Complete Solution of the Lady in the Lake Scenario
arxiv(2024)
摘要
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.
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