The main theoretic justifications for the use of expected utility—hence of means—are the von Neumann Morgenstern axioms. Using the median obeys the completeness and transitivity axioms, but not the continuity and independence ones. It does obey weaker forms of continuity; but in a sense, this doesn’t matter. You can avoid all these issues by making a single ‘ultra-choice’. Simply list all the possible policies you could follow, compute their median return, and choose the one with the best median return. Since you’re making a single choice, independence doesn’t apply.
I think you misunderstand the von Neumann-Morgenstern axioms. Von Neumann-Morgenstern theory refers to one-shot decision making, not iterated decision making, hence there is nothing you can fix by taking decisions over policies.
Median utility maximization satisfy the axioms of completeness, transitivity. It does not satisfy continuity and independence of irrelevant alternatives.
The independence axiom derives most of it intuitive strength from the fact that if you violate it, you can be money pumped when presented with a sequence of decisions. When making a single decision over policy, independence has far less intuitive strength, as violating it has no actual cost.
The independence axiom derives most of it intuitive strength from the fact that if you violate it, you can be money pumped when presented with a sequence of decisions.
If your preferences aren’t transitive, then even your one-shot decision making system is completely broken, since it can’t even yield an action that is “preferred” in a meaningful sense. Vulnerability to money pumping would be the last of your concerns in this case.
Money pumping is an issue in sequential decision making with time-discounting and/or time horizons: any method to aggregate future utilities other than exponential discounting ( * ) over an infinite time horizon yields dynamic inconsistency which could, in principle, be exploited for money pumping.
The intuitive justification for the independence axiom is the following:
What would you like for dessert, sir? Ice cream or cake?
Ice cream.
Oh sorry, I forgot! We also have fruit.
Then cake.
This decision making example looks intuitively irrational. If you prefer ice cream to cake when they are the only two alternatives, then why would you prefer cake to ice cream when a third, inferior, alternative is included? The independence axiom formalizes this intuition about rational behavior.
( * with no discounting being a special case of exponential discounting)
If you prefer ice cream to cake when they are the only two alternatives, then why would you prefer cake to ice cream when a third, inferior, alternative is included?
You’re thinking of a different meaning of “independence”. A violation of the independence axiom of VNM would look more like this:
What would you like for dessert, sir? Ice cream or cake?
Ice cream.
Oh sorry, I forgot! There is a 50% chance that we are out of both ice cream and cake (I know we have either both or neither). But I’ll go check, and if we’re not out of dessert, I’ll get you your ice cream.
Intransitivity breaks your decision system with a single decision point; dependence does not. Hence a single policy decision has to be transitive, but need not be independent.
Except that median-maximising respects independence for options that are available to you (or can be trivially tweaked to do so). It only violates independence for hypothetical bad options that will never be available to you.
If you prefer ice cream to cake when they are the only two alternatives, then why would you prefer cake to ice cream when a third, inferior, alternative is included?
I think you misunderstand the von Neumann-Morgenstern axioms. Von Neumann-Morgenstern theory refers to one-shot decision making, not iterated decision making, hence there is nothing you can fix by taking decisions over policies.
Median utility maximization satisfy the axioms of completeness, transitivity. It does not satisfy continuity and independence of irrelevant alternatives.
The independence axiom derives most of it intuitive strength from the fact that if you violate it, you can be money pumped when presented with a sequence of decisions. When making a single decision over policy, independence has far less intuitive strength, as violating it has no actual cost.
If your preferences aren’t transitive, then even your one-shot decision making system is completely broken, since it can’t even yield an action that is “preferred” in a meaningful sense. Vulnerability to money pumping would be the last of your concerns in this case.
Money pumping is an issue in sequential decision making with time-discounting and/or time horizons: any method to aggregate future utilities other than exponential discounting ( * ) over an infinite time horizon yields dynamic inconsistency which could, in principle, be exploited for money pumping.
The intuitive justification for the independence axiom is the following:
What would you like for dessert, sir? Ice cream or cake?
Ice cream.
Oh sorry, I forgot! We also have fruit.
Then cake.
This decision making example looks intuitively irrational. If you prefer ice cream to cake when they are the only two alternatives, then why would you prefer cake to ice cream when a third, inferior, alternative is included? The independence axiom formalizes this intuition about rational behavior.
( * with no discounting being a special case of exponential discounting)
You’re thinking of a different meaning of “independence”. A violation of the independence axiom of VNM would look more like this:
What would you like for dessert, sir? Ice cream or cake?
Ice cream.
Oh sorry, I forgot! There is a 50% chance that we are out of both ice cream and cake (I know we have either both or neither). But I’ll go check, and if we’re not out of dessert, I’ll get you your ice cream.
Oh, in that case I’ll have cake instead.
Yes, I believe that this is a stronger version. Median utility satisfies the weaker version of the axiom but not the stronger one.
But notice you had two decision points there.
Intransitivity breaks your decision system with a single decision point; dependence does not. Hence a single policy decision has to be transitive, but need not be independent.
The first decision is immediately canceled and has no effect on your utility, hence it isn’t really a relevant decision point.
More generally, the independence axiom makes sure that the outcome of your decision process is not affected by bad options that are available to you.
Except that median-maximising respects independence for options that are available to you (or can be trivially tweaked to do so). It only violates independence for hypothetical bad options that will never be available to you.
It can be rational to do this. There’s a paradox publicized by Martin Gardner demonstrating how. Unfortunately the best link I could easily find was a Reddit comment, but try https://www.reddit.com/r/fffffffuuuuuuuuuuuu/comments/gxwqe/why_i_hate_people/c1r5203 .