5.3 Duels (Optional)
Duels are used to model not only the actual dueling situation but also many problems in other fields. For example, a battle between two companies for control of a third company or asset can be regarded as a duel in which the accuracy functions could represent the probability of success. Duels can be used to model competitive auctions between two bidders. So there is ample motivation to study a theory of duels.
In earlier chapters we considered discrete versions of a duel in which the players were allowed to fire only at certain distances. In reality, a player can shoot at any distance (or time) once the duel begins. That was only one of our simplifications. The theory of duels includes multiple bullets, machine gun duels, silent and noisy, and so on.18
Here are the precise rules that we use here. There are two participants, I and II, each with a gun, and each has exactly one bullet. They will fire their guns at the opponent at a moment of their own choosing. The players each have functions representing their accuracy or probability of killing the opponent, say, pI(x) for player I and pII(y) for player II, with x, y [0, 1]. The choice of strategies is a time in [0, 1] at which to shoot. Assume that pI(0) = pII(0) = 0 and pI(1) = pII(1) = 1. So, in the setup here you may assume that they are farthest apart at time 0 or x = y = 0 and closest together when x = ...