Indeed, but my whole point is that when there is such a period of uncertainty (which is the case of most real world decisions, usually in a minor way, but sometimes in a very significant way), the independence principle doesn’t hold, specifically because of that period of uncertainty.
I’d rephrase it as, when there is such a period of uncertainty, you can no longer factor the problem into neat little chunks and everything gets way way more complicated to work with.
So, my point is that the independence axiom still holds. (.5A + .5B) is preferable to (.5A + .5C), where A, B, and C are world-histories where you know what’s going to happen in advance. And (.5A″ + .5C″) is preferable to (.5A’ + .5B’), where A’, B’, A″, and C″ are world-histories that involve periods of uncertainty. There is no violation of the VNM axioms.
A’, B’: You’re uncertain for a while whether you’ll go to Ecuador or whether you’ll get a laptop.
A″, C″: You’re uncertain where you will go.
A steelmanning of your position would be that a good decision-making heuristic should take into account not simply the assets that will eventually be made available to you, but the assets you’ll realistically be able to take advantage of.
So a rephrasing of your point that takes Nisan et al.’s objections into account would be: Risk-aversion is a valuable heuristic because usually uncertainty may lead to suboptimal decisions (take days off when it turns out you don’t need to, or the other way around). The Allais paradox is a case where such risk aversion isn’t justified, but some careless phrasing of the paradox (or of other situations in decision theory) the risk aversion may be justified.
The Allais paradox (both Eliezer version and the Wikipedia article) seems to not specify at all if the reward is instantaneous or delayed, so I wouldn’t say the risk aversion isn’t justified—if offered the paradox with no precision, I would say there is a chance for the reward to be instant, a chance for it to be delayed, so the risk aversion should be partially considered. It’s a bit of nitpicking, but not that much. There is no mention of time/delay in either of the paradox nor the axiom, and that seems to be a weakness to me.
And even without time, information can still have some value. If you chose 1B in the Allais paradox and lose, you can regret your choice (leading to, in fact, a <0 outcome) more than in you chose 2B and lose. Because of the information that it’s purely because of your choice you lose money. Or people can have a lower opinion of you (which may have negative consequences on your life) if they have the information too. Regretting what was a rational decision can be considered irrational, so I don’t see a problem with it being incompatible with VNM axioms. But the reaction of other people being irrational is not something you can discard the same way—if a VNM rational agent is unable to deal with human beings who aren’t perfectly rational, then there is a problem.
In short : the value of information is more important when it creates uncertainty—when the results of the lottery are delayed. But even when the results are instantaneous, information still have some value (positive or negative) that can make choosing 1A over 1B and 2B over 2A the rational choice to do in some situations.
Indeed, but my whole point is that when there is such a period of uncertainty (which is the case of most real world decisions, usually in a minor way, but sometimes in a very significant way), the independence principle doesn’t hold, specifically because of that period of uncertainty.
I’d rephrase it as, when there is such a period of uncertainty, you can no longer factor the problem into neat little chunks and everything gets way way more complicated to work with.
So, my point is that the independence axiom still holds. (.5A + .5B) is preferable to (.5A + .5C), where A, B, and C are world-histories where you know what’s going to happen in advance. And (.5A″ + .5C″) is preferable to (.5A’ + .5B’), where A’, B’, A″, and C″ are world-histories that involve periods of uncertainty. There is no violation of the VNM axioms.
A’, B’: You’re uncertain for a while whether you’ll go to Ecuador or whether you’ll get a laptop.
A″, C″: You’re uncertain where you will go.
A steelmanning of your position would be that a good decision-making heuristic should take into account not simply the assets that will eventually be made available to you, but the assets you’ll realistically be able to take advantage of.
So a rephrasing of your point that takes Nisan et al.’s objections into account would be: Risk-aversion is a valuable heuristic because usually uncertainty may lead to suboptimal decisions (take days off when it turns out you don’t need to, or the other way around). The Allais paradox is a case where such risk aversion isn’t justified, but some careless phrasing of the paradox (or of other situations in decision theory) the risk aversion may be justified.
The Allais paradox (both Eliezer version and the Wikipedia article) seems to not specify at all if the reward is instantaneous or delayed, so I wouldn’t say the risk aversion isn’t justified—if offered the paradox with no precision, I would say there is a chance for the reward to be instant, a chance for it to be delayed, so the risk aversion should be partially considered. It’s a bit of nitpicking, but not that much. There is no mention of time/delay in either of the paradox nor the axiom, and that seems to be a weakness to me.
And even without time, information can still have some value. If you chose 1B in the Allais paradox and lose, you can regret your choice (leading to, in fact, a <0 outcome) more than in you chose 2B and lose. Because of the information that it’s purely because of your choice you lose money. Or people can have a lower opinion of you (which may have negative consequences on your life) if they have the information too. Regretting what was a rational decision can be considered irrational, so I don’t see a problem with it being incompatible with VNM axioms. But the reaction of other people being irrational is not something you can discard the same way—if a VNM rational agent is unable to deal with human beings who aren’t perfectly rational, then there is a problem.
In short : the value of information is more important when it creates uncertainty—when the results of the lottery are delayed. But even when the results are instantaneous, information still have some value (positive or negative) that can make choosing 1A over 1B and 2B over 2A the rational choice to do in some situations.
The independence principle still holds, because (.5A’ + .5B’) is preferable to (.5A’ + .5C’).