Thursday, July 16, 2015

Infinite expectations

Mark Thoma linked to Chris Dillow, who linked to the "two-envelope problem", which links to the St. Petersburg paradox.

The St. Petersburg paradox involves the expected value of a gambling game where you put up X dollars to enter a game where the k-th time you flip a fair coin and it comes up heads, the pot grows to 2ᵏ (the casino has infinite resources). The first time it comes up tails, you get whatever is in the pot. The question is how much money X should you put up to try your hand at this game?

The naive calculation of the expected value is infinite:

E = (1/2) · 2 + (1/4) · 4 + (1/8) · 8 + ...
E = 1 + 1 + 1 + ...
E = ∞

I was surprised the real answer wasn't listed in the Wikipedia article (but is available on Wikipedia):

E = (1/2) · 2 + (1/4) · 4 + (1/8) · 8 + ...
E = 1 + 1 + 1 + ...
E = ζ(0)
E = -1/2

... so you shouldn't enter the game at any ante.

[Update: This is a bit tongue-in-cheek. But only a bit.]


  1. So, is the blogging hiatus officially over?

    1. Technically these are posts, so maybe I should just call it "light blogging" ...

  2. So, is the blogging hiatus officially over?

  3. It is unfortunate that the term "expectation" (or even worse, "expected value") got picked by the mathematical literature. The so-called expectation is not necessarily the value you should expect, not at all.

  4. OK, that was a mind blower... but I learned a little about a lot of different things. I'm content to leave it there for now.


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