Eliezer, I see from this example that the Axiom of Independence is related to the notion of dynamic consistency. But, the logical implication goes only one way. That is, the Axiom of Independence implies dynamic consistency, but not vice versa. If we were to replace the Axiom of Independence with some sort of Axiom of Dynamic Consistency, we would no longer be able to derive expected utility theory. (Similarly with dutch book/money pump arguments, there are many ways to avoid them besides being an expected utility maximizer.)
I’m afraid that the Axiom of Independence cannot really be justified as a basic principle of rationality. Von Neumann and Morgenstern probably came up with it because it was mathematically necessary to derive Expected Utility Theory, then they and others tried to justify it afterward because Expected Utility turned out to be such an elegant and useful idea. Has anyone seen Independence proposed as a principle of rationality prior to the invention of Expected Utility Theory?
I’m equally afraid ;). The Axiom of Independence is intuitively appealing to me, but I don’t posit it to be a basic principle of rationality, because that smells like a mind projection fallacy. I suspect you’re right, also, about dutch book/money pump arguments.
I tentatively conclude that a rational agent need not evince preferences that can be represented as an attempt to maximize such a utility function. That doesn’t mean Expected Utility Theory can’t be useful in many circumstances or for many agents, but this still seems like important news, which merits more discussion on Less Wrong.
Eliezer, I see from this example that the Axiom of Independence is related to the notion of dynamic consistency. But, the logical implication goes only one way. That is, the Axiom of Independence implies dynamic consistency, but not vice versa. If we were to replace the Axiom of Independence with some sort of Axiom of Dynamic Consistency, we would no longer be able to derive expected utility theory. (Similarly with dutch book/money pump arguments, there are many ways to avoid them besides being an expected utility maximizer.)
I’m afraid that the Axiom of Independence cannot really be justified as a basic principle of rationality. Von Neumann and Morgenstern probably came up with it because it was mathematically necessary to derive Expected Utility Theory, then they and others tried to justify it afterward because Expected Utility turned out to be such an elegant and useful idea. Has anyone seen Independence proposed as a principle of rationality prior to the invention of Expected Utility Theory?
I’m equally afraid ;). The Axiom of Independence is intuitively appealing to me, but I don’t posit it to be a basic principle of rationality, because that smells like a mind projection fallacy. I suspect you’re right, also, about dutch book/money pump arguments.
I tentatively conclude that a rational agent need not evince preferences that can be represented as an attempt to maximize such a utility function. That doesn’t mean Expected Utility Theory can’t be useful in many circumstances or for many agents, but this still seems like important news, which merits more discussion on Less Wrong.
Have you read these posts?
indexical uncertainty and the Axiom of Independence
Towards a New Decision Theory