A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints
DOI:
https://doi.org/10.20535/SRIT.2308-8893.2026.1.06Keywords:
conflict confrontation, optimal allocation of resources, two-person game, Colonel Blotto’s game, probabilistic payoff model, efficiency of resource use, 1-parametrization, Nash equilibriumAbstract
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.
References
E. Borel, “La théorie du jeu les équations intégrales á noyau symétrique,” Comptes Rendus de l’Académie, vol. 173. pp. 1304–1308, 1921.
Enric Boix-Adserà, Benjamin L. Edelman, Siddhartha Jayanti, The Multiplayer Colonel Blotto Game. doi: https://doi.org/10.48550/arXiv.2002.05240
L. Friedman, “Game-theory Models in the Allocation of Advertising Expenditure,” Op-erations Research, vol. 6, pp. 699–709, 1958.
R.W. Robson, Multi-Item Contest; Working Paper No. 446. Australian National Uni-versity, 2005, 27 p. Available: https://www.researchgate.net/publication/4980074_Multi-Item_Contests
B. Roberson, “The Colonel Blotto Game,” Economic Theory, vol. 29, pp. 1–24, 2006. doi: https://doi.org/10.1007/s00199-005-0071-5
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
