INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
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DECISION MAKING UNDER RISK
Mamatova Zilolaxon Xabibulloxonovna
Associate Professor at Fergana State University,
Doctor of Philosophy in Pedagogical Sciences (PhD)
E-mail:
mamatova.zilolakhon@gmail.com
Shovkatjonov Komiljon Qahramonjonovich
Student at Fergana State University
E-
mail:
shavkatjonovkomiljon0506@gmail.com
Annotation:
Decision-making under risk and variable-sum games are significant areas in
game theory and decision-making processes. This topic explores the development of optimal
strategies in environments characterized by uncertainty and risks. Unlike traditional fixed-
sum games, variable-sum games consider scenarios where the total benefit to participants
may vary depending on the game’s outcome. In these games, participants strive to maximize
their own interests, but their decisions depend on the actions of other participants and
external uncertainties. Decision-making under risk employs methods such as probability
theory, statistical analysis, and scenario modeling. In such conditions, decision-makers must
balance expected gains and potential losses based on uncertain information. In variable-sum
games, cooperative and competitive strategies play a crucial role, with collaboration or
competition among participants influencing the game’s outcomes.
Keywords:
Risk, variable-sum games, decision-making, uncertainty, game theory, strategic
decisions, cooperation, competition.
Introduction
Decision-making under risk and variable-sum games are among the important research
areas in modern decision theory and practice. In environments where uncertainty and risks
are present, the decision-making processes become complex due to the strategic interactions
between participants and their dependence on external factors. Unlike constant-sum games,
variable-sum games are characterized by the fact that the total payoff for participants can
vary depending on the outcome of the game. In such games, participants aim not only to
maximize their own interests but are also compelled to consider the actions of other
participants and the uncertain conditions.
Solution Methods:
The following are key methods used in decision-making under risk and uncertainty:
Hurwicz Criterion
Expected Value Maximization Method
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ISSN: 2692-5206, Impact Factor: 12,23
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Laplace Criterion
Minimax–Maximin Methods
Savage Criterion (Minimax Regret)
Hoede–Lehmann Method
Objective:
The specific objectives are as follows:
1. To systematically examine the main methods of decision-making under risk,
including probability analysis, scenario modeling, and risk management approaches.
2. To analyze the characteristics of variable-sum games, their differences from constant-
sum games, and the role of cooperative and competitive strategies.
3. To explore the applicability of core game theory concepts—such as Nash equilibrium,
Pareto efficiency, and the Shapley value—to variable-sum games.
4. To evaluate the effectiveness of these methods in solving real-world problems in
fields such as economics, management, political science, and artificial intelligence.
5. To develop recommendations for applying the research findings to practical scenarios,
such as resource allocation, international negotiations, or business strategies.
Sample Case Study:Problem:
Entrepreneur Sarvar is planning to open a small eatery in the city, specializing in fast food.
He is considering three options:
1. A burger shop
2. A shawarma shop
3. A hot dog shop
Depending on market conditions, demand may fall into one of the following three
categories:
Demand Level
Probability
High demand
0.3
Moderate
demand
0.5
Low demand
0.2
The entrepreneur sells the products at the following prices (
1 USD = 12,000 UZS
):
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Product
Price (UZS)
Price (USD)
Burger
30,000
$ 2.50
Shawarma
25,000
$ 2.08
Hot dog
18,000
$ 1.50
The estimated monthly sales vary according to each demand level.
Profit matrix for 1 month (in USD):
PRODUCT
HIGH DEMAND
MODERATE
DEMAND
LOW DEMAND
Burger
$2,500
$2,000
$750
Shawarma
$1,875
$1,458
$417
Hot dog
$1,500
$1,200
$750
β = 0.7 γ = 0.3,
Task:
Which Product to Choose?
Solve the decision problem using the following 6 methods:
1.Hurwicz (Gurvits) Criterion:
This method calculates a weighted average between the best and worst payoffs, where α
represents the level of optimism.
2.Expected Value Maximization Method:
This method involves calculating the expected payoff for each option, considering the
probabilities of each demand level, and selecting the product with the highest expected value.
3.Laplace Criterion:
Under the assumption of equal probabilities for all possible outcomes, the Laplace
criterion calculates the average payoff for each option and selects the one with the highest
average.
4.Minimax and Maximin Methods:
These methods focus on minimizing the maximum possible loss (Minimax) or
maximizing the minimum gain (Maximin) for each option under different demand scenarios.
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5.Savage (Minimax Regret) Criterion:
This method calculates the regret for each option under each scenario (i.e., the difference
between the chosen outcome and the best possible outcome) and selects the product with the
minimum maximum regret.
6.Hoede-Lehmann Method:
This method evaluates the overall utility of each option by accounting for both the
potential gains and losses under varying demand levels, considering a balanced risk approach.
