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.