Markov Duels

Markov duels are a general class of stochastic duels in which each weapon has Markov-dependent fire, that is, the outcomes of shots by each weapon form a Markov process. This paper develops duel models for the situation in which the outcomes form a finite stationary Markov chain and both weapons have an unlimited supply of ammunition, fire at constant intervals of time, and duel until one is killed. Based on these assumptions, the probability of a given side winning the duel is obtained for two sets of starting conditions: (1) both weapons begin with unloaded weapons and have tactical equity, and (2) one weapon has the advantage of surprise and can fire y rounds at the other before the two-sided duel begins, where y is a random variable with a geometric distribution. The mean and variance of the number of rounds to kill a passive target are also derived and two example duels are solved. Finally, methods are indicated for obtaining the solution to a Markov duel between weapons having exponential firing times and either fixed, limited ammunition supplies or infinite supplies.