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MATRIX GAME EVALUATION IN GAME THEORY
Mamatova Zilolakhon Khabibullokhonovna
Fergana state university associate professor ,
pedagogy sciences according to philosophy Doctor of Philosophy (PhD)
E-mail:
mamatova.zilolakhon@gmail.com
ORCID ID
Habibjanov Behruz Bahodur Zoda
Fergana State University Practical mathematics direction
3rd year student, student of group 22-08
Email:
habibjonovbehruz0gmail.com
Abstract:
Article games theory matrix games to the department dedicated to be , to work
release and economy in the field conflicted processes mathematician modeling and solution
methods seeing It turns out . Matrix of games main concepts , including achievement
matrix , game lower and high prices , saddle point , maxmin and minimum strategies is
explained . If the game saddle to the point has if not , mix strategies method using solution
find process is described . Matrix games linear programming to the issue to bring methods
and this optimal strategies in the process determination algorithms The article is cited . two
example through theoretical concepts practical solutions with is strengthened . Used literature
list
the topic deeper study for additional sources presented Article
mathematician
programming and optimization in the field professionals , students and researchers for useful
is considered .
Keywords:
Games
theory , matrix games , conflict processes , achievement matrix ,
saddle point , maxmin strategy , minimax strategy , game price , mixed strategies , linear
programming , optimal strategy , mathematics modeling , economic issues , work release
Introduction .
Processes research and optimal management – decision acceptance to
do and systems to optimize scientific fields oriented .
1- Process research resources effective distribution for mathematician models , linear
programming , games theory and networks optimization such as from methods uses .
2-Optimal management systems the most good management strategies determination
with He is engaged in his work . main methods Pontryagin's Maximum principle and
Bellman's dynamic programming .
Research methodology
In the article cited research methodology analysis to do for his/her content and from
the structure come out , following main aspects seeing
Let's go out . Article games
theory matrix games to the department dedicated is a research methodology theoretical
analysis , mathematics modeling and practical examples through solution to find is based on .
Below methodology in detail analysis cited : Research general approach Theoretical basis :
Article games
theory main concepts ( achievement) matrix , saddle point , maxmin and
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minimax strategies , mixed strategies ) from explanation begins . These concepts conflicted
processes economy and working release in the field modeling for basis to be service does .
Mathematician Modeling : Research central methodology conflicted processes simplified
mathematician as models ( games )
to express is based on . In this process main factors
into account is taken , the second level factors and out of consideration aside is left . Practical
orientation : Article
theoretical concepts practical through examples (example 1 and
example 2) strengthen and help students this apply methods to real problems application
opportunity gives .
Literature analysis
In the article cited used literature list games theory , mathematics programming and
optimization in the field important sources cover takes . Below this of literature briefly
analysis brought by : Akulich I.
L.
Mathematical programming in in examples and
zadachax - M .: Vysshaya school , 1996. Analysis : This book mathematician
programming in the field practical issues and their to the solutions dedicated
games
theory and linear to program related examples and exercises own inside takes . Book students
and practitioners for comfortable theoretically
knowledge practical skills with
connects . In the article cited linear programming methods and matrix games
solution
algorithms this to the source based to be possible.Relevance : Article linear programming to
the issues to bring process in explanation important contribution adds.Badalov FB
Optimization theory and mathematician programming . “ Teacher ” , T. 1989. Analysis :
Uzbek published in
this book optimization theory and mathematician programming main
concepts own inside takes . Local students
for customized games
theory main
principles in explanation important source as service does . The book Uzbek
in the
language in the environment written of the article local to the context compatibility
provides.Relevance : In the article cited of the game price , strategies and their mathematician
modeling such as concepts this from the book taken to be possible . Kuznetsov A. V. ,
Novikova
G. I. , Kholod
N. I. Collection
task by mathematically speaking
programming . Minsk , Vysheishaya school , 1985. Analysis : This source mathematician
programming according to wide extensive issues set presented it will , that's it including
games theory and linear programming with related issues own inside takes . The set practical
orientation in the article cited examples and solutions to the structure Relevance : In the
article like example 1 and example 2 given practical issues this in the set to approaches based
to be probability high .
Analyses and results
Matrix games . Production
release and economy in the field many
practical
issues in solution conflicted processes ( situations ) come comes out . Conflicting processes
own inside very many factors takes . Many in cases the process study comfortable to be
for main factors into account take , second level factors into account unable to his/her
mathematician model We will make such conflict of the process shortened model game It
is called . Game clear one to the rule according to take will go .
Game
meaning from that consists of every one participant so one the solution
acceptance does that game at the end the most good to the result Let it be . Play . the result
( iskhod ) is this one how many functions value to him achievement function or payment
function If the players achievement amount to zero equal if , then to the game zero total
game It is called .
Any pair the game matrix in appearance expression possible
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=
mn
m
m
n
n
a
a
a
a
a
a
a
a
a
A
...
...
...
...
...
...
...
