Minimizing the distance to one evader while chasing another

To approach a simple game Δ2 of P and E = {E1, E2} with no a priori evadersʹ role assignment and the payoff equal to the distance to one evader at an instant of catching another, we introduce a concept of casting and study the games Δ1,2 and Δ2,1 for preassigned and Δp2 for open-loop casting procedures. Since Δp2 is reduced to Δ1,2 or Δ2,1 which, in turn, are distinguished only by their notations, we focus attention mainly on Δ1,2. According to the tenet of transition, Δ1,2 is divided into a concatenation of Δ1,2b (basic) and Δ1,2a (auxiliary) games that model the problem before and after the first instant of E1 capture. The games Δ1,2a, Δ1,2b, Δ1,2 are studied one after another with use of the Isaacsʹ approach extended by Berkowitz, Breakwell, Bernhard et al.