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volume 4, issue 3, 2025
633
DECISION MAKING UNDER UNCERTAINTY
Mamatova Zilolaxon Xabibulloxonovna
Associate Professor at Fergana State University
Doctor of Philosophy (PhD) in Pedagogical Sciences
E-mail:
Abdusamadova Vasilakhon Elyorjon daughter
Student at Fergana State University
E-
mail:
Annotation:
In this article, we will explore various criteria encountered in decision-making
problems involving nature. These include the expected value (mathematical expectation)
criterion, the Laplace criterion, Wald’s minimax (maximin) criterion, the Savage criterion, the
Hurwicz criterion, and the Hodges-Lehmann criterion.
Keywords:
Game against nature, expected monetary value, Laplace criterion, Wald’s
minimax (maximin) criterion, Savage criterion, Hurwicz criterion, Hodges-Lehmann criterion.
Annotatsiya:
Biz bu maqolada tabiat bilan o‘yin masalasida uchraydigan turli kriteriyalar bilan
tanishib chiqamiz. Bular yutuqning matematik kutilmasi kriteriyasi, Laplas kriteriyasi, Valdning
minimaks (maksimin) kriteriyasi, Sevidj kriteriyasi, Gurvits kriteriyasi va Xodja-Leman
kriteriyasi.
Kalit so‘zlar:
Tabiat bilan o‘yin, yutuqning matematik kutilmasi, Laplas kriteriyasi, Valdning
minimaks (maksimin) kriteriyasi, Sevidj kriteriyasi, Gurvits kriteriyasi, Xodja-Leman kriteriyasi.
Аннотация:
В данной статье мы ознакомимся с различными критериями,
возникающими при решении задач взаимодействия с природой. Это критерий
математического ожидания выигрыша, критерий Лапласа, минимаксный (максиминный)
критерий Вальда, критерий Сэвиджа, критерий Гурвица и критерий Ходжа-Лемана.
Ключевые слова:
Игра с природой, математическое ожидание выигрыша, критерий
Лапласа, критерий Вальда минимакс (максимин), критерий Савиджа, критерий Гурвица,
критерий Ходжа-Лемана.
Introduction
Decision Making under Risk - A Game with Nature
The states are known and defined by
�
1
, �
2
, …, �
�
. Let the decisions (solutions) we make be
�
1
, �
2
, …, �
�
. Suppose that when we make the
�
�
decision, nature brings about the
�
�
state. In this
case, the benefit (profit, income, gain) we receive will be equal to
�
��
. This can be expressed in
the following table:
�
1
�
2
…
�
�
�
1
�
11
�
12
…
�
1�
�
2
�
21
�
22
…
�
2�
⋮
⋮
⋮
⋮
⋮
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volume 4, issue 3, 2025
634
�
�
�
�1
�
�2
⋯
�
��
Objective:
The goal in a game with nature is to choose one of the possible solutions
�
1
, �
2
, …, �
�
, without knowing which state
�
1
, �
2
, …, �
�
nature will bring about, in such a way
that the resulting gain is maximized. To achieve this objective, several methods—mentioned
above—have been proposed. We will now examine each of these methods one by one.
1. Hurwicz Method: This method depends on a
0 ≤ � ≤ 1
parameter, which indicates the degree
of "optimism" of the decision-maker. First, based on the value of
�
, the differences
� = 1,2, …, �
are calculated for all values of
�
�
= �max
�=1�
�
��
+ (1 − �)min
��
�
��
.
Then, the value of
�
�
that maximizes
�
is determined, and the corresponding
�
�
is selected.
2. Method of Maximizing the Expected Value: In this method, it is assumed that
�
1
, �
2
, …, �
�
the
probabilities of the possible states occurring are known, and let them be
�
1
, �
2
, …, �
�
accordingly.
In that case, by choosing decision
�
�
, one obtains an average gain of
�
�
= ∑
�=1
�
�
�
�
��
. The
maximum among these
�
�
values determines the decision
�
�
that should be selected.
3. Laplace Method: This method is a special case of the method of maximizing the expected
value
�
1
= �
2
= … = �
�
= 1/�
in which
�
1
, �
2
, …, �
�
the probabilities of the possible states are
assumed to be equal.
4. Minimax and Maximin Methods:
�
the decision determined by the minimum of the row-wise
maximum values in the payoff table is called the minimax decision.
�
�
the decision determined
by the maximum of the row-wise minimum values in the payoff table is called the maximin
decision.
5. Savage Method: In the Savage method, a regret table R is constructed based on the following
rule:
�
��
= max
�=1�
�
��
− �
��
. The maximin method is then applied to this table to determine the
optimal decision
�
�
.
