Rate of Loss Characterization That Resolves the Dilemma of the Wall Pursuit Game Solution

The scope of this article is the well-known wall pursuit game, which has been used in the literature to illustrate the existence of a singular surface (dispersal line) and the associated game dilemma. We derive an analytical expression for the value function of the game, which is the viscosity solution of the Hamilton–Jacobi–Isaacs equation. Then, we introduce a hold time analysis and the rate of change for the loss of time to capture along the dispersal line, and show that the rate has a well-defined saddle point along the dispersal line, which can be used to resolve the dilemma. Moreover, we prove that the saddle point of the rate characterizes optimal game actions not only on the dispersal line, but also for all other states of the game. Finally, we analyze the same game in a version with a nonzero hold time and show that in that case, the actions from the dispersal line have to be applied both on the dispersal line and in a narrow band around it. To illustrate that, we use an example to compute the band around the line.