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 zerosum 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.

Download paper here