The “Princess and Monster” Game on an Interval

Abstract

A minimizing searcher S and a maximizing hider H move at unit speed on a closed interval until the first (capture, or payoff) time $T=\min \{ t:S(t)=H(t)\}$ that they meet. This zero-sum princess and monster game or less colorfully search game with mobile hider was proposed by Rufus Isaacs for general networks $Q.$ While the existence and finiteness of the value $V=V(Q)$ has been established for such games, only the circle network has been solved (value and optimal mixed strategies). It seems that the interval network $Q=[-1,1]$ had not been studied because it was assumed to be trivial, with value $3/2$ and “obvious” searcher mixed strategy going equiprobably from one end to the other. We establish that this game is in fact nontrivial by showing that $V<3/2.$ Using a combination of continuous and discrete mixed strategies for both players, we show that $15/11\leq V\leq 13/9.$ The full solution of this very simple game is still open and appears difficult, though many properties of the optimal strategies are derived here.