关于Labouchere系统最大赌注的期望值
On the Expected Value of the Maximal Bet in the Labouchere System
摘要：
考虑一个由独立回合组成的游戏，每个回合的收益都是偶数，参与者以固定的概率$p \geq 1/3$获胜，以概率$1 - p$失败。Labouchere系统是一种投注策略，它需要保持一个正实数列表，并在每一轮投注列表上的第一个和最后一个数字的总和。在获胜的情况下，这两个数字将从列表中删除，并且，在失败的情况下，投注金额将附加到列表的末尾。当列表变为空时，玩家结束游戏。众所周知，在一个有$p \leq 1/2$的Labouchere系统的博弈中，赌注和最大赤字都有无穷大的期望。Grimmett和Stirzaker提出了一个问题，即最大赌注是否也是如此。本文证明了$p > c$的最大赌注的期望是有限的，其中$c \approx 0.613763$解决了$(c-1)^2 c = \frac{8}{27(1+\sqrt{5})}$，从而部分回答了这个问题。
英文摘要：
Consider a game consisting of independent turns with even money payoffs in which the player wins with a fixed probability $p \geq 1/3$ and loses with probability $1 - p$. The Labouchere system is a betting strategy which entails keeping a list of positive real numbers and betting the sum of the first and the last number on the list at every turn. In case of a victory, those two numbers are erased from the list, and, in case of a loss, the bet amount is appended to the end of the list. The player finishes the game when the list becomes empty. It is known that, in a game played with the Labouchere system with $p \leq 1/2$, both the sum of the bets and the maximal deficit have infinite expectation. Grimmett and Stirzaker raised the question of whether the same is true for the maximal bet. In this paper we show the expectation of the maximal bet is finite for $p > c$, where $c \approx 0.613763$ solves $(c-1)^2 c = \frac{8}{27(1+\sqrt{5})}$, thus partially answering this question.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