2
1
2
22
21
1
12
11
Let's say the matrix rows first player 's possible was A
1
, A
2
, …, A
m
marches ,
columns and second player's possible was B
1
, B
2
, …, B
m
their walks Let us define . A
matrix payment or achievement matrix is called . The matrix every one a
job
element first
Player A
to walk choose the second player B
j
to walk when you choose first player's
achievement ( second player's means "loss " .
The game purpose first the player maximum to success and second to minimize the
player's loss to achieve provision for the most acceptable strategy from choosing consists of .
If the first player some A
i
strategy choose , he never unless
ij
j
i
a
min
=
a
to success
achieves this . into account take this
player
his/her own the most less achievements
maximizer , i.e.
ij
j
i
a
min
max
=
a
PCB provider to walk chooses . Here
a
size of the game
lower price and to him/her suitable strategy maxmin It is called .
Second player , own in turn , his the most big possible was losses minimizing , that
is
ij
i
j
a
max
min
=
b
PCB provider to walk chooses . Here
b
size of the game high price and
to him/her suitable The strategy is called minimax .
If
a
=
b
if , that is
ij
j
i
ij
i
j
a
a
V
min
max
max
min
=
=
equality If it is done , then V of the
game price This condition is called of the satisfying matrix A a
job
to the element of the game
saddle point It is called .
So , matrix game saddle to the point has if so , its solution maxmin and minimax
methods with is found .
Example 1.
Given matrix game
for lower and high grades and Find the optimal
price of the game .
-
=
6
7
5
1
4
2
2
1
3
A
Matrix in the row the most small elements of the following consists of :
5
)
6
,
7
,
5
(
min
1
)
1
,
4
,
2
(
min
1
)
2
,1
,
3
(
min
=
-
=
-
=
j
j
j
So , the game lower price
5
)
5
,1
,1
(
max
)
min
(
max
=
-
=
=
i
ij
j
i
a
a
It will be . Now every one on the column the most big element we will find .
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
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Journal:
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page 1276
6
)
6
,1
,
2
(
min
7
)
5
,
4
,1
(
max
5
)
5
,
2
,
3
(
max
=
-
=
=
i
i
i
In that case , the game high price to the following equal will be .
5
)
6
,
7
,
5
(
)
max
(
min
=
-
=
=
j
ij
i
j
mix
a
b
This game lower and high grades mutual equal happened for optimal price of the game V=
b
=
a
=5 . This is the estimate . provider a
31
element game saddle point and A
3
and B
1
strategies optimal strategy will be .
If the achievements matrix saddle to the point has if not , then maxmin and minimax
methods with of the game solution found In this case of the game solution in finding mixture
strategies from the method is used .
First player 's mixture strategy , components following
=
=
=
m
i
i
i
m
i
x
x
1
,1
,
0
,1
the conditions to the vector X =( x
1
, x
2
, … , x
m
) satisfying It is said . In this every one x
i
first Player A
walk choice probability indicates .
Second player 's mixture strategy , components following
=
=
=
n
j
j
i
n
j
y
y
1
,1
,
0
,1
the conditions satisfactory Y =( y
1
, y
2
, … , y
n
) to the vector It is said . In this every one y
j
second player's B
j
walk choice probability indicates .
Mix strategies in the way first Player A
walk choose the second
player
B
j
walk when you choose first player's achievement as his/her of winning
mathematician expectation is taken , that is, it is taken to equal will be
=
=
=
m
i
n
j
j
i
ij
y
x
a
y
x
V
1
)
,
(
Function V( x,y ) payment or achievement function It is called .
If the first player his/her own
)
...,
,
,
(
*
*
2
*
1
*
m
x
x
x
X
=
optimal strategy if it is used ,
then second player how strategy from choosing strict look , its achievement from the game's
V rating less it won't happen , that is
=
=
m
i
i
ij
n
j
V
x
a
1
*
,1
,
.
Same also , if the second player his/her own
)
...,
,
,
(
*
*
2
*
1
*
m
y
y
y
Y
=
optimal strategy if it is
used , then first player how strategy from choosing strict look , its loser from the game's V
rating does not exceed , that is
=
=
n
j
j
ij
m
i
V
y
a
1
*
,1
,
.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
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page 1277
Matrix the game linear programming to the issue to bring
Matrix the game linear programming to the issue to bring process seeing We will go
out . The most before optimal player mix strategy and of the game price We find . Its for
inequalities system and conditions To summarize , the following in appearance we write :
+
+
+
+
+
+
+
+
+
V
x
a
x
a
x
a
V
x
a
x
a
x
a
V
x
a
x
a
x
a
m
mn
n
n
m
m
m
m
...
......
..........
..........
..........
..........
...
...
2
2
1
1
2
2
22
1
12
1
2
21
1
11
1
...
2
1
=
+
+
+
m
x
x
x
)
,1
(
0
1
m
i
x
=
Game price what Considering the equation , (
V
0
>
) of the system everyone
inequalities to become below system harvest we do :
+
+
+
+
+
+
+
+
+
1
...
..
..........
..........
..........
..........
1
...
1
...
2
2
1
1
2
2
21
1
12
1
2
21
1
11
m
mn
n
n
m
m
m
m
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
V
t
t
t
m
1
...
