OK, thank you a lot @IosifPinelis !

Thank you very much for your answer @IosifPinelis. By the way, I meant that $p$ is given and fixed in the problem (say $p=9/10$). I guess that there is not closed form solution working for any given $p\in(1/2,1)$, right?

I was also wondering how does this problem change if, whenever $X\le k$, we decrease $t$ by $x+1$ (instead of by $1$), and the stopping condition is $t\le 0$? I guess in this case the "symmetry" of the problem could make it easier to solve.

Thank you @NateEldredge for your comments. I see. Let $T$ be the random variable equal to total number of rolls (taking value $\tau\in[1,\infty]$), and let $D(r)~~ (:=D_{k,n}(r))$ be the total number of dollars won at round $r$. $D(r)<0$ means that we are loosing money at round $r$. Instead of asking for $\mathbb{E}[D(T)]$ as in the original problem version, would it be more meaningful to ask for $\mathbb{E}[\lim\sup_{r\to T}D(r)]$ and $\mathbb{E}[\lim\inf_{r\to T}D(r)]$?

Is Wald's equation useful in this case?

Yes @kodlu . The meaning basically is "When is convenient to play this game? (without loosing time)" Edited.

@TimothyChow this is something I was thinking about. I will add it when I will find it. Thank you for your message. By the way, do you think that the problem becomes more interesting if each player can always distinguish between an attempt to occupy the same location chosen by another player during the same round, and a location which was already occupied in the previous rounds?