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Anna Karenina and the two envelopes problem

机译：Anna Karenina和两个信封问题

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摘要

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) emphasises 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.
机译：Anna Karenina原则以同名小说的开幕式命名：幸福的家庭都是相似的;每个不开心的家庭都以自己的方式不满意。这两个信封问题（Tep）是概率理论，数学经济学，逻辑和哲学中的多学习悖论。时间和再次发布了一个新的分析，其中提交人声称终于解释了这个悖论中实际出错的问题。每个作者（包括当前提交人）强调他们的方法是什么，并结束了早期的方法没有得到此事的根源。我们观察到，尽管只有一个逻辑论点是正确的，但如果每一步都是正确的，那么一个明显的逻辑论点就可以被认为是在不同的地方误入歧途。这导致TEP的文献与一部成功的电影特许经营权之间的比较：它产生了一系列续集，甚至是前列的，每个前任者都有一个以个人方式接近相同的基本前提。我们在文献中进行了调查决议，以综合，正确的常见错误，并在大多数TEP变体和解释的核心处提供交换对随机变量的订单属性的新定理。无论是更小或更大的信封金额与问题之间的渐近独立性的定理表明，由于我们仅接近这种情况，不正当的前锋或无限期望值的病理情况会产生后果。

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