Multi-Step Game of Reserves Management in the Attack-Defense Model

Abstract

The authors describe a multi-step generalization of the “attack-defense” model, defined and studied by Germeier. It is a modification of the Gross’ model. The similar model was proposed by Gorelik for the gasoline production. In the military models the points are usually interpreted as directions and characterize the spatial distribution of defense resources across the width of the defense front. The dynamics of the average number of parties described by the “attack-defense” game can be described by finite-difference Osipov-Lanchester’ equations. Therefore, it would also be interesting to obtain a generalization of Germeyer’s classical model to the dynamic case when the “attack-defense” game is played many times. On this basis, in the present work, a dynamic expansion of the model is constructed in the form of a positional game with opposing interests of the distribution of parties’ reserves with complete information. The authors studied the simplest multi-step extension of the attack-defense model, which consists in the fact that the corresponding game is played repeatedly. Multi-step game with the complete information of the parties’ reserves management was built on this basis. It is assumed that the defense party makes the first move at each step and the attack party became aware about this move. The functional equation for the best guaranteed result of the defense, which is the value of the positional game due to the parties’ adopted sequence of moves was written out. Its analytical solution for a two-step game was obtained and it was shown that it is advantageous for an attack party to enter all reserves simultaneously, as in the classic attack-defense game.