Imagine two people going through this experiment, Alice and Bob. When Alice wakes up, she decides to pick red, because she always picks red which is her favori

Imagine two people going through this experiment, Alice and Bob. When Alice wakes up, she decides to pick red, because she always picks red which is her favorite color. Lo and behold, the red envelope is empty with >90% probability (because if Alice picked it on the first day, it will be empty on the next 9, and Alice doesn't know which day it is). When Bob wakes up, he decides to flip a coin, because that's how he always makes decisions. The coin tells him to pick red. Lo and behold, the red envelope is full with 50% probability. Even though Alice and Bob chose the same color. Nerdy explanation: if you maximize expected utility in the Von Neumann and Morgenstern sense, you can prove a theorem saying there's always an optimal strategy that's deterministic. You shouldn't need randomization, even if the world is random. In my experiment, the classical assumptions don't apply (specifically the axiom of independence), so the theorem becomes false and you need randomization to get the best result. The point of the experiment is showing that the axiom of independence isn't reasonable when you have copies, so vNM utility maximization needs to be modified. (For cases like this one, the right fix is modeling utility maximization as a single player game with imperfect information, chance moves, and behavioral strategies.) Bostrom's anthropic problems also show the same weirdness, but my contribution is making a non-anthropic scenario with fixed number of observers that still shows the weirdness.