赌博的格子
A Lattice of Gambles
摘要：
一个赌徒带着一张非常真实的美元钞票走进一个假设的公平赌场，但当他离开时，他已经把美元换成了随机的钱。在这个过程中失去了什么？可能是赌徒在一天结束的时候，在经历了过山车般的输赢之后，他的一美元仍然完好无损，甚至可能是两美元。但是赌徒在下第一注的时候输的是位置。他用一种货币分配交换一种质量较低的分配，他不能从中返回。我们在这项工作中的第一个讨论连接已知的结果，经济不平等和优势的概率论赌博和鞅。我们提供了一个简单的证明，公平赌博不能增加洛伦兹曲线，我们还建设性地证明，任何序列的非增加洛伦兹曲线对应于至少一个鞅。接下来，我们考虑赌博的效率。如果所有的公平赌博都是可用的，那么我们可以沿着洛伦兹序定义的分布格向下移动。然而，从一个分布到下一个分布的步骤并不是唯一的。有没有一种效率感，让人可以沿着洛伦兹流往下走？一种方法是尽量减少摆在桌面上的平均资金总量。在这种情况下，事实证明，使用私人随机性实施部分策略可以帮助减少对赌场随机性的需求，从而减少赌场无法获得的桌面上的钱。
英文摘要：
A gambler walks into a hypothetical fair casino with a very real dollar bill, but by the time he leaves he's exchanged the dollar for a random amount of money. What is lost in the process? It may be that the gambler walks out at the end of the day, after a roller-coaster ride of winning and losing, with his dollar still intact, or maybe even with two dollars. But what the gambler loses the moment he places his first bet is position. He exchanges one distribution of money for a distribution of lesser quality, from which he cannot return. Our first discussion in this work connects known results of economic inequality and majorization to the probability theory of gambling and Martingales. We provide a simple proof that fair gambles cannot increase the Lorenz curve, and we also constructively demonstrate that any sequence of non-increasing Lorenz curves corresponds to at least one Martingale. We next consider the efficiency of gambles. If all fair gambles are available then one can move down the lattice of distributions defined by the Lorenz ordering. However, the step from one distribution to the next is not unique. Is there a sense of efficiency with which one can move down the Lorenz stream? One approach would be to minimize the average total volume of money placed on the table. In this case, it turns out that implementing part of the strategy using private randomness can help reduce the need for the casino's randomness, resulting in less money on the table that the casino cannot get its hands on.
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。
