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Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{1, 2, \ldots,n\}\ $ and $\ n\ $ are fixed. Let $t$ be a parameter that varies according to the outcome of each roll. Initially $t=1$ and we stop rolling the die when $t=0$.

Each time when the value $x$ taken by $X$ is less than or equal to $\ k\ $ we win $x$ dollars and we decrease $t$ by $1$. Each time when $x$ is larger than $k$, we lose $x$ dollars and we increase $t$ by adding $x-1$ to its current value. We have an unlimited amount of money.

Question: What is the minimum value of $\ k:= k(n)\ $ such that the expected number of dollars obtained (i.e., the difference between the total number of dollars won and the total number of dollars lost) is strictly larger than $0$?

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    $\begingroup$ Is "larger than" a strict inequality? $\endgroup$
    – kodlu
    Commented Dec 16, 2020 at 21:37
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    $\begingroup$ I feel that the q. about Wald should be moved to a comment. $\endgroup$
    – Wlod AA
    Commented Dec 16, 2020 at 22:28
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    $\begingroup$ Is Wald's equation useful in this case? $\endgroup$
    – Let101
    Commented Dec 16, 2020 at 22:40
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    $\begingroup$ If I'm not mistaken, whenever we take $k < n-1$, the counter $t$ has positive drift and there is a positive probability that the game never terminates. What do we mean by "total number of dollars won" in this case? $\endgroup$
    – Nate Eldredge
    Commented Dec 17, 2020 at 4:32
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    $\begingroup$ Wald does seem helpful because this is a random walk $S_m$ stopped at a stopping time $\tau$. However to apply it, we need to have $E[\tau] < \infty$ and as I note above, in most cases we do not even have $\tau < \infty$ a.s. $\endgroup$
    – Nate Eldredge
    Commented Dec 17, 2020 at 4:35

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