the second player’s optimal
strategies (company IDEA) are uniformly distributed, i.e. all of them have the same, equal
“probability of success.”
Statistics
Min.
1st Qu.
Median
3rd Qu.
Max
Mean
St.Dev.
Table 1. Summary statistics of the optimal strategies and the game’s value.
Optimal strategies: 1. PLAYER
Game’s
Optimal strategies: 2. PLAYER
value
P1*
P2*
P3*
q1*
q2*
q3*
0.1580
0.1207
0.2768
0.1112
0.3333
0.3333
0.3333
0.4126
0.1740
0.3779
0.3158
0.3333
0.3333
0.3333
0.4699
0.2044
0.4106
0.3804
0.3333
0.3333
0.3333
0.5244
0.2340
0.4692
0.4347
0.3333
0.3333
0.3333
0.6867
0.3429
0.6687
0.5499
0.3333
0.3333
0.3333
0.4600
0.2067
0.4290
0.3643
0.3333
0.3333
0.3333
0.1058
4.23E-02
8.59E-02
8.93E-02
1.51E-08
1.91E-08
1.28E-08
Similar conclusion we can get from the summary statistics which are shown in Table 1.
Here, for the optimal strategies sequences p*[k], q*[k], as well as the sequence υ∗[k], we
computed some of the “usual” statistics: minimums, maximums, median, quartiles, means and
standard deviations. Especially, we took the means of these series as estimated, optimal
values, i.e. percentage amount of funds that each company should invest in proper form
advertisements. The optimal strategy of the firm RODA can be expressed as vector
p*=(0.2067, 0.4290, 0.3643)T
which components represent the percentage amount of funds that the company needs to invest,
respectively, in advertising in newspapers, on radio and television. On the other hand, the
optimal strategy of the other player, companies IDEA, as a solution of the dual problem is
q*=(0.3333, 0.3333, 0.3333)T
So, this company with equal probability should be chosen by any pure strategy, i.e. with the
same means and the same advertising investments in each of these media. Finally, in the
realized model obtained the average optimal value games υ* = 0.4600. This means that the
company RODA, in the "ideal" case, obtained 0.46% of the market held by its rival. In
contrast, for the other company IDEA the same obtained value of game υ* is the minimum
value of the loss of market, expressed as percentages.
LITERATURE
[1] Binmore K. Game Theory-A Very Short Introduction. Oxford University Press Inc, New
York. 2007.
[2] Božinović M., Stojanović V. Mathematical Methods and Models in Economics of
Companies. VEŠ, Leposavić. 2005.
[3] Božinović M., Stojanović V. The Appliance of the Games Theory in Solution of the
Ecological Conflict. Proceeding by Conference SYM-OP-IS 2007. P.27-30.
[4] Božinović M.. Researsh Operation, Faculty of Economics, Kosovska Mitrovica. 2012.
[5] Carmichael F. A Guide to Game Theory. Pearson Education Limited. Harlow. 2005.
[6] Jones A. J. Games Theory-Mathematical Models of Conflict. Mathematics and its
Application. Chichester, E. Horwood. 1980.
[7] Воробьев, Н. Основы теории игр. Бескоалиционные игр. Наука, Москва. 1984.