Solution:
Hurwicz Criterion
We will solve this problem using the Hurwicz criterion. The Hurwicz criterion: 0≤β≤10 The
value of β is a parameter, and based on the value of β, the weights wiw_iwi for all i=1,2,...,m
are determined.
w
i
= β × max × w
ij
+ (1 − β) × min × w
ij
1.We will solve using the formula.
We will create matrix A.
A =
2500 2000 750
1875 1458 417
1500 1200 750
Now, based on this, we will find it.
max × w
ij
w
1
∗
= max × w
1j
= max(2500,2000,750) = 2500
w
2
∗
= max × w
2j
= max(1875,1458,417) = 1875
w
3
∗
= max × w
3j
= max(1500,1200,750) = 1500
Now, based on this, we will find it.
min × w
ij
w
1∗
= min × w
1j
= min(2500,2000,750) = 750
w
2∗
= min × w
2j
= min(1875,1458,417) = 417
w
3∗
= min × w
3j
= min(1500,1200,750) = 750
We will apply the main formula and obtain the result.
w
1
= β × max × w
ij
+ (1 − β) × min × w
ij
== 0.7 ∗ 2500 + (1 − 0.7) ∗ 750 = 1975
w
2
= β × max × w
ij
+ (1 − β) × min × w
ij
== 0.7 ∗ 1875 + (1 − 0.7) ∗ 417 = 1437,6
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w
3
= β × max × w
ij
+ (1 − β) × min × w
ij
== 0.7 ∗ 1500 + (1 − 0.7) ∗ 750 = 1275
We will obtain the answer from the results we have.
max = w
i
= max(1975, 1437.6, 1275) = 1975
Thus, it is clear from the results that the answer is w
Answer: The solution α1 should be chosen according to the Hurwicz criterion. Therefore,
Sarvar should choose the burger.
2.Maximum Expected Value Method
Let the states of nature be θ1,θ2,...,θn with probabilities p1,p2,...,pn.Then, to find the
solution αk we use:
w
i
=
j=1
n
p
j
× w
ij
Maximum Expected Value Method
max w
i
va
α
k
.
w
1
= p
1
× w
11
+ p
2
× w
12
+ p
3
× w
13
= 0.3 × 2500 + 0.5 × 2000 + 0.2 × 750 = 1900
w
2
= p
1
× w
21
+ p
2
× w
22
+ p
3
× w
23
= 0.3 × 1875 + 0.5 × 1458 + 0.2 × 417
= 1374,9
w
3
= p
1
× w
31
+ p
2
× w
32
+ p
3
× w
33
= 0.3 × 1500 + 0.5 × 1200 + 0.2 × 750 = 1200
Now let's find the
max w
i
.
max w
i
= w
i
(1900,1374.9, 1200) = 1900
Therefore, since the
max w
i
= w
1
solution
α
1
should be selected.
Answer:
According to the method of maximizing the expected value, Sarvar should choose
the Burger.
3. Laplace method
In the Laplace method, the probabilities of the states
θ
1
, θ
2
, . . . , θ
n
are assumed to be
equal. That is,
p
1
= p
2
= . . . = p
n
=
1
n
.
w
i
=
j=1
n
p × w
ij
=
1
n ×
j=1
n
w
ij
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Using the formula, we determine the solution
α
k
.
We find w
i
p =
1
3
w
i
w
1
=
1
n ×
j=1
n
w
1j
=
1
3 × (2500 + 2000 + 750) = 1750
w
2
=
1
n ×
j=1
n
w
2j
=
1
3
× (1875 + 1458 + 417) = 1250
w
3
=
1
n ×
j=1
n
w
3j
=
1
3 × (1500 + 1200 + 750) = 1150
max w
i
= max (1750, 1250, 1150) = 1750
So, since
α
k
= α
1
, Sarvar should choose the Burger."
Answer:
According to the Laplace method, the solution
α
1
should be selected. Therefore, the
entrepreneur Sarvar should choose the Burger.
4.Minimax and Maximin methods
The solution
α
k
, which is determined by the minimum of the row-wise maximum values in
the
θ
table, is called the minimax solution.
The solution
α
k
, which is determined by the maximum of the row-wise minimum
values in the
θ
table, is called the maximin solution.
w
1
∗
= max × w
1j
= max(2500,2000,750) = 2500
w
2
∗
= max × w
2j
= max(1875,1458,417) = 1875
w
3
∗
= max × w
3j
= max(1500,1200,750) = 1500
Now, from this, we determine
min × w
i
∗
.
α
k
= minimax(2500, 1875, 1500) = 1500
Therefore, according to the minimax method,
α
k
= α
3
, meaning that the Hot-dog
should be chosen.
w
∗1
= min × w
1j
= min(2500,2000,750) = 750
w
∗2
= min × w
2j
= min(1875,1458,417) = 417
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ISSN: 2692-5206, Impact Factor: 12,23
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w
∗3
= min × w
3j
= min(1500,1200,750) = 750
Now, from this, we determine
max × w
∗i
.
α
k
= maxmini(750, 417, 750) = 750
Therefore, according to the maximin method,
α
k
= α
1
= α
3
, meaning that both the
Burger and the Hot-dog should be chosen.
5.Savage method
In the Savage method, a table called regret (R table) is constructed based on the following
rule.
The elements of the R table are
r
ij
= max w
lj
− w
ij
.
maxmini α
k
By applying the maximin method to the table
generated by αk\alpha_kαk , we determine the solution
α
k
.
max w
i1
= max(2500, 1875,1500) = 2500
max w
i2
= max(2000, 1458, 1200) = 2000
max w
i3
= max(750, 417, 750) = 750
We construct the R-table.
r
11
= w
i1
− w
11
= 2500 − 2500 = 0
r
21
= w
i1
− w
21
= 2500 − 1875 = 625
r
31
= w
i1
− w
31
= 2500 − 1500 = 1000
r
12
= w
i2
− w
12
= 2000 − 2000 = 0
r
22
= w
i2
− w
22
= 2000 − 1458 = 542
r
32
= w
i2
− w
32
= 2000 − 1200 = 800
r
13
= w
i3
− w
13
= 750 − 750 = 0
r
23
= w
i3
− w
23
= 750 − 417 = 333
r
33
= w
i3
− w
33
= 750 − 750 = 0
R =
0
0
0
625 542 333
1000 800
0
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r
∗1
= min(0, 0, 0) = 0
r
∗2
= min(625, 542, 333) = 333
r
∗3
= min(1000, 800, 0) = 0
Therefore, according to the Savage method,
α
1
and
α
3
should be selected as the
solutions.
Answer:
According to this method, if the entrepreneur Sarvar chooses the Burger and Hot-
dog, his regret will be 0.
6. Hodja-Lemann method
In the Hodja-Lemann method, the parameter
0 ≤ γ ≤ 1
is involved, and for the
probabilities of the states
θ
1
, θ
2
, . . . , θ
n
, denoted as
p
1
, p
2
, . . . , p
n
,
w
i
= γ × p
j
× w
ij
+ (1 − γ) × min w
ij
We find the
max w
i
and
α
k
using the formula.
w
1
= γ × w
1
+ (1 − γ) × w
1∗
= 0.3 × 1900 + 0.7 × 750 = 1095
w
2
= γ × w
2
+ (1 − γ) × w
2∗
= 0.3 × 1374.9 + 0.7 × 417 = 704.37
w
3
= γ × w
3
+ (1 − γ) × w
3∗
= 0.3 × 1200 + 0.7 × 750 = 885
Therefore, it is clear that the solution is
α
k
= α
1
.
Answer:
Even after using the Hodja-Lemann method to find the solution, the
entrepreneur Sarvar should choose the Burger.
Based on all the answers to this problem, we can conclude that it would be advisable for
Sarvar to open a kitchen that prepares burgers. This is because we approached the problem in
a real-life context, analyzing it from three different perspectives. In reaching this conclusion,
we have reinforced our skills in making decisions under uncertainty, applied theoretical
knowledge in practice, and learned to connect it to real-life problems.
Conclusion
Decision-making under uncertainty and variable-sum games are important research
areas in modern decision theory and practice, helping to find optimal solutions in
environments where uncertainty and strategic interaction exist. This research analyzed the
characteristics of variable-sum games, their differences from constant-sum games, and the
importance of cooperative and competitive strategies.
In decision-making under uncertainty, methods such as probability analysis, scenario
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
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page 488
modeling, game theory techniques (Nash equilibrium, Pareto efficiency, Shapley value), risk
management, and artificial intelligence-based approaches were discussed. The findings of the
research show that these methods are crucial in solving real-life problems such as resource
allocation, negotiations, and strategic planning in fields such as economics, management,
political science, ecology, and artificial intelligence. Variable-sum games allow for balancing
cooperation and competition between participants, while risk management methods help
make stable and effective decisions in conditions of uncertainty.
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and Complexity. Dover Publications.
2. Applegate, D., Bixby, R., Chvátal, V., & Cook, W. (2006). The Traveling Salesman
Problem: A Computational Study. Princeton University Press.
3. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1985). The
Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley.
4. Gutin, G., & Punnen, A. P. (2002). The Traveling Salesman Problem and Its Variations.
Springer.
5. Reinelt, G. (1994). The Traveling Salesman: Computational Solutions for TSP
Applications. Springer.