6. Hodges–Lehmann Method: In this method, a parameter
0 ≤ � ≤ 1
is involved, and its value
determines the confidence level of the probabilitie
�
1
, �
2
, …, �
�
that represent the likelihood of
the different states
�
1
, �
2
, …, �
�
occurring. The corresponding decision
�
�
is determined by
finding the maximum of the values based on
�
�
= �∑
�=1
�
�
�
�
��
+ (1 − �)min
�=1�
�
��
.
Sample Problem:
Let’s consider the example given in the table below using the six
different methods discussed above.
Given Probabilities: p
1
=
1
3
, p
2
=
1
6
, p
3
=
1
4
, p
4
=
1
4
Coefficients:
�
=
2
3
,
�
=
2
3
�
1
�
2
�
3
�
4
�
1
4
0
5
2
�
2
2
3
1
4
�
3
3
2
6
1
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1. Hurwics Method
w
*1
=
max
�=1�
�
1�
=
max
�=1�
(4,0,5,2)=5
w
*2
=
max
�=1�
�
2�
=
max
�=1�
(2,3,1,4)=4
w
*3
=
max
�=1�
�
3�
=
max
�=1�
(3,2,6,1)=6
w
1*
=
min
�=1�
�
1�
=
min
�=1�
(4,0,5,2)=0
w
2*
=
min
�=1�
�
2�
=
min
�=1�
(2,3,1,4)=1
w
3*
=
min
�=1�
�
3�
=
min
�=1�
(3,2,6,1)=1
the following formula of the Hurwicz criterion
�
�
= �max
�=1�
�
��
+ (1 − �)min
��
�
��
according to ,
w
1
=
�
w
*1
+(1-
�
)w
1*
=
2
3
*5+
1
3
*0=
10
3
w
2
=
�
w
*2
+(1-
�
)w
2*
=
2
3
*4+
1
3
*1=3
w
3
=
�
w
*3
+(1-
�
)w
3*
=
2
3
*6+
1
3
*1=
13
3
These can also be written in general form as follows:
max
2
3
(5,4,6) +
1
3
(0,1,1) = max
10
3
, 3,
13
3
=
13
3
,
Thus, it follows that the decision-maker should choose strategy
�
3
2. Method of Maximizing the Expected Value:
In this method, the decision is determined by finding the maximum
w
i
of the formula for solution
�
k
�
�
= ∑
�=1
�
�
�
�
��
w
1
=p
1
w
11
+p
2
w
12
+p
3
w
13
+p
4
w
14
=
1
3
*4+
1
6
*0+
1
4
*5+
1
4
*2=
37
12
w
2
=p
1
w
21
+p
2
w
22
+p
3
w
23
+p
4
w
24
=
1
3
*2+
1
6
*3+
1
4
*1+
1
4
*4=
29
12
w
3
=p
1
w
31
+p
2
w
32
+p
3
w
33
+p
4
w
34
=
1
3
*3+
1
6
*2+
1
4
*6+
1
4
*1=
37
12
max
�=1,2,3
�
�
=
max
�=1,2,3
(
37
12
,
29
12
,
37
12
)=
37
12
3. Laplace Method:
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636
In this method, the decision-maker's
�
�
strategy provides an average gain of
1
�
�=1
�
�
��
o.
Therefore, they will choose the
�
�
strategy that maximizes this average gain,
max
�
�
1
�
�=1
�
�
��
=
1
�
�=1
�
�
��
In our given example, based on the Laplace method,
max
�
�
1
4
�=1
3
�
��
= max
1
4 (11,10,12) = max(
11
4 ,
5
2 , 3) = 3
Therefore, according to the Laplace criterion, the decision-maker should apply strategy
�
3
.
4. Maximin (Minimax) Method:
In this criterion, if the decision-maker applies strategy
�
�
and an unfavorable state occurs due to
nature, their gain will be
�
�
= min
�
�
��
.
Therefore, they will try to apply such a strategy
�
i
that the maximum of the minimum gains is
determined,
�
∗
= max
�
�
�
= max
�
min
�
�
ı
�
the decision that ensures the maximum value of
�
�
is considered the optimal strategy of the
decision-maker.
Based on the above, by applying strategy
�
�
, the decision-maker is guaranteed to achieve at least
�
∗
−
in guaranteed gain.
Now, for each given
� = 1,2,3
in the example, let’s determine
min
�
�
��
:
min
�
�
1�
= min 4,0,5,2 = 0, min
�
�
2�
= min(2,3,1,4) = 1, min
�
�
3�
= min(3,2,6,1) = 1.
This
implies that,
�
�
= max
�
min
�
�
��
= max min
�
�
1�
, min
�
�
2�
, min
�
�
3�
= max 0,1,1 = 1
Therefore, the decision-maker's optimal maximin strategy is
�
2
and
�
3
, and their guaranteed
gain is equal to 1.
