Faster Symmetric Rendezvous on Four or More Locations

TL;DR

This paper demonstrates that the classic Anderson-Weber strategy for symmetric rendezvous is suboptimal for four or more locations, by constructing strategies with lower expected meeting times.

Contribution

The authors prove the Anderson-Weber strategy is not optimal for any number of locations greater than or equal to four, introducing improved strategies.

Findings

Anderson-Weber strategy is not optimal for n ≥ 4

Constructed strategies outperform the classic approach for these cases

Expected meeting times are reduced with the new strategies

Abstract

In the symmetric rendezvous problem two players follow the same (randomized) strategy to visit one of $n$ locations in each time step $t=0,1,2,\dots$. Their goal is to minimize the expected time until they visit the same location and thus meet. Anderson and Weber [J. Appl. Prob., 1990] proposed a strategy that operates in rounds of $n-1$ steps: a player either remains in one location for $n-1$ steps or visits the other $n-1$ locations in random order; the choice between these two options is made with a probability that depends only on $n$. The strategy is known to be optimal for $n=2$ and $n=3$, and there is convincing evidence that it is not optimal for $n=4$. We show that it is not optimal for any $n\geq 4$, by constructing a strategy with a smaller expected meeting time.