Differential Game Theory Lady in the Lake numerical approximation
Abstract:
The Lady in the Lake problem in differential game theory has been classically solved via the Isaacs minimax equation. The first half of this paper takes a different approach to give a proof without using the Isaacs Equation. First, we will set up an ODE model using a geometric construction in a straight-forward manner. Then, by using some nice tricks and proved intuitions, we can make the model a lot more workable, hence simplifying the problem from a 2D minimax problem to a 1D minimization problem. Finally, the game is naturally divided into two stages, and this paper solves for the optimal strategy of the lady in each of the two stages using calculus and change of variables. An expression for the terminal payoff is provided in the end as well. The second half of this paper develops a numerical algorithm to look for the fastest optimal strategy of this game, after proving that the optimal strategy is non-unique. The algorithm is a recursive nonlinear programming problem, which yields a numerical approximation of the lady's fastest optimal strategy in stage 1.
