The Anderson-Weber strategy is not optimal for symmetric rendezvous search on K4

We consider the symmetric rendezvous search game on a complete graph of n locations. In 1990, Anderson and Weber proposed a strategy in which, over successive blocks of n−1 steps, the players independently choose either to stay at their initial location or to tour the other n− 1 locations, with probabilities p and 1− p, respectively. Their strategy has been proved optimal for n = 2 with p = 1/2, and for n = 3 with p = 1/3. The proof for n = 3 is very complicated and it has been difficult to guess what might be true for n > 3. Anderson and Weber suspected that their strategy might not be optimal for n > 3, but they had no particular reason to believe this and no one has been able to find anything better. This paper describes a strategy that is better than Anderson–Weber for n = 4. However, it is better by only a tiny fraction of a percent. 1 The Anderson–Weber strategy In the symmetric rendezvous search game on Kn (the completely connected graph on n vertices) two players are initially placed at two distinct vertices (called locations). The game is played in discrete steps and at each step each player can either stay where he is or move to a different location. The players share no common labelling of the locations. Our aim is to find a (randomizing) strategy such that if both players independently follow this strategy then they minimize the expected number of steps until they first meet. Rendezvous search games of this type were first proposed by Steve Alpern in 1976. They are simple to describe, and have received considerable attention in the popular press as they model problems that are familiar in real life. They are notoriously difficult to analyse. The Anderson–Weber strategy is a mixed strategy that proceeds in blocks of n− 1 steps. Players begin at distinct locations, called their home locations. In each successive block a player either stays at his home location, with probability p, or makes a randomly chosen tour of his n − 1 non-home locations, doing this with probability 1 − p. The motivation for the strategy comes from the wait-for-mommy strategy that is optimal in an asymmetric version of the problem. With probability 2p(1 − p) the players play the wait-for-mommy strategy over the first n− 1 steps and so rendezvous in expected time (n+ 1)/2. Anderson and Weber (1990) proved that the above strategy is optimal for the game on K2, with p = 1/2, and conjectured that it should be optimal for K3, with p = 1/3. This Statistical Laboratory, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB2 0WB, [email protected]