RobinZ comments on Expected utility without the independence axiom - Less Wrong

9 Post author: Stuart_Armstrong 28 October 2009 02:40PM

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Tags: decision_theory probability rationality expected independence utility

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Comment author: SilasBarta 28 October 2009 08:50:29PM *  1 point [-]

The problem is that you're losing money doing it once.

Again, if suddenly being offered the choice of 1A/1B then 2A/2B as described here, but being "inconsistent", is what you call "losing money", then I don't want to gain money!

If they are willing to trade A for B in a one-shot game, they shouldn't be willing to pay more for A than for B in a one-shot

But that's not what's happening the paradox. They're (doing something isomorphic to) preferring A to B once and then p*B to p*A once. At no point do they "pay" more for B than A while preferring A to B. At no point does anyone make or offer the money-pumping trades with the subjects, nor have they obligated themselves to do so!

Comment author: RobinZ 28 October 2009 09:57:28PM *  1 point [-]

Consider Eliezer's final remarks in The Allais Paradox (I link purely for the convenience of those coming in in the middle):

Suppose that at 12:00PM I roll a hundred-sided die. If the die shows a number greater than 34, the game terminates. Otherwise, at 12:05PM I consult a switch with two settings, A and B. If the setting is A, I pay you $24,000. If the setting is B, I roll a 34-sided die and pay you $27,000 unless the die shows "34", in which case I pay you nothing.

Let's say you prefer 1A over 1B, and 2B over 2A, and you would pay a single penny to indulge each preference. The switch starts in state A. Before 12:00PM, you pay me a penny to throw the switch to B. The die comes up 12. After 12:00PM and before 12:05PM, you pay me a penny to throw the switch to A.

I have taken your two cents on the subject.

You're right insofar as Eliezer invokes the Axiom of Independence when he resolves the Allais Paradox using expected value; I do not yet see any way in which Stuart_Armstrong's criteria rule out the preferences (1A > 1B)u(2A < 2B). However, in the scenario Eliezer describes, an agent with those preferences either loses one cent or two cents relative to the agent with (1A > 1B)u(2A > 2B).

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