论文标题

Anna Karenina和两个信封问题

Anna Karenina and The Two Envelopes Problem

论文作者

Gill, R. D.

论文摘要

安娜·卡雷尼娜(Anna Karenina)的原则是以同名小说中的开头句子命名的:幸福的家庭都是一样的;每个不快乐的家庭都以自己的方式不高兴。两个信封问题(TEP)是概率理论,数学经济学,逻辑和哲学的一个备受研究的悖论。一次又一次地发表了一项新的分析,其中作者声称最终解释了这种悖论中实际出现的问题。每个作者(包括本作者)都强调了他们的方法的新事物,并得出结论,早期的方法没有达到问题的根源。我们观察到,虽然逻辑上的论点只有在每个步骤都是正确的情况下才是正确的,但是可以将误入歧途的逻辑论点视为在不同的地方误入歧途。这导致了关于TEP的文献与成功电影系列的文献之间的比较:它产生了一系列续集,甚至是前传,每个续集都与不同的导演,他们以个人方式接近相同的基本前提。我们调查了文献中的决议,以考虑合成,纠正常见错误并为大多数TEP变体和解释的核心提供有关可交换随机变量对的顺序属性的新定理。关于信封中的渐近独立性与它是较小还是较大的问题的定理表明,不当先验或无限期望值的病理状况会产生后果,因为我们只是在处理这种情况。

The Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The Two Envelopes Problem (TEP) is a much-studied paradox in probability theory, mathematical economics, logic, and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasizes what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.

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