Rendezvous on the Line with Different Speeds and Markers that can be Dropped at Chosen Time

TL;DR

This paper models a rendezvous game on an infinite line with markers and different speeds, revealing optimal strategies and the impact of markers and waiting tactics through an LP-based approach.

Contribution

Introduces an LP formulation for a rendezvous game with markers and analyzes optimal strategies considering different player speeds and marker usage.

Findings

Waiting can be optimal for slow-moving players.

Marker utility depends on which player holds it and their speed.

Strategies differ significantly based on player roles and marker possession.

Abstract

In this paper we introduce a Linear Program (LP) based formulation of a Rendezvous game with markers on the infinite line and solve it. In this game one player moves at unit speed while the second player moves at a speed bounded by vmax smaller than 1. We observe that in this setting a slow moving player may have interest to rest still instead of moving. This shows that in some conditions the wait-for-mummy strategy is optimal. We observe as well that the strategies are completely different if the player that holds the marker is the fast or slow one. Interestingly, the marker is not useful when the player without marker moves slowly, i.e. the fast moving player holds the marker.