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Complete Solution of the Lady in the Lake Scenario

Alexander Von Moll, Meir Pachter

TL;DR

This paper provides a unique strategy by solving an auxiliary zero-sum differential game based on the Lady in the Lake scenario where a mobile agent is pitted against an agent who is constrained to move along the perimeter of a circle.

Abstract

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.

Complete Solution of the Lady in the Lake Scenario

TL;DR

This paper provides a unique strategy by solving an auxiliary zero-sum differential game based on the Lady in the Lake scenario where a mobile agent is pitted against an agent who is constrained to move along the perimeter of a circle.

Abstract

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.
Paper Structure (12 sections, 11 theorems, 68 equations, 3 figures)

This paper contains 12 sections, 11 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Classical Lady in the Lake Scenario
  3. Min-Max Time to Reach the Antipodal Point
  4. Focal Line
  5. Equilibrium Heading for FL Tributaries
  6. Equilibrium Flowfield
  7. Computation of the FL Entry Point
  8. Universal Line
  9. Equilibrium Heading for UL Tributaries
  10. Equilibrium Flowfield
  11. Full Solution
  12. Conclusion

Key Result

Proposition 1

There is a Focal Line (FL) given by wherein $L$'s equilibrium control keeps the state of the state of the system on the line $θ = π$ (i.e., she chooses the heading, $ψ$, s.t. $\dot{θ} = 0$): and $M$'s equilibrium control is

Figures (3)

  • Figure 1: Equilibrium flowfield for the classical Lady in the Lake differential game for $μ = 0.3$.
  • Figure 2: Equilibrium trajectories of the complete Lady in the Lake game with $μ = 0.3$.
  • Figure 3: Focal Line trajectories starting from the tributaries in the non-rotating Cartesian coordinate system. In (a), $L$ initially heads towards the tangent of the circle of radius $\frac{s^2}{μ}$ (Case 1), while in (b) $L$ only heads away from the tangent (Case 2). Open markers indicate initial positions, triangles designate positions at the moment the FL is reached, and closed markers indicate terminal positions.

Theorems & Definitions (21)

  • Proposition 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 11 more