Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності
The classic Colonel Blotto game for two players was considered. The probabilistic model of the payoff functions of the specified problem was investigated, and the game conditions are not subject to the restrictions of symmetry and homogeneity. The system of equations obtained using the method of Lag...
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| description | The classic Colonel Blotto game for two players was considered. The probabilistic model of the payoff functions of the specified problem was investigated, and the game conditions are not subject to the restrictions of symmetry and homogeneity. The system of equations obtained using the method of Lagrange multipliers has a large dimension. In order to find a solution, a way to reduce the dimension was found. The found ratio between the resources of both players, distributed over the courts, made it possible to identify a parameter determined by the ratio of Lagrange multipliers from the corresponding functions for both players. For such a parameter, an interval constraint that it satisfies was found, and an equation is formulated to find it, which is solved numerically. The found value of the parameter makes it possible to calculate individual Lagrange multipliers and obtain the optimal distribution of players’ resources in the form of a Nash equilibrium in pure game strategies. An example of a game under significantly different conditions for players was studied. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2026.1.06 |
| first_indexed | 2026-04-20T01:00:21Z |
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S.A. Smirnov, I.M. Tereshchenko, 2026
92 ISSN 1681–6048 System Research & Information Technologies, 2026, № 1
TIÄC
МЕТОДИ ОПТИМІЗАЦІЇ, ОПТИМАЛЬНЕ
УПРАВЛІННЯ І ТЕОРІЯ ІГОР
UDC 519.83
DOI: 10.20535/SRIT.2308-8893.2026.1.06
A PROBABILISTIC MODEL OF THE COLONEL
BLOTTO GAME WITHOUT SYMMETRY
AND HOMOGENEITY CONSTRAINTS
S.A. SMIRNOV, I.M. TERESHCHENKO
Abstract. The classic Colonel Blotto game for two players was considered. The
probabilistic model of the payoff functions of the specified problem was investigat-
ed, and the game conditions are not subject to the restrictions of symmetry and ho-
mogeneity. The system of equations obtained using the method of Lagrange multi-
pliers has a large dimension. In order to find a solution, a way to reduce the
dimension was found. The found ratio between the resources of both players, dis-
tributed over the courts, made it possible to identify a parameter determined by the
ratio of Lagrange multipliers from the corresponding functions for both players. For
such a parameter, an interval constraint that it satisfies was found, and an equation is
formulated to find it, which is solved numerically. The found value of the parameter
makes it possible to calculate individual Lagrange multipliers and obtain the optimal
distribution of players’ resources in the form of a Nash equilibrium in pure game
strategies. An example of a game under significantly different conditions for players
was studied.
Keywords: conflict confrontation, optimal allocation of resources, two-person
game, Colonel Blotto’s game, probabilistic payoff model, Nash equilibrium, effi-
ciency of resource use, 1-parametrization.
INTRODUCTION
Colonel Blotto’s problem as a model of competitive struggle between two players
on several platforms has long become a classic. It was first presented by Borel [1]
in 1921, but despite its century-old history of applications, it still remains relevant
and attracts the attention of researchers [2]. As a model of competitive struggle, it
finds numerous applications in the social, economic spheres of activity, etc.
The game is described as follows. Two players compete against each other
on several sites, which can be battlefields, election areas, product markets, etc.
Each participant has a limited resource that must be distributed across the sites in
such a way as to maximize their winnings, taking into account that each site has
its own value for each of the rivals. In the auction model, the winner of a given
site is the player who has allocated more resources to it than the opponent. In the
case of equal allocated resources, a draw occurs.
A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints
Системні дослідження та інформаційні технології, 2026, № 1 93
More interesting for us is the case of the probabilistic model. Here, the prob-
ability of winning on a certain site is directly proportional to the resource allocat-
ed to it and inversely proportional to the sum of the resources allocated to it by
both players. It should be noted that the following constraints are usually imposed
on the game: symmetry, when the total resources of each player are the same, and
homogeneity, when the worth of victory for any site coincides with that of its op-
ponent. Thus, when searching for equilibrium in Colonel Blotto’s problem, re-
searchers limited themselves to symmetric or homogeneous cases, with the latter
condition being used quite often [3–5].