2
1
=
+
+
+
here
V
x
t
1
1
=
The first player tries to maximize his payoff, i.e. the value of the game. So, for the
first player,
V
t
t
t
m
1
...
2
1
=
+
+
+
the most small (minimum) value acceptance to do This is necessary . in requirements system
following linear programming to the issue turns into :
+
+
+
+
+
+
+
+
+
1
...
..
..........
..........
..........
..........
1
...
1
...
2
2
1
1
2
2
21
1
12
1
2
21
1
11
m
mn
n
n
m
m
m
m
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
t
a
0
,...,
0
,
0
2
1
m
t
t
t
min
...
2
1
®
+
+
+
=
m
t
t
t
Z
In a similar way, to find the optimal mixed strategy of the second player and the cost
of the game, the following linear programming problem must be solved.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
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page 1278
+
+
+
+
+
+
+
+
+
V
y
a
y
a
y
a
V
y
a
y
a
y
a
V
y
a
y
a
y
a
n
mn
m
m
n
n
n
n
...
......
..........
..........
..........
..........
...
...
2
2
1
1
2
2
22
1
21
1
2
12
1
11
0
,...,
0
,
0
2
1
n
y
y
y
max
...
1
2
1
®
+
+
+
=
=
n
y
y
y
V
F
Problems ( 7.15)-( 7.17) and (7.18-7.20) mutual hesitant linear programming from
issues consists of will be . Of them optional one undressing , both of them solution easily find
possible .
Example 2
=
4
3
6
5
5
3
2
3
5
A
matrix the game mixture in strategies Find the solution .
Solution . First player for the game linear programming to the issue We will turn it
around . for the most before following the system harvest we will do .
+
+
+
+
+
+
V
x
x
x
V
x
x
x
V
x
x
x
3
2
1
3
2
1
3
2
1
4
5
2
3
5
3
6
3
5
1
3
2
1
=
+
+
x
x
x
0
,
0
,
0
3
2
1
x
x
x
( 7. 21) We divide both sides of each inequality in the system
0
>
V
by ( ) and
V
x
t
1
1
=
introduce the notation to form the following system:
+
+
+
+
+
+
1
4
5
2
1
3
5
3
1
6
3
5
3
2
1
3
2
1
3
2
1
t
t
t
t
t
t
t
t
t
V
t
t
t
1
3
2
1
=
+
+
0
,
0
,
0
3
2
1
t
t
t
This system can be written as the following linear programming problem:
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ISSN: 2692-5206, Impact Factor: 12,23
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Journal:
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page 1279
+
+
+
+
+
+
1
4
3
6
1
5
5
3
1
2
3
5
3
2
1
3
2
1
3
2
1
u
u
u
u
u
u
u
u
u
0
,
0
,
0
3
2
1
u
u
u
max
1
3
2
1
®
+
+
=
=
u
u
u
V
F
The given matrix game for the second player becomes the following linear programming
problem.
+
+
+
+
+
+
1
4
3
6
1
5
5
3
1
2
3
5
3
2
1
3
2
1
3
2
1
u
u
u
u
u
u
u
u
u
0
,
0
,
0
3
2
1
u
u
u
max
1
3
2
1
®
+
+
=
=
u
u
u
V
F
Issues each other hesitant are issues . Therefore for from them optional one take off ,
the other one solution easily find possible .
Conclusion
Article games
theory matrix games department analysis to do dedicated is
conflicting
processes economy and working release in the field mathematician modeling
and solution methods illuminates . Research methodology theoretical analysis , mathematics
modeling , linear programming and practical to examples is based on . The game lower and
high prices , saddle point , maxmin and minimax strategies , as well as mixed strategies such
as main concepts clear explained . Matrix games linear programming to the issue optimal
strategies are presented and of the game price is determined . Practical examples through
theoretical knowledge is strengthened . References list
local and international sources
comprehensive , scientific the basis strengthens , but modern technological approaches
absence restriction as record Methodology
article to their goals complete
suitable
comes , systematic and practical solutions presented However , empirically information
and software tools input through the research further enrichment possible Overall , the
article
games
theory economic to issues in use important guidance students ,
researchers and experts for useful source is considered .
Literature:
1. Akulich IL Mathematical programming in examples and problems. - M.: Higher school,
1996.
2. Badalov FB Optimallash Nazarius and mat or matic dasturlash . “ O ' q ituvchi ”, Vol.
3. Kuznetsov A.V., Novikova G.I., Kholod N.I. Collection of problems in mathematical
programming. Minsk, Higher School, 1985.
4. Kuritsky B.Ya. Search for optimal solutions using Excel . “Saint Petersburg ”, 1997.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 1280
5. Safaeva K., Beknazarova N. Operatsiyalarni tekshirishning matеmatik usullari.
“O'qituvchi”, 1984y. 1 qism.
6. Lesin V.V., Lisovets Yu.P. Fundamentals of optimization methods. Moscow, MAI
Publishing House, 1998.
7. Khazanova L.E. Mathematical modeling in economics . M. BEK, 1998.
