Here is a classic math paradox for your amusement, and perhaps your edification: (edification is a fancy word for: beware, I’m trying to learn you something).

You are on a TV game show where you will be asked to choose between two, identical-looking envelopes. All you know about the envelopes is that one of them has twice as much money as the other. The envelopes are shuffled, and you pick one. You peak in and see that your envelope contains $400, and you feel pretty good. But then you are given a choice: you can switch your envelope with the other one; the one you didn’t take. You reason that the other envelope either has $800 or $200 with equal probability. That is, a switch will either net you a $400 gain, or loose you $200. Since $400 is bigger than $200, you switch. Did that decision make sense. It seems that, at this game, every contestant should switch envelopes. Hmm.

The solution follows: The problem with this analysis is an error common in children and politicians — the confusion between your lack of knowledge of a thing, and actual variability in the system. In this case, the contestant is confusing his (or her) lack of knowledge of whether he/she has the big envelope or the smaller, with the fixed fact that the total between the two envelopes has already been set. It is some known total, in this case it is either $600 or $1200. Lets call this unknown sum y. There is a 50% chance that you now are holding 2/3 y and a 50% chance you are holding only 1/3y. therefore, the value of your current envelope is 1/3 y + 1/6y = 1/2 y. Similarly, the other envelope has a value 1/2y; there is no advantage is switching once it is accepted that the total, y had already been set before you got to choose an envelope.

And here, unfortunately is the lesson:The same issue applies in reverse when it comes to government taxation. If you assume that the total amount of goods produced by the economy is always fixed to some amount, then there is no fundamental problem with high taxes. You can print money, or redistribute it to anyone you think is worthy — more worthy than the person who has it now – and you won’t affect the usable wealth of the society. Some will gain others will lose, and likely you’ll find you have more friends than before. On the other hand, if you assume that government redistribution will affect the total: that there is some relationship between reward and the amount produced, then to the extent that you diminish the relation between work and income, or savings and wealth, you diminish the total output and wealth of your society. While some balance is needed, a redistribution that aims at identical outcomes will result in total poverty.

This is a variant of the “two-envelopes problem,” originally posed in 1912 by German, Jewish mathematician, Edmund Landau. It is described, with related problems, by Prakash Gorroochurn, Classic Problems of Probability. Wiley, 314pp. ISBN: 978-1-118-06325-5. Wikipedia article: Two Envelopes Problem.

Robert Buxbaum, February 27, 2019