The 'princess and monster' game on an interval

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 zerosum 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 non-trivial by showing that V < 3/2. Using a combination of continuous and discrete mixed strategies for both players, we show that 15/11 V 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.