Discrete versus continuous operational calculus in antagonistic stochastic games

Abstract

We consider a class of antagonistic stochastic games in real time between two players A and B, formalized by two marked point processes. The players attack each other at random times with random impacts. Either player can sustain casualties up to a fixed threshold. A player is defeated when its underlying threshold is crossed. Upon that time (referred to as the first passage time), the game is over. We introduce a joint functional of the first passage time, along with the status of each player upon this time. The latter are the cumulative magnitudes of casualties to each player upon the end of the game, obtained in an analytically tractable form. We then use discrete and continuous operational calculus for the transform inversion. We demonstrate in a special case that the discrete operational calculus is more efficient, allowing us to avoid numerical inversion. It leads to explicit formulas for the joint distribution of associated random variables (first passage time and the status of cumulative casualties to the players upon the end of the game).