Complete Solution of the Lady in the Lake Scenario

In the Lady in the Lake scenario, a mobile agent, L, is pitted against an agent, M, who is constrained to move along the perimeter of a circle. L is assumed to begin inside the circle and wishes to escape to the perimeter with some finite angular separation from M at the perimeter. This scenario has, in the past, been formulated as a zero-sum differential game wherein L seeks to maximize terminal separation and M 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, L seeks to reach a point opposite of M at a radius for which their maximum angular speeds are equal (i.e., the antipodal point). L wishes to minimize the time to reach this point while M 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 tributaries' equilibrium strategy for L is semi-analytic, while the Universal Line tributaries' equilibrium strategy is obtained in closed form.