# Rendezvous on the Line with Different Speeds and Markers That Can Be Dropped at Chosen Time
## 6 Solution of the Problem with Marker Held by the Fast Player  
The family of linear programs to be solved when the marker is held by the fast player (I) is very similar to the one defined by Equations (15)-(17). The changes are that the  
Figure 5: Optimal strategy when player I holds the marker and is the slower player.  
Figure 6: Optimal solutions with (18) and without marker. The slow player holds the marker.  
coefficients \(a_{i}\) are no longer multiplied by the maximal speed \(v\), as are now the coefficients \(d_{i}\). The system of equations is not reproduced here to save some space.  
The results are plotted in Figure 7. We observe that for speeds slower than \(v\approx 0.805\) the marker is not useful and the optimal solution is given by the optimal solutions without marker stated in Theorem 1. For speeds faster than \(v\approx 0.805\), the marker starts to be useful and the strategy is similar to the optimal one when the marker is held by the slow player, see Figure 5. When the fast player holds the marker, we obtain that the rendezvous times are given by:  
\[\begin{split} z&=\frac{1}{3v+1},\ t_{1}=\frac{3}{3v +1},\ t_{2}=\frac{3v+5}{(v+1)(3v+1)},\\ t_{3}&=\frac{9v+5}{(v+1)(3v+1)},\ t_{4}=\frac{3v+7} {(v+1)^{2}},\\ \end{split} \tag{19}\]  
\[\begin{split} R&=\frac{24v^{2}+52v+20}{(v+1)^{2}(3v +1)}.\\ \end{split}\]  
The optimal rendezvous value is decreasing with the maximal speed \(v\) because a strategy for maximal speed \(v\) is a strategy for maximal speed \(v^{\prime}\geq v\). Moreover, if we denote \(opt(v)\) the optimal rendezvous value for maximal speed \(v\), and \(nextToopt(v)\) the next-to-optimal rendezvous value, it follows that if \(nextToopt(v+dv)\geq opt(v)\) and the strategy leading to \(opt(v)\) and \(opt(v+dv)\) is the same, then the strategy is optimal on the entire interval \([v,v+dv]\). By numerical computation and using the two stated observations we obtain the following Theorem.  
Theorem 3.: _For \(v\in[1/1000,0.618]\) the rendezvous value of the game is given by (13) and the optimal strategy is plotted in Figure 3; for \(v\in[0.619,0.805]\), the rendezvous value of the game is given by (14) and the optimal strategy is plotted in Figure 4; for \(v\in[0.807,0.966]\), the rendezvous value of the game is given by (19), the rendezvous values are plotted in Figure 7 and the optimal strategy is plotted in Figure 8._  
Figure 7: Optimal solutions with marker, i.e., (13) for \(v\leq(\sqrt{5}-1)/2\), (14) for \(v\geq~{}(\sqrt{5}-1)/2\), (19) for \(v>0.805\), and without marker. The fast player holds the marker.  
It is relevant to point out the difference between the optimal strategies when the marker is held by the slow (Figure 5) or fast player (Figure 8). After the rendezvous time \(t_{2}\) in Figure 5 the slow player (who holds the marker) turns, while in Figure 8, the fast player (who holds the marker) continues on their way. The transition from the two strategies is \"continuous\" in the sense that when the speeds are equal at time \(t_{2}\), the two remaining agents to be found are at equal distance from player I. Hence, both strategies are optimal (turning or continuing).