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- Title
ON THE COOKIE-CUTTER GAME: SEARCH AND EVASION ON A DISC.
- Authors
Danskin, John M.
- Abstract
In the cookie-cutter game, there is a trapping circle, of radius 1, in which an evader hides. A searcher has a "cookie-cutter", a disk of radius r < 1. If, when he places the cookie-cutter on the trapping circle, the evader is within it, the evader is caught and the searcher wins. Otherwise the evader wins. If r = √2/2, the problem is trivial. The evader should choose a point from the uniform distribution on the outer circumference of the trapping circle, and the searcher a point from the uniform distribution on the circle of radius r* = (1 - r²)1/2 concentric to that circle; this choice gives him maximum coverage of the outer circumference. For the case r ≧ 1/2, an easy and elegant solution was given by Gale and Glassey in 1974. Both players should go to the center with probability 1/7. The minimizer should go to the outer circumference, and the maximizer to r* = √3/2, both with probability 6/7. For other r the problem is difficult. This paper proves that there are no solutions based on finitely many radii if r < r0 ... 0.476, where r0 solves a cubic equation, finds two-point solutions on [r0, r1], where r1 ... 0.515 solves a trigonometric equation, and proves qualitative facts for r ∈ (r1, √2/2).
- Subjects
PROBABILITY theory; COOKIE cutters; DISTRIBUTION (Probability theory); TRIGONOMETRIC functions; EQUATIONS; UNIFORM distribution (Probability theory)
- Publication
Mathematics of Operations Research, 1990, Vol 15, Issue 4, p573
- ISSN
0364-765X
- Publication type
Article
- DOI
10.1287/moor.15.4.573