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MATRIX GAMES-DOMINATION
Mamatova Zilolaxon Khabibullokhonovna
Associate Professor, Fergana State University Doctor of Philosophy
(PhD) in Pedagogical Sciences
Email:
mamatova.zilolakhon@gmail.com
Qakhramonova Mukhlisa Jumahoja kizi
Student at Fergana State University
Email:
qahramomovakmalxuja@gmail.com
Annotatsiya:
Ushbu maqolada o‘yinlar nazariyasining asosiy tamoyillari, xususan, ikki shaxsli
nol yig‘indi o‘yinlar, bluff strategiyalari va ko‘p bosqichli o‘yinlar tahlil qilinadi. Bellman,
Blackwell, Dresher, Ferguson, Kuhn va Nash kabi yetakchi olimlarning ishlari asosida strategik
qaror qabul qilishning nazariy va amaliy jihatlari ko‘rib chiqiladi. Optimal strategiyalarni
aniqlashda ehtimollik, muvozanat (Gleichgewichtspunkt) va dominatsiya kontseptsiyalariga
alohida e’tibor qaratilgan. Tadqiqotlar raqobatbardosh va kooperativ o‘yinlarda optimal
natijalarga erishish usullarini ochib beradi.
Kalit so‘zlar:
o‘yinlar nazariyasi, nol yig‘indi o‘yinlar, bluff strategiyasi, muvozanat nuqtasi,
dominatsiya, ko‘p bosqichli o‘yinlar, strategik qaror qabul qilish.
Annotation:
This article analyzes the fundamental principles of game theory, focusing
particularly on two-person zero-sum games, bluff strategies, and multi-stage games. Based on
the works of leading scholars such as Bellman, Blackwell, Dresher, Ferguson, Kuhn, and Nash,
the theoretical and practical aspects of strategic decision-making are examined. Special
attention is given to concepts such as probability, equilibrium points (Gleichgewichtspunkt),
and domination in identifying optimal strategies. The research reveals methods for achieving
optimal outcomes in competitive and cooperative games.
Keywords:
game theory, zero-sum games, bluff strategy, equilibrium point, domination, multi-
stage games, strategic decision-making.
Аннотация:
В данной статье анализируются основные принципы теории игр, в частности,
двухличные нулевые игры, стратегии блефа и многоэтапные игры. На основе работ
ведущих ученых, таких как Беллман, Блэквелл, Дрешер, Фергюсон, Кун и Нэш,
рассматриваются теоретические и практические аспекты принятия стратегических
решений. Особое внимание уделяется таким концепциям, как вероятность, точка
равновесия (Gleichgewichtspunkt) и доминирование при определении оптимальных
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стратегий. Исследование раскрывает методы достижения оптимальных результатов в
конкурентных и кооперативных играх.
Ключевые слова:
теория игр, нулевые игры, стратегия блефа, точка равновесия,
доминирование, многоэтапные игры, принятие стратегических решений.
A finite two-player zero-sum game in strategic form
( , , )
X Y L
, sometimes called a matrix game
because the payoff function L can be represented by a matrix. If
{
}
1
,...,
m
X
x
x
=
and
{
}
1
,..., y
n
Y
y
=
then the game matrix or payoff matrix is expressed as:
11
1
1
n
m
mn
a
a
A
a
a
=
L
M O
M
L
, here
(x y )
ij
i j
a
L
=
,
In this form, player 1 chooses a row, player 2 chooses a column, and player 2 pays player 1 the
value in the selected row and column. Note: the values in the matrix are considered as the
winnings of the player who chose from the row and the losings of the player who chose from
the column.
Mixed strategy for player 1
1
2
(p ,p ,...,p )
m
p
=
1in the form of, probabilities that sum to 1is
represented as a set of . If player 1uses a mixed strategy and player 2If he chooses the column,
then the (average) payoff for player 1 will be:
1
m
i ij
i
p a
=
Also, the mixed strategy of player 2
1
2
(q
)
,q ,...,q
n
q
=
2will look like this. If player 2strategy
and player 1If he chooses the -row, the winnings for player 1 are:
1
n
ij j
j
a q
=
More generally, if player 1strategy and 2nd playerstrategy, then the (average) payoff for player
1 is:
1
1
m
n
T
i ij j
i
j
p Aq
p a q
=
=
=
It is also worth mentioning that Player 1's pure strategy, i.e., only
i
-row selection,
i
e
unit
vector (only
i
-position and 0s in the remaining positions). Similarly, player 2's
j
-column
selection
i
e
is represented by a unit vector.
In the following sections, we will try to "solve" games, that is, find the value and determine at
least one optimal strategy for each player. Sometimes we are also interested in determining all
optimal strategies.
Balance points.
Sometimes the game is easy to solve. If the matrix
ij
a
if it has the following
properties:
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1.
ij
a
-
i
is the smallest element of the array, and
2.
ij
a
-
j
-if the largest element in the column is
Then to us
ij
a
We call it the equilibrium point. If
ij
a
if the equilibrium point is 1,then player 1
i
-select row at least
ij
a
can win the value and player 2
j
choose a column and lose
ij
a
may not
exceed the value of. Thus,
ij
a
The game will be worth it .
Example 1.
4 1
3
3 2 5
0 1 6
A
-
=
In this matrix, the middle element 2 is the equilibrium point, since it is the minimum in its row
and the maximum in its column. Therefore, it is optimal for player 1 to choose from the second
row and for player 2 to choose from the second column. The cost of the game is 2, and the
vector (0,1,0) is the optimal mixed strategy for both players.
Large size
m n
For matrices, checking each element can be tedious. So, as an easier way:
calculate the minimum of each row and the maximum of each column and check if there is a
match between them.
3 2 1 0
0 1 2 0
1 0 2 1
3 1 2 2
A
=
and
3 1 1 0
0 1 2 0
1 0 2 1
3 2 2 2
B
=
For matrix A:
Row minimums: 0,0,0,1
Column maximums:3,2,2,2
For matrix B:
Row minimums: 0,0,0,1
Column maximums: 3,1,2,2
In matrix A, no row minimum is equal to the column maximum, so there is no equilibrium
point. However, if in matrix A
12
a
f the element is 1 instead of 2, we get matrix B. In this case,
the minimum of the fourth row will be equal to the maximum of the second column. So,
42
b
The
inflection point is
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Solution for 2×2 games.
Let's generalize Let's consider a 2 × 2 game matrix.
a b
A
d c
=
To solve this game (i.e., find the value of the game and at least one optimal strategy for each
player), we proceed as follows:
1.
Checking the balance point.
2.
If an equilibrium point does not exist, solve by finding equalizing strategies.
Now, we will show how to use the method of equalizing strategies, which works when there is
no equilibrium point. Through this method, we can extract the value of the game and the
optimal strategies.
Let's assume there is no equilibrium point. If
a b
so
b c
<
, otherwise the equilibrium point
will be.
c b
<
because it is not
c a
>
must be , otherwise there will be an equilibrium point
again. Continuing like this, we will see that ,
d a
<
and
d b
>
should be. In other words ,if
a b
then
a b
>
and
b c d a
< > <
. It can be shown by symmetry that if
a b
then
a b c d a
< > < >
.
This is what I want.shows that: If there is no equilibrium point, then either
,
a b b c
>
<
and
c d
>
and
,
d a
<
or
,
,
a b b c c d
<
>
<
and
d a
>
.
In equations (1), (2) and (3) in Qui, the general
2 2 We will derive optimal strategies and
game value formulas for the game. If the player chooses the first row
p
chooses with
probability(i.e., mixedstrategy (
,1
p
p
-
),), and if player ll uses columns 1 and 2, we equate
player l's average utility to:
(1 p) bp c(1 p)
ap d
+
- =
+
-
p
If we solve for:
(a b) (c d)
c d
p
-
=
- + -
(1)
Since there is no equilibrium point,
(a b)
-
and
(c d)
-
have the same sign (both positive
or negative), so
0
1
p
< <
. If player ll takes the first column
q
chooses with probability (i.e.,,
( ,1 )
q
q
-
aralash strategiya ishlatsa), uses a mixed strategy), then we equate the average loss of
player ll:
(1 q) dq c(1 q)
aq b
+
- = +
-
Solving for Q:
c b
q
a b c d
-
=
- + -
(2)
Again, because there is no equilibrium point
0
1
q
< <
.
The average loss of player ll under this strategy is:
(1 q)
ac bd
aq b
v
a b c d
-
+
- =
=
- + -
This result shows that the game is worthwhile, and both players have optimal strategies
(in which case the minimax theorem always guarantees).
Example 2:
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Let's calculate:
q
=same as (
7
12
q p
= =
)
8 9
1
2 3 4 3 12
v
-
=
=
- - - -
Example 3:
0
10
1
2
A
-
=
2 1
1
2 10 1 0 11
p
-
=
=
+
- -
2 ( 10)
12
2 10 1 0 11
q
- -
=
=
+
- -
Here q is greater than 1. What happened? The mistake was that we forgot to check the lower
left element (1,1) — it was an equilibrium point. So, there is an equilibrium point in the game.
Removing dominant strategies.
Sometimes large matrix games can be reduced in size
(hopefully down to 2x2 size) by removing rows and columns that clearly harm the player.
