Complete Solution of the Lady in the Lake Scenario
A. Von Moll, M. Pachter
Published in Dynamic Games and Applications, 2024
In the Lady in the Lake scenario, a mobile agent, ๐ฟ, is pitted against an agent, ๐, who is constrained to move along the perimeter of a circle. ๐ฟ is assumed to begin inside the circle and wishes to escape to the perimeter with some finite angular separation from ๐ at the perimeter. This scenario has, in the past, been formulated as a zerosum differential game wherein ๐ฟ seeks to maximize terminal separation and ๐ seeks to minimize it. Its solution is well-known. However, there is a large portion of the state space for which the canonical solution does not yield a unique equilibrium strategy. This paper provides such a unique strategy by solving an auxiliary zero-sum differential game. In the auxiliary differential game, ๐ฟ seeks to reach a point opposite of ๐ at a radius for which their maximum angular speeds are equal (i.e., the antipodal point). ๐ฟ wishes to minimize the time to reach this point while ๐ wishes to maximize it. The solution of the auxiliary differential game is comprised of a Focal Line, a Universal Line, and their tributaries. The Focal Line tributariesโ equilibrium strategy for ๐ฟ is semi-analytic, while the Universal Line tributariesโ equilibrium strategy is obtained in closed form.