Hence, the case where these two conditions are not met is of not only scien-
tific interest. When trying to find equilibrium strategies for such a problem, it be-
comes necessary to solve systems of equations of high dimension. Using the ap-
proach proposed in [6], the system of equations is reduced to the one-dimensional
case and represents by a single variable equation. In this case, we arrive at the
precize solution of the asymmetric Colonel Blotto game without homogeneity
constraints.
PROBLEM STATEMENT
Consider the probabilistic model of Colonel Blotto’s game. Two opposing parties
distribute their resources ix and iy , 1,i n , across n sites. The resource con-
straints are determined by the following inequalities,
1
n
i x
i
x R
,
1
n
i y
i
y R
. The
probability of the first player winning on the i -th court is given as follows:
( , )
i
i i
r
x i i
i i i r r
i i i i
xp x y
x y
,
where (0,1]ir , and 0i , 0i are the efficiency coefficients of resource use
on the corresponding sites for the first and second players. Similarly, the formula
for the probability of winning ( , )y
i i ip x y for the second player is:
( , )
i
i i
r
y i i
i i i r r
i i i i
yp x y
x y
.
In this case, the payoff functions take the form:
1
( , ) ( , )
n
x
x i i i i
i
F x y X p x y
,
1
( , ) ( , )
n
y
y i i i i
i
F x y Y p x y
,
where iX , iY is the value of winning on the i -th court for each of the two play-
ers. A Nash equilibrium in pure strategies ( , )x y is a pair of vectors that satis-
fies for any ),( yx satisfies: ( , ) ( , )x xF x y F x y , ( , ) ( , )y yF x y F x y .
S.A. Smirnov, I.M. Tereshchenko
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 94
Asymmetric game when x yR R , with the following parameter values 1ir ,
1i , i N was considered in [3] for the case i iX Y const . The only Nash
equilibrium in this case is the use of such pure strategies, when resources must be
evenly distributed between sites. A continuation of these studies was the work [4],
where an equilibrium in pure strategies was found for the case i i iX Y V with arbi-
trary (0,1]ir , 0i , i N . This paper considers the situation when i iX Y .
The aim of the work is to find the Nash equilibrium in pure strategies for
the asymmetric and heterogeneous Colonel Blotto game for the case of the proba-
bilistic model.
PROCEDURE FOR CONSTRUCTING AN OPTIMAL SOLUTION
To find a solution to the problem, we write the Lagrange function for each of the
players:
1
( )
n
x x x z i
i
L F R x
,
1
( )
n
y y y y i
i
L F R y
.
Then we get the system of equations:
0x x
x
i i
L F
x x
,
0y y
y
i i
L F
y y
,
1
0
n
x
x i
ix
L R x
,
1
0
n
y
y i
iy
L
R y
.
Starting from the system, we get:
1
2( )
i i
i i
r r
x i i i i i
i xr r
i i i i i
L r x yX
x x y
,
1
2( )
i i
i i
r r
y i i i i i
i yr r
i i i i i
L r x yY
y x y
.
We have (2 2)n equations with (2 2)n variables: 1, ,x 1, , , , ,n n x yx y y ,
that is, the system can be solved. Consider the ratio x
y
. We will get:
1 2
12
( )1
( )
i i i i
i i i i
r r r r
x i i i i i i i i i i i
i r r r r
y i i ii i i i i i i i i
r x y x y X yX
Y Y xx y r x y
.
A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints
Системні дослідження та інформаційні технології, 2026, № 1 95
From here x i
i i
y i
Yy x
X
. Let’s mark x
y
, i
i
i
YC
X
. Then
i
i
i
y C
x
. (1)
We will use expression (1) to go from the multidimensional case with (2 2)n
equations and (2 2)n variables to the one-dimensional case. To do this, we will find
an equation for searching . Consider the expressions x xR and y yR :
1
2
1 1 ( )
i i
i i
r rn n
i i i i i
x x i x i i r r
i i i i i i
r x yR x x X
x y
2
2 2
21 1
( )
/( )
( ) /( ) ( ( ) )
i
i i i
i i i
i
ri
i i ir r rn n
i i i i i i i
i ir r r
rii ii i i i i
i i
i
yr
r x y x xX X yx y x
x
(2)
2
1
( ) .
