Rendezvous-Evasion as a Multistage Game with Observed Actions

Abstract

We consider a rendezvous-evasion zero-sum game that examines how two agents of team R can optimally rendezvous while evading an enemy searcher S. We investigate the game in a discrete framework where the search region consists of n identical locations along a directed cycle D known to all players. The agents R ₁ R ₂, and the searcher S start at distinct locations and, at each integer time, they can move to any one of the other locations or stay still. The game ends when some location is occupied by more than one player. If S is at this location, S (maximizer) wins and the payoff is 1; otherwise R (minimizer) wins and the payoff is 0. The value of the game v _n is the probability that S wins under optimal play. We model the rendezvous-evasion problem as a multistage game where all players are obliged to announce their actions truthfully at the end of each step. We also assume that the agents R ₁ and R ₂ can jointly randomize their strategies. We prove that we need only consider a special class of strategies for R and show that ${\upsilon _3} = \frac{5}{9} \approx 0.55556$ and ${\upsilon _4} = \frac{{17}}{{32}} \approx 0.53125$. We also prove that if the players have no common knowledge of D, the value of the game is $\frac{{47}}{{76}} \approx 0.61842$ when there are three locations.