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 1199
SOLUTION UNDER RISK CONDITIONS: LAPLAS CRITERION. MINIMAX AND
MAXMIN CRITERION. SAVAGE AND HURWITZ CRITERIA
Mamatova Zilolaxon Khabibullokhonovna
Associate Professor, Doctor of Philosophy (PhD) in Pedagogical Sciences,
Fergana State University
Orchid : 0009-0009-9247-3510
E-mail:
Azimjonova Mohinur Asiljon kizi
Fergana State University Practical mathematics 3rd year student , group 22-08 student
Email:
azimjonovamohinur88@gmail.com
Abstract
: Decision-making under risk refers to the consideration of risk in decision-making
processes. The Laplace criterion is a method that considers probabilities equal and calculates
the average value of each option for each possible situation. The minimax criterion aims to
minimize the worst-case scenario under risk. The maxmin criterion is the inverse of the
minimax approach. The Savage criterion aims to consider the average and worst-case
scenarios in terms of risk in decision-making. The Hurwitz criterion aims to reduce the
uncertainty between the average and worst-case scenarios under risk. The approaches
presented in this article show what methods can be used in decision-making under risk.
Keywords:
Risk, decision making, Laplace criterion, minimax criterion, maxmin criterion,
Savage criterion, Hurwitz criterion, uncertainty, probabilities, decision making.
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 .
Risk-taking under the circumstances solution acceptance to do , to do a lot in fields including
economics , management , engineering and other technician in the fields wide This topic is
used according to literature various methods and approaches offer it will , this and decision
acceptance to do in the process risk-taking how to manage to study help gives . References
analysis following criteria based on take Let's go : Laplace criterion , Minimax criterion ,
Maximin criterion , Savage criterion and Hurwitz criterion .. Laplace criterion risk under the
circumstances decision acceptance to do the most simple and wide widespread from the
methods Laplace (1814) defined risk as equal distribution hypothesis does and all of
circumstances probabilities He considers all
possible was
situations for of values
average value to calculate and the most high average to value has was option to choose
provides . Minimax criterion decision acceptance in doing the most inconvenient the situation
to minimize This approach is aimed at main purpose , decision acceptance doer for the most
bad the results the most less to the level von Neumann and Morgenstern (1944) in their "
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 1200
Theory in the work " Games " minimax approach game theory within current those who have
reached.Maxmin criterion minimax approach reverse approach decided acceptance doer
every one option the most good in the state value This method main advantage is that the
decision acceptance doer the most good the situation to take strives . Savage criterion
decision acceptance in doing average and bad situations in consideration to take goal This
method is mainly used to determine acceptance the one who does risk level to balance Savage
( 1950 ) in his method risk relief , average the value maximum to do aimed at strategy 5.
Hurwitz criterion risk to minimize aimed at from methods He is a risk taker . relief for
average and bad situations between the difference tries to alleviate . Gurvitz (1956) in his
criterion risk and uncertainty alleviate for working came out .
This study aims to explore effective methods for making decisions in risky situations. The
study methodology theoretical analysis , mathematics modeling , comparison analysis and
experimental analysis own
inside Research
during literature analysis through risk
under the circumstances decision acceptance to do according to there is scientific sources and
methods Laplace , Minimax , Maxmin , Savage
and Hurwitz criteria such as various
approaches analysis will be done and their risk-taking in management efficiency is studied. In
the study mathematician modeling methods using risk under the circumstances decision
acceptance to do processes simulation In these processes probability distributions , risk and
uncertainties in consideration is taken . Simplex method , graphic method and other modeling
methods using decision acceptance to do process optimization opportunities analysis
Comparative
analysis through various methods compared and their efficiency is
determined . Experimental analysis methodology theoretical shaped risk under the
circumstances solution acceptance to do of the plan to practice implementation to be and the
results to study In this process methods using solutions learned , risk - taking management
the most effective methods working This is
methods using risk under the circumstances
decision acceptance to do process improve and economic efficiency increase possible .
Analyses and results
Optional the process research to grow during decision ( solution ) to do right comes . In
this case , the solution acceptance to do how in the situation : m data full
or full it's not
being important are , they are two opposite situations represents . Of these in the middle was
again one situation There is something called risk . In this event
to give possible was of
circumstances probability , or their distribution functions given It can also be . For example ,
in the market any to the product was
price accuracy under the circumstances unchanged
stands , risk under the circumstances and it is random quantity become , change law
distribution function through given will be , uncertainty under the circumstances and it is
random quantity is , its change legality or not given , or in general clearly It won't be . But
uncertainty under the circumstances information no
means , this only random of the
amount surrender to the law relatively said is , its acceptance to do values ( cases) complex ,
situations set ) in advance known will be . The data surrender to the level Depending on the
issue , the given issue is also different. For example , the above in the example : clear under
the circumstances , accepted made solution quality gone cost express through ( criterion )
possible , risk under the circumstances and price random quantity to be , to be into account
received without criterion to compose need will be , uncertainty under the circumstances and
the price about information absolutely unknown is , above clear generally support the
criteria It will not be . These considerations shows that the data indeterminacy increasing
progress criterion both choice and solution acceptance to do making the issue even more
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 1201
difficult , and generally , firmly reliable to the results take doesn't come . So so , accuracy
under the circumstances criterion to compose no how difficulty It doesn't work , but it's risky.
and uncertainty under the circumstances and various criteria are used . Their no one flawless ,
without flaw without counting , only various in appearance was requirements satisfaction
with The issue under consideration is twofold . participatory to the game cited is the second
participant " unconscious " opponent as acceptance if done , such game nature with is called a
game . Such of the game the most important side : nature various situations brought release
with , it is this of circumstances which one event from giving interested not . Nature with
game
in the matter main problem-to-goal directed criterion determination and to him/her
relatively optimal solution is to find . If the in the matter second participant conscious
opponent , suppose if done , this situation controversial with him games theory is engaged
in .
To the topic issue
If an investor buys shares or other to assets investment want to enter If so , he is taking the
risk. in consideration take it , take it financial status and of the market future status guess For
example , high
risk-taking has was
startup company's to shares investment input is
decision , investor risk acceptance to do depends .
Investments risk under the circumstances decision acceptance to do methods various kind
methodologies own inside takes.Ush this issue Laplace , Minimax , Maxmin , Savage and
Hurwitz criteria using analysis We try to do it . Every criterion risk and risks management
methods to oneself typical in a way illuminates . Now and every one method seeing we go
out :
1. Laplace criterion:
Laplace criterion is all possible was situations for probabilities equal
if , decision acceptance to do method . That is , every one of the situation probability one
suppose that different will be done .
Issue: Investor to shares investment input according to decision acceptance The investor is
choice :
Action 1: Year 60% increase at the end or 20% reduction probabilities there is .
Actions 2: Year 30% increase at the end or 10% reduction probabilities there is .
This is an investor situations for probability considers equal ( i.e. , every
one 0.5
probability for ). So , Laplace to the criterion According to the average investor income
calculating comes out .
Action 1: 1
1
(
60%) (
( 20)) 30% 10% 20%
2
2
*
+
* -
=
-
=
( Average income )
Actions 2: 1
1
(
30%) (
( 10)) 15% 5% 10%
2
2
*
+
* -
=
-
=
( Average income )
Laplace criterion The investor chooses Stock 1 because
this average income high
does .
2. Minimax criterion
: Minimax criterion the decision the most bad in case acceptance to do
emphasizes , that is the most bad probably based on , most less damage causing option
selection .
Issue:
Action 1: Year 60% increase at the end or a 20% decrease .
Actions 2: Year 30% increase at the end or a 10% decrease .
Each option the most bad to the state let's see :
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 1202
Action 1: The most bad status - 20% decrease .
Actions 2: The most bad status - 10% decrease .
Minimax to the criterion Therefore , the investor chooses Stock 2 because
this the most
bad in case less damage brings (-10%).
3. Maxmin criterion
: Maximin criterion is decision acceptance in doing , every for one
option the most good the situation choice and from them the most above choice is a method .
Issue:
Action 1: Year 60% increase at the end or a 20% decrease .
Actions 2: Year 30% increase at the end or a 10% decrease .
Each option the most good status seeing we go out :
Action 1: The most good status - 60 % increase .
Actions 2: The most good status - 30 % increase .
Maxmin to the criterion Therefore , the investor chooses Stock 1 because
this the most
good in case the most high growth presented reaches (60%).
4. Savage Criterion
: Savage criterion risk-taking relatively conservative approach In this
method , the investor himself acceptance to do possible was the danger into account takes and
probability into account take the most cautious option chooses .
Issue:
Action 1: Year 60% increase at the end or a 20% decrease .
Actions 2: Year 30% increase at the end or a 10% decrease .
this method, the investor calculates probabilities seeing comes out and from them the most
conservative the situation For example , select Actions 2 possible , because this decrease
probability smaller and danger decreased .
5. Hurwitz criterion:
Hurwitz criterion probabilities optimistic and pessimistic values with
compare this two the situation average This is mainly due to the risk relatively balanced
approach provides .
Issue:
Action 1: Year 60% increase at the end or a 20% decrease .
Actions 2: Year 30% increase at the end or a 10% decrease .
Hurwitz criterion according to , each one situation for the most optimistic ( good ) and
pessimistic ( bad ) probabilities seeing we will go out and average to take let's count .
Action 1: Average value = 60% ( 20%) 20%
2
+ -
=
Actions 2: Average value = 30% ( 10%) 10%
2
+ -
=
Hurwitz criterion by , Select Actions 1
acceptable will be , because his/her average
value high (20%).
Final analysis
Laplace Criteria : Promotions 1
Minimax Criteria : Promotions 2
Maxmin Criteria : Promotions 1
Savage Criteria : Actions 2
Hurwitz Criteria : Promotions 1
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 1203
Decision acceptance in doing every one criterion various kind approaches presented will be ,
and the investor will risks how acceptance to do and probabilities how evaluation choice
need .
Conclusion
Risk-taking under the circumstances decision acceptance in doing used methods and criteria
about many
research available . Laplace criterion simple and equal probabilities into
account received without effective works , but there are inaccuracies high was
in cases
limitation possible . Minimax and Maxmin criteria decision acceptance the one who does the
most inconvenient or the most good situations into account to take focused on . Savage
criterion and Hurwitz criterion and , risk into account take , average
and bad situations
between the difference to minimize is focused on . Each of the method to oneself typical
advantages and disadvantages there is are , they are various under the circumstances decision
acceptance in doing usage possible .
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and make it real release to the conditions application issues illuminated .
3. R. Dorfman, P. Samuelson, R. Solow - " Linear programming and economic analysis "
(1958). Optimal planning , constraints and goal function based on decision acceptance to
do discussion made .
4. IG Bashmakov - "Optimal working release systems " (2005). Modern working release
processes to optimize related theoretical and practical approaches showing given .
5. GN Nemchinov - " Linear " economic models " (1972). Economic in systems linear
programming and analysis methods to use dedicated .
