Rabbit and hunter game: Two discrete stochastic formulations
This work concerns different versions of the same basic game. A rabbit R can move back and forth along a finite wall in a discrete world \(\{\) 1, 2,..., \(N\}=I_ N\). If \(x_ t\in I_ N\) is the position of R at time t, and \(u_ t\) its jump at time t, the dynamics of R are simply \(x_{t+1}=x_ t+u_ t\). A Hunter H tries to shoot it at each discrete time. If \(v_ t\in I_ N\) is the position at which H aims at time t the two first versions suppose that the bullet reaches the wall in one step of time, so a bullet hits the wall at \(Z_ t\) at time t with \(Z_{t+1}=v_ t\). Capture is defined by \(t_ 1=\inf (t|\) \(x_ t=Z_ t)\). The third version assumes that the bullet takes several time steps to reach the wall. Mixed strategies for H and R are introduced and so \(t_ 1\) is a stopping time. In the first model the payoff that H (resp. R) looks for minimize (resp. maximize) is \(E(t_ 1)\). In the second and third models a fixed time T is given when the game warden forces the hunter to leave, and the payoff is then \(P(t_ 1\leq T)\) that of course H (resp. R) tries to maximize (resp. minimize). In these three models, the authors prove the existence of value and saddle points, and give algorithms to calculate them.
scientific article; zbMATH DE number 4012366