Anna Karenina and The Two Envelopes Problem

by   R. D. Gill, et al.

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.


page 1

page 2

page 3

page 4


Corrections to "Wyner's Common Information under Rényi Divergence Measures"

In this correspondence, we correct an erroneous argument in the proof of...

Arrow's Theorem Through a Fixpoint Argument

We present a proof of Arrow's theorem from social choice theory that use...

Generalizing the de Finetti–Hewitt–Savage theorem

The original formulation of de Finetti's theorem says that an exchangeab...

Effective Unsupervised Author Disambiguation with Relative Frequencies

This work addresses the problem of author name homonymy in the Web of Sc...

A General Counterexample to Any Decision Theory and Some Responses

In this paper I present an argument and a general schema which can be us...

Zero-One Laws and Almost Sure Valuations of First-Order Logic in Semiring Semantics

Semiring semantics evaluates logical statements by values in some commut...

Padovan heaps

We analyze priority queues of Fibonacci family. The paper is inspired by...

Please sign up or login with your details

Forgot password? Click here to reset