( ( ) )
i
i
rn
i i i i
i r
i i i i
r CX
C
Similarly,
2
1
( )
( ( ) )
i
i
rn
i i i i
y y i r
i i i i
r CR Y
C
. (3)
From formulas (2) and (3) we determine the relationship between x xR
and y yR :
2 2
1 1
( ) ( )/
( ( ) ) ( ( ) )
i i
i i
r rn n
x x x i i i i i i i i
i ir r
i iy y y i i i i i i
R R r C r CX Y
R R C C
2 2
1 1
/
( ( ) ) ( ( ) )
i i
i i
r rn n
i i i i i i i i
i ir r
i ii i i i i i
rC rCX Y
C C
.
Let’s rewrite this formula in the following form:
2 21 1( ( ) ) ( ( ) )
i i
i i
r ri i
i i i in n
x i i
i i
r ri ii iy
i i
i i
rC rC
R Y X
R C C
. (4)
Let’s find the limits within which the value lies . From (1) we have
i i iy C x . Then
1 1
n n
i y i i
i i
y R C x
. Hence
1 1 1
min max
n n n
i i i i i ii ii i i
C x C x C x
.
S.A. Smirnov, I.M. Tereshchenko
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 96
Since
1
n
x i
i
R x
, and
1
n
y
i i
i
R
C x
, then
min maxy
x i x ii i
R
R C R C
,
1 1(max ) (min )y y
i iii
x x
R R
C C
R R
,
min maxy yi i
i i
x i x i
R RX X
R Y R Y
.
Since, except for , all other parameters of equation (4) are known, its solu-
tion can be found on the specified segment by an appropriate numerical method,
for example, the dichotomy method.
Let’s find the optimal values for ix , iy .
1 2
2 2 2
/( )
( ) ( ) /( )
i i i i i
i i i i i
r r r r r
i i i i i i i i i i i i
x i i i ir r r r r
i i i i i i i i i i
r x y r x y x
x X x X
x y x y x
2 2
( ) ( )
( ( ) ) ( ( ) )
i i
i i
r ri i i
i i i
i i i
i i i
r ri i i
i
i i i
yr r C
xX X ay C
x
.
Similarly,
2
( )
( ( ) )
i
i
ri
i i
i
y i i i
ri
i
i
r C
y Y b
C
.
In the right-hand sides of the equalities, all parameters are known, therefore
ia and ib take specific values. Now we express x in terms of xR and ia , and y
in terms of yR and ib , after which we find ix and iy .
x i ix a
1 1
n n
x i i
i i
x a
1 1
n n
x i i
i i
x a
1
n
x x i
i
R a
1
1
x
x n
i
i
R
a
.
1
i x i
i n
x
i
i
a R ax
a
. (5)
Similarly, for y :
A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints
Системні дослідження та інформаційні технології, 2026, № 1 97
y i iy b 1
1
y
y n
i
i
R
b
.
1
y ii
i n
y
i
i
R bby
b
. (6)
Formulas (5) and (6) express the optimal solution to the problem — the Nash
equilibrium in pure strategies. Then the payoff functions take the following form:
1 1
( , )
( )
i
i i
i
i
rn n
i i i
x i ir r
rii ii i i i
i
i
xF x y X X
x y C
,
1 1
( )( , )
( )
i i
i i
i
r rn n
i i i
y i ir r
rii ii i i i
i
i
y CF x y Y Y
x y C
.
RESULTS OF THE NUMERICAL EXPERIMENT
We will conduct a numerical experiment based on theoretical calculations. We
will consider Colonel Blotto’s game on five platforms and also record the values
of some parameters:
100xR .
1 0.1r ; 2 0.25r ; 3 0.5r ; 4 0.75r ; 5 1r .
1 1.4 ; 2 1.9 ; 3 3.1 ; 4 3.7 ; 5 4.1 .
1 1.3 ; 2 1.8 ; 3 2.8 ; 4 3.5 ; 5 5 .
We will consider three cases, where 150yR , 250yR , 350yR . For
each case, we will calculate two options.
The first case.
T a b l e 1 . Option No.1 ( 150yR )
i 1 2 3 4 5
iX / iY 1.1/1.5 1.1/2.1 1.1/3.4 1.1/4.2 1.1/4.9
ix / iy 5.01/2.96 12.56/10.38 24.22/32.42 29.02/47.98 29.16/56.24
x
ip / y
ip 0.53/0.46 0.52/0.47 0.48/0.51 0.42/0.57 0.29/0.7
xF / yF 2.44/9.22
Calculated value 0.407 .
