On relative effectiveness in duels

We study the relative effectiveness of two multi-unit forces, a and B, fighting a successive series of duels between randomly selected pairs of opposing units. a duel consists of a series of games - each of at most s units of time - and the duel terminates as soon as one of the two duelists is hit. a new independent duel then starts, and so on. Each force is characterized by a common probability distribution function representing the time required by a single unit of the force to hit a non-firing unit of the other force in a game of unbounded duration. the relative effectiveness of force a with respect to force B is expressed by the exchange rate, R(s), defined as the expected number of units of force B hit by a single unit of force a until unit a itself is hit. We derive a general characterization of R(s) and develop analytic results for several choices of pairs of families of time-to-hit distributions. We show that the shape of the distributions can strongly affect the outcome of the duels and that this effect changes with the duration of the duel. We further show that duels are not necessarily transitive. Some Numerical results are presented.