What if we redefine rationality to be relative to choice sets? We might not have to depart too far from vNM-rationality this way.
The axioms of vNM-rationality are justified by Dutch books/money pumps and stochastic dominance, but the latter can be weakened, too, since many outcomes are indeed irrelevant, so there’s no need to compare to them all. For example, there’s no Dutch book or money pump that only involves changing the probabilities for the size of the universe, and there isn’t one that only involves changing the probabilities for logical statements in standard mathematics (ZFC); it doesn’t make sense to ask me to pay you to change the probability that the universe is finite. We don’t need to consider such lotteries. So, if we can generalize stochastic dominance to be relative to a set of possible choices, then we just need to make sure we never choose an option which is stochastically dominated by another, relative to that choice set. That would be our new definition of rationality.
Here’s a first attempt:
Let C be a set of choices or probabilistic lotteries over outcomes (random variables), and let O be the set of all possible outcomes which have nonzero probability in some choice from C (or something more general to accommodate general probability measures). Then for X,Y∈C , we say X stochastically dominates Y with respect to C if:
P[X<Cz]≤P[Y<Cz]
for all z∈O, and the inequality is strict for some z∈O. This can lift comparisons using <C, a relation ⊆O×O, between elements of O to random variables over the elements of O. <C need not even be complete over O or transitive, but stochastic dominance thus defined will be transitive (perhaps at the cost of losing some comparisons). <C could also actually be specific to C, not just to O.
We could play around with the definition of O here.
When we consider choices to make now, we need to model the future and consider what new choices we will have to make, and this is how we would avoid Dutch books and money pumps. Perhaps this would be better done in terms of decision policies rather than a single decision at a time, though.
Interesting!
What if we redefine rationality to be relative to choice sets? We might not have to depart too far from vNM-rationality this way.
The axioms of vNM-rationality are justified by Dutch books/money pumps and stochastic dominance, but the latter can be weakened, too, since many outcomes are indeed irrelevant, so there’s no need to compare to them all. For example, there’s no Dutch book or money pump that only involves changing the probabilities for the size of the universe, and there isn’t one that only involves changing the probabilities for logical statements in standard mathematics (ZFC); it doesn’t make sense to ask me to pay you to change the probability that the universe is finite. We don’t need to consider such lotteries. So, if we can generalize stochastic dominance to be relative to a set of possible choices, then we just need to make sure we never choose an option which is stochastically dominated by another, relative to that choice set. That would be our new definition of rationality.
Here’s a first attempt:
Let C be a set of choices or probabilistic lotteries over outcomes (random variables), and let O be the set of all possible outcomes which have nonzero probability in some choice from C (or something more general to accommodate general probability measures). Then for X,Y∈C , we say X stochastically dominates Y with respect to C if:
for all z∈O, and the inequality is strict for some z∈O. This can lift comparisons using <C, a relation ⊆O×O, between elements of O to random variables over the elements of O. <C need not even be complete over O or transitive, but stochastic dominance thus defined will be transitive (perhaps at the cost of losing some comparisons). <C could also actually be specific to C, not just to O.
We could play around with the definition of O here.
When we consider choices to make now, we need to model the future and consider what new choices we will have to make, and this is how we would avoid Dutch books and money pumps. Perhaps this would be better done in terms of decision policies rather than a single decision at a time, though.
(This approach is based in part on “Exceeding Expectations: Stochastic Dominance as a General Decision Theory” by Christian Tarsney, which also helps to deal with Pascal’s wager and Pascal’s mugging.)