Mathematical models of the battle between parties for electorate or between companies for markets

A model of optimization of strategies for any number of political parties (companies) in election (advertising) campaigns is proposed. We formulated the problem of choosing an optimal strategy for allocating scarce resources among regions (markets) in which political parties (companies) have different influences. The optimization model is analyzed and the existence of a unique solution is proved for different functions of dependencies between investments and winning. An analytical solution to the model is found for the case of two parties (companies). A direction for further development is the study of the sensitivity of an optimal solution to model parameters and formation of a stochastic model.     

Analysis of Matrix-Convex Functions

The properties of matrix-convex functions, which may take values from an expanded numerical axis, are analyzed. Matrix convexity is specified by a pair of matrices, a final set of matrices, or an infinite family of matrices. The results obtained are used in the theory of differential games.     

Operator Constructions and Volterra Mappings in Differential Games

It is proved that B. N. Pshenichnyi's -strategies and strategies based on Volterra mappings are equivalent. It is shown that Volterra mapping sought for is a boundary for some directednesses of prelimit mappings.     

Invariant Sets in the Discrete Game of Retention

Differential games of retention are considered. Problems in a new game formulation under full sweeping condition are generalized. The problem of constructing minimum and maximum invariant sets of dynamic systems is analyzed. Two diametrically opposite examples are considered depending on the matrix of the discrete linear equation.__________Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 150–154, March–April 2005.     

Linear pursuit-and-evasion games with impulse evader control

Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 120–126, July–August, 1995.