Rendezvous on a planar lattice

We analyze the optimal behavior of two players who are lost on a planar surface and who want to meet each other in least expected time. They each know the initial distribution of the other's location, but have no common labeling of points, and so cannot simply go to a location agreed to in advance. They have no compasses, so do not even have a common notion of North. For simplicity, we restrict their motions to the integer lattice Z2 (graph paper) and their motions to horizontal and vertical directions, as in the original work of Anderson and Fekete.