Tuesday, November 12, 2013

Two Envelopes Paradox

Just for spice, let's look at academic philosophy getting it right.
"We want to compare the amount that we would gain by switching if we would gain by switching, with the amount we would lose by switching if we would indeed lose by switching. However, we cannot both gain and lose by switching at the same time. We are asked to compare two incompatible situations." 
And yet,
"God help us if, after the fourth round of drinks, someone brings up the two envelopes paradox." (Here, via
La Wik's phrasing needs work, so in my own words:

When first looking at switching, it seems I must switch, as the expected payoff is 5/4 A. However, now I've switched, I can't re-define the amount in the new envelope as A - it's still 5/4 A, and thus switching back must be a loss. At first this feels like I've unjustifiably broken the symmetry - before I pick an envelope, the expected value must be equal - but since I've added an assumption asymmetrically, it would instead be weird if the symmetry remained.

The second version, where I actually open the envelope, also breaks the symmetry. Sadly this one is still best analyzed by throwing all the academics in a lake.

Your daughter has some terrible eye-melting but curable disease, and you are poor. The opened envelope has $10 000 in it. Do you switch? Obviously not - it's not risking five grand, it's risking your daughter.

Your daughter is fine. You're thinking of buying a new car. The opened envelope has $10 000 in it. Do you switch? Obviously so - no matter what, you get a nicer car, we're just haggling about whether you also get to pad your retirement fund.

Seems clear to me. But why the lake? Have fun trying to find even one academic who will make this clear. The real world and applications are low status, don't ye know.

Second, I've tried to respect intuition. Nobody analyzes these things entirely abstractly. It is passed to the subconscious and the subcon uses concrete examples, usually whatever the availability bias spits out. (E.g. think of a cat - no no, not a cat with fur, an abstract cat. [Mine's an adult tabby. {Side view, facing left, tail erect, front right paw (white) lifted, looking a bit surprised.}]) This will naturally lead to very different intuitions based on things like how rich you currently are or how secure you currently feel.

Additionally, expected value is not the whole story, and one other consideration is whether you can absorb the possible loss. Your intuition will consider this and you can't tell it not to without training.

Completely unrelated, this article taught me a new rule of thumb. "Though Bayesian probability theory ..." The translation: "We are now going to talk out of our ass." Any paragraph or section explicitly calling itself Bayesian is not worth reading. Presumably one calling itself frequentist would be just as bad, but I haven't seen one of those yet.

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