Note: If the elements of the table
� = �
��
represent the decision-maker's cost (loss, defeat),
then, using the above reasoning, the guaranteed cost will be equal to the
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637
min
�
max
�
�
��
= min max
�
�
�1
, max
�
�
�2
, max
�
�
�3
= min(5,4,6) = 4
In this case, the strategy to be chosen will be
�
2
, but
min
�
max
�
�
��
≠ max
�
min
�
�
��
.
5. Savage Method:
In the Savage method, a table called "regret" is constructed based on the following rule:
�
��
=
max
�=1�
�
��
− �
��
. The maximin method is applied to the resulting table, and the decision
�
�
is
determined.
It is known that the number
max
�
min
�
�
ıȷ
represents the guaranteed gain of the decision-maker.
max
�=1,2,3
�
�1
= max
�=1,2,3
4,2,3 = 4
,
max
�=1,2,3
�
�2
= max
�=1,2,3
0,3,2 = 3
max
�=1,2,3
�
�3
= max
�=1,2,3
5,1,6 = 6
,
max
�=1,2,3
�
�4
= max
�=1,2,3
2,4,1 = 4
The elements of the table
� = �
��
are obtained by subtracting each column element of the table
� = �
��
from the largest element in that column (as previously derived).
r
11
=
max
�=1,2,3
�
�1
−
w
11
=4-4=0 ,
r
21=
max
�=1,2,3
�
�1
−
w
21
=4-2=2,
r
31
=
max
�=1,2,3
�
�1
−
w
31
=4-3=1
r
12
=
max
�=1,2,3
�
�2
− �
12
=3-0=3 ,
r
22
=
max
�=1,2,3
�
�2
− �
22
=3-3=0
r
32
=
max
�=1,2,3
�
�2
− �
32
=3-2=1
r
13
=
max
�=1,2,3
�
�3
− �
13
=6-5=1 ,
r
23
=
max
�=1,2,3
�
�3
− �
23
=6-1=5
r
33
=
max
�=1,2,3
�
�3
− �
33
=6-6=0
r
14
=
max
�=1,2,3
�
�4
− �
14
=4-2=2 ,
r
24
=
max
�=1,2,3
�
�4
− �
24
=4-4=0
r
34
=
max
�=1,2,3
�
�4
− �
34
=4-1=3
The general form of the "regret" table R will be as follows:
�
1
�
2
�
3
�
4
�
1
r
11
r
12
r
13
r
14
�
2
r
21
r
22
r
23
r
24
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638
�
3
r
31
r
32
r
33
r
3
If the obtained values are placed in the table accordingly, it will look as follows:
�
1
�
2
�
3
�
4
�
1
0
3
1
2
�
2
2
0
5
0
�
3
1
1
0
3
r
1*
=min(0,3,1,2)=0 , r
2*
=min(2,0,5,0)=0 , r
3*
=min(2,1,0,3)=0
It is known that the number
max
�
min
�
�
��
represents the guaranteed gain of the decision-maker.
max
�
min
�
�
��
= max 0,0,0 = 0
Therefore, in the Savage method, the decision-maker's strategy is
�
1
, �
2
,
�
3
.
6. Hodges–Lehmann Method:
The decision corresponding to this method is determined by finding the maximum of the values
�
�
= �∑
�=1
�
�
�
�
��
+ (1 − �)min
�=1�
�
��
of the solution
�
�
It is given that it is equal to
�
=
2
3
.
w
1
= � �
1
+ 1 − � �
1∗
=
2
3
∗
37
12
+
1
3
∗ 0 =
37
18
w
2
= � �
2
+ 1 − � �
2∗
=
2
3
∗
29
12
+
1
3
∗ 1 =
35
18
w
3
= � �
3
+ 1 − � �
3∗
=
2
3
∗
37
12
+
1
3
∗ 1 =
43
18
max
�=1,2,3
w
i
=max(
37
18
,
35
18
,
43
18
)=
43
18
Therefore, according to the Hodges-Lehmann criterion, the decision-maker should apply strategy
�
3
.
Conclusion
The problem under consideration has been framed as a two-player game, where the second
player is considered the "unaware" opponent. This type of game is called a game with nature.
The most important aspect of such a game is that nature, by bringing about various states, is not
interested in which of these states occurs. The main issue in the game with nature is to define a
goal-oriented criterion and find the optimal solution relative to it.
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Management under Risk
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2. Mamatqulov N.M., Xasanov B.X.
Economic Analysis and Decision-Making
. – Tashkent:
Economics, 2020.
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3. Abdurahmonov Q.X., Tadjibayev B.O.
Fundamentals of Economic Analysis and Business
Planning
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4. Akmalov A.T.
Theory of Decision Making under Risk
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Game Theory and Its Practical Applications
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Decision Analysis: Introductory Lectures on Choices under Uncertainty
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