From: Prophet Steering on 28 Mar 2010 01:23 In Section 3, this problem is generalized to tbe one where the gambler does not persist in the very largest, but rather feels satisfaction provided that his fortune upon stopping is within $m$ units from the largest. More specifically, when the allowance measure is $m$ $($ $m$ is a given positive integer), the gambler seeks an optimal stopping time $\sigma_{m}^{*}$ such that $P_{r} \{X_{\sigma_{m}^{*}}\geq 0\leq n\leq \mathrm{m}\mathrm{a} \mathrm{x}TX_{n}-(m-1)|X_{0}=i\}=\max_{\sigma\in C}P_{r}\{x_{\sigma} \geq 0\leq n\leq T\mathrm{m}\mathrm{a}\mathrm{x}X_{n}-(m-1)|X_{0}=i\}$ . $(1.3)$ Obviously when $m=1$ , this problem is reduced to the one considered in Section 2. It can be shown that there exists an optimal stopping time $\sigma_{m}^{*}$ of the following form; $\sigma_{m}^{*}=\min\{n:X_{n}=0\leq j\leq n\mathrm{m}\mathrm{a} \mathrm{x}X_{j}-(m-1)$ , \$X_{n}