Centre for Discrete and Applicable Mathematics

CDAM Research Report, LSE-CDAM-2006-17

November 2006

The `Princess and Monster' Game on an Interval

Steve Alpern, Robbert Fokkink, Roy Lindelauf, and Geert Jan Olsder

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. While the existence and finiteness of the value V has been established for such games, only the circle network has been solved (value and optimal mixed strategies). It seems that the interval network I=[-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

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		Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk