Stochastic processes in a discrete model of ground combat

Discrete models of combat are a rare part of the combat modelling literature. Our work introduces a stochastic version of a discrete ground combat based on Epstein theory featuring two adversarial sides, namely an attacker and a defender. Noticeably, the Epstein model of ground combat features an evolving battle front through a withdrawal mechanism to capture the connection between attrition and movement of the front historically prevalent in ground war. Our extension from the deterministic setting of the Epstein model to the stochastic setting is achieved by taking the exchange ratio of attackers lost to defenders to be a mean-reverting stochastic process. The extension of the exchange ratio to a stochastic process is interrupted to be the result of changing strategies and engagements by either side as well of the generally uncertainty of warfare known as the Fog of War upon the outcome of combat. In the deterministic setting of our model, our toy numerical example results in an attacker victory. In the extension of the exchange ratio to a stochastic process, the attackers are no longer assured victory. However, the variations in the exchange ratio can be of benefit to the attackers in that they may achieve victory in a shorter combat duration and as a consequence suffer less attrition. Thus we interpret the stochastic process as introducing a risk vs reward scenario for the attackers where the risk is quantified through the volatility of the process. Our numerical simulations explore the shift in the outcome of combat for the attackers as they take on additional risk and more uncertainty is introduced into combat. We observe the probability that the attacker is victorious, the time till victory when the attacker is victorious, and the remaining ground force strength of the attacking forces for varying volatility. Our results show that for increasing values of the volatility of the exchange ratio process, the probability of an attacker victory increases but the combat duration decreases and the remaining combat power of the attacker forces increases