Optimal Symmetric Rendezvous Search on Three Locations

In the symmetric rendezvous search game played on n locations two players are initially placed at two distinct locations. The game is played in discrete steps, at each of which each player can either stay where he is or move to a different location. The players share no common labelling of the locations. We wish to find a strategy such that, if both players follow it independently, then the expected number of steps until they are in the same location is minimized. Informal versions of the rendezvous problem have received considerable attention in the popular press. The problem was proposed by Alpern in 1976 [Alpern, S. 1976. Hide and seek games. Seminar at the Institut fur Hohere Studien, Vienna], and it has proved notoriously difficult to analyse. In this paper we prove a 20-year-old conjecture that the following strategy is optimal for the game on three locations: in each block of two steps, stay where you are, or search the other two locations in random order, doing these with probabilities 1/3 and 2/3, respectively. This is now the first nontrivial symmetric rendezvous game of any type to be fully solved.