Rendezvous Search on the Line with Indistinguishable Players

• Authors: Edward J. Anderson;Skander Essegaier
• Venue: SIAM Journal on Control and Optimization
• Year: 1995

Alpern introduced a problem in which two players are placed on the real line at a distance drawn from a bounded distribution $F$ known to both. They can move at maximum velocity one and wish to meet as soon as possible. Neither knows the direction of the other, nor do they have a common notion of a positive direction on the line. It is required to find the symmetric rendezvous value $R^{s}(F)$, which is the minimum expected meeting time achievable by players using the same mixed strategy. This corresponds to the case where the players are indistinguishable they both take directions from a controller who does not know their names. In this paper we give a mixed strategy which has an expected meeting time of $1.78D+\mu /2$, where $D$ is the maximum of $F$ and $\mu$ its mean. This leads to an upper bound $R^{s}(F)\le 1.78D+\mu /2$ on the symmetric rendezvous value, which is better than the upper bound $R^s(F)\le 2D+\mu /2$ obtained by Alpern.