S.A. Smirnov, I.M. Tereshchenko
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 98
Fig. 1. Optimal solutions of players Fig. 2. Probabilities of players’ winnings
T a b l e 2 . Option No. 2 ( 150yR )
i 1 2 3 4 5
iX / iY 5.5/1.5 5.5/2.1 5.5/3.4 5.5/4.2 5.5/4.9
ix / iy 3.92/2.19 9.18/7.19 18.69/23.68 29.11/45.57 39.07/71.35
x
ip / y
ip 0.53/0.46 0.52/0.47 0.49/0.5 0.43/0.56 0.3/0.69
xF / yF 12.47/9.1
Calculated value 2.171 .
Fig. 3. Optimal solutions of players Fig. 4. Probabilities of players’ winnings
The second case.
T a b l e 3 . Option No. 1 ( 250yR )
i 1 2 3 4 5
iX / iY 1.1/1.5 1.1/2.1 1.1/3.4 1.1/4.2 1.1/4.9
ix / iy 4.35/4.16 10.76/14.41 21.69/47.04 29.36/78.66 33.82/105.7
x
ip / y
ip 0.51/0.48 0.49/0.5 0.42/0.57 0.33/0.66 0.2/0.79
xF / yF 2.13/10.34
A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints
Системні дослідження та інформаційні технології, 2026, № 1 99
Calculated value 0.684 .
Fig. 5. Optimal solutions of players Fig. 6. Probabilities of players’ winnings
T a b l e 4 . Option No. 2 ( 250yR )
i 1 2 3 4 5
iX / iY 5.5/1.5 5.5/2.1 5.5/3.4 5.5/4.2 5.5/4.9
ix / iy 4.28/4.01 9.64/12.67 18.84/40.07 29.14/76.55 38.07/116.67
x
ip / y
ip 0.52/0.47 0.49/0.5 0.43/0.56 0.33/0.66 0.21/0.78
xF / yF 10.87/10.24
Calculated value 3.842 .
Fig. 7. Optimal solutions of players Fig. 8. Probabilities of players’ winnings
The third case.
T a b l e 5 . Option No. 1 ( 350yR )
i 1 2 3 4 5
iX / iY 1.1/1.5 1.1/2.1 1.1/3.4 1.1/4.2 1.1/4.9
ix / iy 4.06/5.37 9.93/18.38 20.34/60.94 29.33/108.53 36.31/156.75
x
ip / y
ip 0.51/0.49 0.47/0.52 0.39/0.6 0.28/0.71 0.15/0.84
xF / yF 1.95/10.94
S.A. Smirnov, I.M. Tereshchenko
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 100
Calculated value 0.967 .
Fig. 9. Optimal solutions of players Fig. 10. Probabilities of players’ winnings
T a b l e 6 . Option No. 2 ( 350yR )
i 1 2 3 4 5
iX / iY 5.5/1.5 5.5/2.1 5.5/3.4 5.5/4.2 5.5/4.9
ix / iy 4.77/6.34 10.42/19.4 19.5/58.78 29.19/108.67 36.1/156.78
x
ip / y
ip 0.51/0.48 0.47/0.52 0.38/0.61 0.28/0.71 0.15/0.84
xF / yF 9.83/10.97
Calculated value 5.689 .
Fig. 11. Optimal solutions of players Fig. 12. Probabilities of players’ winnings
Let us analyze the first case. From Table 1 it can be seen that the optimal
values of the vector components *x are greater than the corresponding vector
components *y at the first and second sites. At the same time, as follows from
Fig. 1 and Fig. 2, despite the lack of advantage of the first player on the third
court, the probability of his victory on the indicated court is almost equal to this
value for the opponent. If the values of all the first player’s sites increase fivefold,
the overall picture presented in Table 2, Fig. 3 and Fig. 4 remains the same.
Changes occur for the parameter .
A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints
Системні дослідження та інформаційні технології, 2026, № 1 101
Let’s move on to the second case. The opponent increases his resources by
100. As can be seen from Table 3 and Fig. 5, the first player has a small ad-
vantage on the first site among the components of the optimal solution. A similar
situation persists in the probability of winnings. However, as shown in Fig. 6, on
the second court the probability of winning for the second player is slightly higher
than that for the first. Increasing the value of each site fivefold, as follows from
Table 4, Fig. 7 and Fig. 8, again does not change the situation as a whole, except
for the value of the parameter . In general, the increase in the opponent’s re-
sources led to the loss of one site where the first player had previously won.
