A probabilistic model of the Colonel Blotto game without symmetry and homogeneity constraints

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.