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

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Complete Solution of the Lady in the Lake Scenario

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Alexander Von Moll,
  • Meir Pachter

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.