On the Cookie-Cutter Game: Search and Evasion on a Disc

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).