In the third case, we will again increase the opponent’s resources by 100. As
follows from Table 5, Fig. 9, Fig. 10 and Table 6, Fig. 11, Fig. 12, the compo-
nents of the optimal vector of the first player are smaller than similar components
of the second, but the probability of winning on the first and slightly smaller on
the second platforms for the first player remains higher. Thus, the overall picture
remains the same compared to the second case. It is clear that with a further in-
crease in the opponent’s resources, his winning probabilities will exceed those of
the first player. Hence, the problem of the ratio of players’ resources when the
opponent’s winning probabilities become larger on all platforms is of interest.
CONCLUSIONS
The case of a probabilistic model for two players is considered. A precise solution
for the asymmetric Colonel Blotto game without the homogeneity constraint is
found. The optimal allocation of players’ resources is obtained in the form of a
Nash equilibrium in pure strategies of the game.
A method for reducing the dimensionality of a system of equations obtained
using the Lagrange multiplier method is proposed.
Of further interest is the study of solutions under conditions of incomplete
information, that is, situations where the values of the coefficients are not precise-
ly known.
REFERENCES
1. E. Borel, “La théorie du jeu les équations intégrales á noyau symétrique,” Comptes
Rendus de l’Académie, vol. 173. pp. 1304–1308, 1921.
2. Enric Boix-Adserà, Benjamin L. Edelman, Siddhartha Jayanti, The Multiplayer Colo-
nel Blotto Game. doi: https://doi.org/10.48550/arXiv.2002.05240
3. L. Friedman, “Game-theory Models in the Allocation of Advertising Expenditure,”
Operations Research, vol. 6, pp. 699–709, 1958.
4. R.W. Robson, Multi-Item Contest; Working Paper No. 446. Australian National University,
2005, 27 p. Available: https://www.researchgate.net/publication/4980074_Multi-Item_Contests
5. B. Roberson, “The Colonel Blotto Game,” Economic Theory, vol. 29, pp. 1–24,
2006. doi: https://doi.org/10.1007/s00199-005-0071-5
6. S. Smirnov, O. Glushchenko, K. Ilchuk, I. Makeenko, N. Oriekhova, “Assignments of
factors levels for design of experiments with resource constraints,” Continuous and Dis-
tributed Systems. Theory and Applications. Ser. Solid Mechanics and Its Applications, vol.
211. Springer, 2014. doi: https://doi.org/10.1007/978-3-319-03146-0_6
S.A. Smirnov, I.M. Tereshchenko
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 102
Received 26.04.2024
INFORMATION ON THE ARTICLE
Ivan M. Tereshchenko, ORCID: 0000-0003-0823-7507, National Technical University
of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail:
ivan78ter@gmail.com
Sergey A. Smirnov, ORCID: 0000-0002-7233-5315, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: sergsmr@gmail.com
ЙМОВІРНІСНА МОДЕЛЬ ГРИ ПОЛКОВНИКА БЛОТТО БЕЗ ОБМЕЖЕНЬ
СИМЕТРИЧНОСТІ ТА ОДНОРІДНОСТІ / С.А. Смирнов, І.М. Терещенко
Анотація. Розглянуто класичну гру полковника Блотто для двох гравців. Дос-
ліджено ймовірнісну модель вказаної задачі, причому на гру не накладаються
обмеження симетричності та однорідності. З метою пошуку розв’язку знайде-
но спосіб пониження розмірності, оскільки одержана за допомогою методу
множників Лагранжа система рівнянь має велику розмірність. Знайдене спів-
відношення між ресурсами обох гравців, що розподілені по майданчиках, дало
змогу виділити параметр, який визначається співвідношенням множників Лаг-
ранжа з відповідних функцій для обох гравців. Для такого параметра знайдено
інтервальне обмеження, яке він задовольняє, та для його пошуку сформульо-
вано рівняння, яке розв’язується чисельно. Знайдене значення параметру дає
можливість розрахувати окремі множники Лагранжа та отримати оптимальний
розподіл ресурсів гравців у вигляді рівноваги Неша в чистих стратегіях гри.
