If you really have preferences over all possible histories of the universe, then technically you can do anything.
Money pumping thus only makes sense in a context where your preferences are over a limited subset of reality.
Suppose you go to a pizza place. The only 2 things you care about are which kind of pizza you end up eating, and how much money you leave with. And you have cyclic preferences about pizza flavor A<B<C<A.
Your waiter offers you a 3 way choice between pizza flavors A, B, C. Then they offer to let you change your choice for $1, repeating this offer N times. Then they make your pizza.
Without loss of generality, you originally choose A.
For N=1, you change your choice to B, having been money pumped for $1. For N=2, you know that if you change to B the first time, you will then change to C, so you refuse the first offer, and then change to B. The same goes for constant known N>2, repeatedly refuse, then switch at the last minute.
Suppose the waiter will keep asking and keep collecting $1 until you refuse to switch. Then, you will wish you could commit yourself to paying $1 exactly once, and then stopping. But if you are the sort of agent that switches, you will keep switching forever, paying infinity $ and never getting any pizza.
Suppose the waiter rolls a dice. If they get a 6, they let you change your pizza choice for $1, and roll the dice again. As soon as they get some other number, they stop rolling dice and make your pizza. Under slight strengthening of the idea of cyclical preferences to cover decisions under uncertainty, you will keep going around in a cycle until the dice stops rolling 6′s.
So some small chance of being money pumped.
Money pumping is an agent with irrational cyclic preferences is quite tricky, if the agent isn’t looking myopically 1 step ahead but can forsee how the money pumping ends long term.
If you really have preferences over all possible histories of the universe, then technically you can do anything.
Money pumping thus only makes sense in a context where your preferences are over a limited subset of reality.
Suppose you go to a pizza place. The only 2 things you care about are which kind of pizza you end up eating, and how much money you leave with. And you have cyclic preferences about pizza flavor A<B<C<A.
Your waiter offers you a 3 way choice between pizza flavors A, B, C. Then they offer to let you change your choice for $1, repeating this offer N times. Then they make your pizza.
Without loss of generality, you originally choose A.
For N=1, you change your choice to B, having been money pumped for $1. For N=2, you know that if you change to B the first time, you will then change to C, so you refuse the first offer, and then change to B. The same goes for constant known N>2, repeatedly refuse, then switch at the last minute.
Suppose the waiter will keep asking and keep collecting $1 until you refuse to switch. Then, you will wish you could commit yourself to paying $1 exactly once, and then stopping. But if you are the sort of agent that switches, you will keep switching forever, paying infinity $ and never getting any pizza.
Suppose the waiter rolls a dice. If they get a 6, they let you change your pizza choice for $1, and roll the dice again. As soon as they get some other number, they stop rolling dice and make your pizza. Under slight strengthening of the idea of cyclical preferences to cover decisions under uncertainty, you will keep going around in a cycle until the dice stops rolling 6′s.
So some small chance of being money pumped.
Money pumping is an agent with irrational cyclic preferences is quite tricky, if the agent isn’t looking myopically 1 step ahead but can forsee how the money pumping ends long term.