If
(a )
ij
A
=
in the matrix
ij
kj
a
a
all
j
For s,then we say,
i
-string
k
-takes precedence over -
line . If
ij
kj
a
a
>
all
j
For s,then we say,
i
-string
k
-is in a much greater position than the line.
Similarly, if
ij
kj
a
a
(or strictly
ij
kj
a
a
>
) all
i
If it’s for you , then
j
-column
k
-overrides the
column.
Player 1 can achieve any outcome using the dominant row, so dominant rows can be removed.
Similarly, dominant columns can be removed. More precisely, removing a dominant row or
column does not change the value of the game. However, sometimes the optimal strategy relies
on a dominant row or column. If this is the case, then removing that row or column also
removes that optimal strategy (but at least one optimal strategy remains). However, if a row or
column with a large advantage is removed, the set of optimal strategies does not change.
We can apply this method to multiple rows and columns in a row. For example, consider the
following matrix A:
2 0 4
1 2 3
4 1 2
A
=
We remove the last column and get:
2 0
1 2
4 1
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Now the top row is superimposed by the bottom row. We remove the top row and get:
1 2
4 1
This
2 2 There is no equilibrium point in the matrix. Therefore, we find mixed strategies
using the previous method:
3
1
,
4
4
p
q
=
=
7
4
v
=
. So the optimal strategy in the original game is
For l:
3 1
0, ,
4 4
,
For ll:
1 3
, ,0
4 4
It is also possible to remove column strategies through mixed combinations. That is, if a row or
column is dominated by a possible combination of several rows, it will also be removed.
For 0<p<1:
2
(1 p)a
ij
i j
kj
pa
a
+ -
if,
k
-string
1
i
and
2
i
will be dominated by a mixture of rows. (The same applies to columns.)
Example 4.
0 4 6
5 7 4
9 6 3
A
=
The middle column is a mixture of outer columns (each
1
2
) is preferred over . We remove it.
Then the middle row was dominated by a mixture of the upper and lower rows (
1
3
and
2
3
). We
will remove it too.
It remains:
0 6
9 3
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Game value:
54 9
12 2
V
=
=
Conclusion.
It is not always possible to eliminate dominant strategies. If there is an equilibrium point in the
matrix, then the game cannot be simplified.3 x 3 Even if there is a balance point in the game, it
is still necessary to check the advantage.
Literature:
1. R. Bellman (1952) On the games of "bluffing", Rendiconti del Circolo Mat. di Palermo Ser.
2, Vol. 1, 139–156.
2. R. Bellman and D. Blackwell (1949) "Two-person zero-sum games", Ann. Math. Stat. 20,
600–605, 9/40/49.
3. Bellman and Blackwell (1950) "On Bluffing Games," RAND Memo P-168, 8/01/50.
4. John D. Beasley (1990) The Mathematics of Games, Oxford University Press.
5. Emile Borel (1938) Papers on computational probabilities and games, Gauthier-Villars,
Paris.
6. H. Cutler (1975) "Optimal strategies for poker in the capital", Amer. Math. Monthly 82,
358–367.
7. W. H. Cutler (1976) "Poker Before the End", Perception.
8. Melvin Dresher (1961) Theory and Practice of Strategy, Prentice Hall, New Jersey.
9. RJ Evans (1979) "Silverman's game on intervals", Amer. Math. Mo. 86, 277–281.
10. TS Ferguson (1967) Mathematical Statistics—A Decision Theory Approach, Academic
Press, New York.
11. J. Filar and K. Visse (1997) Competitive Markov Decision Processes, Springer-Verlag,
New York.
12. Filar (1971) "A game about bluff strategies", Math. Sci. 17, 8764–8771.
13. S. Gal (1974) “Discrete search games”, SIAM J. Appl. Math. 27, 641–648.
14. D. B. Gillies, J. P. Mayberry, and J. von Neumann (1953) "Two Variants of the Game of
Poker," Contrib. Theory of Games II, 13–50.
15. J. Goldman and J. J. Stone (1960b) "A symmetric controller game model", J. Res. Nat. Bur.
Standards 64B, 35–40.
16. Heuer and U. Leopold-Wildburger (1991) "Balanced Silverman's Games in Discrete Series
Games", Lecture Notes in Econ. & Math. Syst., No. 365, Springer-Verlag.
17. S. M. Johnson (1964) "On Search Games," Advances in Game Theory, AMS Study #52,
39–48.
18. Samuel Carlin (1959) Economics in Mathematical Methods and Game Theory, Dover
Publications Inc., New York.
19.
S. Karlin and H. Restrepo (1957) "Multi-stage poker games", Contrib. Theory. Games III,
337–363.