Досліджено приклад гри за суттєво відмінних умов для гравців.
Ключові слова: конфліктне протиборство, оптимальний розподіл ресурсів,
гра двох осіб, гра полковника Блотто, ймовірнісна модель виграшу, рівнова-
га Неша, ефективність використання ресурсів, 1-параметризація.
|
| id | journaliasakpiua-article-358074 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-04-20T01:00:21Z |
| publishDate | 2026 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/89/a66d30780f752a56288b541780ccc889.pdf |
| spelling | journaliasakpiua-article-3580742026-04-19T21:53:19Z A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності Smirnov, Sergey Tereshchenko, Ivan conflict confrontation optimal allocation of resources two-person game Colonel Blotto’s game probabilistic payoff model efficiency of resource use 1-parametrization Nash equilibrium конфліктне протиборство оптимальний розподіл ресурсів гра двох осіб гра полковника Блотто ймовірнісна модель виграшу рівновага Неша ефективність використання ресурсів 1-параметризація The classic Colonel Blotto game for two players was considered. The probabilistic model of the payoff functions of the specified problem was investigated, and the game conditions are not subject to the restrictions of symmetry and homogeneity. The system of equations obtained using the method of Lagrange multipliers has a large dimension. In order to find a solution, a way to reduce the dimension was found. The found ratio between the resources of both players, distributed over the courts, made it possible to identify a parameter determined by the ratio of Lagrange multipliers from the corresponding functions for both players. For such a parameter, an interval constraint that it satisfies was found, and an equation is formulated to find it, which is solved numerically. The found value of the parameter makes it possible to calculate individual Lagrange multipliers and obtain the optimal distribution of players’ resources in the form of a Nash equilibrium in pure game strategies. An example of a game under significantly different conditions for players was studied. Розглянуто класичну гру полковника Блотто для двох гравців. Досліджено ймовірнісну модель вказаної задачі, причому на гру не накладаються обмеження симетричності та однорідності. З метою пошуку розв’язку знайдено спосіб пониження розмірності, оскільки одержана за допомогою методу множників Лагранжа система рівнянь має велику розмірність. Знайдене співвідношення між ресурсами обох гравців, що розподілені по майданчиках, дало змогу виділити параметр, який визначається співвідношенням множників Лагранжа з відповідних функцій для обох гравців. Для такого параметра знайдено інтервальне обмеження, яке він задовольняє, та для його пошуку сформульовано рівняння, яке розв’язується чисельно. Знайдене значення параметру дає можливість розрахувати окремі множники Лагранжа та отримати оптимальний розподіл ресурсів гравців у вигляді рівноваги Неша в чистих стратегіях гри. Досліджено приклад гри за суттєво відмінних умов для гравців. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2026-03-31 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/358074 10.20535/SRIT.2308-8893.2026.1.06 System research and information technologies; No. 1 (2026); 92-102 Системные исследования и информационные технологии; № 1 (2026); 92-102 Системні дослідження та інформаційні технології; № 1 (2026); 92-102 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/358074/344000 |
| spellingShingle | конфліктне протиборство оптимальний розподіл ресурсів гра двох осіб гра полковника Блотто ймовірнісна модель виграшу рівновага Неша ефективність використання ресурсів 1-параметризація Smirnov, Sergey Tereshchenko, Ivan Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності |
| title | Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності |
| title_alt | A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints |
| title_full | Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності |
| title_fullStr | Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності |
| title_full_unstemmed | Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності |
| title_short | Ймовірнісна модель гри полковника Блотто без обмежень симетричності та однорідності |
| title_sort | ймовірнісна модель гри полковника блотто без обмежень симетричності та однорідності |
| topic | конфліктне протиборство оптимальний розподіл ресурсів гра двох осіб гра полковника Блотто ймовірнісна модель виграшу рівновага Неша ефективність використання ресурсів 1-параметризація |
| topic_facet | conflict confrontation optimal allocation of resources two-person game Colonel Blotto’s game probabilistic payoff model efficiency of resource use 1-parametrization Nash equilibrium конфліктне протиборство оптимальний розподіл ресурсів гра двох осіб гра полковника Блотто ймовірнісна модель виграшу рівновага Неша ефективність використання ресурсів 1-параметризація |
| url | https://journal.iasa.kpi.ua/article/view/358074 |
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