Intransitive Preferences You Can’t Pump
In the usual argument of money-pumping we take an agent with preferences A>B, B>C and C>A. Then we offer it to exchange C+$1 for B, then B+$1 for A, and finally A+$1 for C. Now the agent paid $3 and ended up where it started.
The assumption here is that not only does this agent prefer A to B, it prefers A to B+$1. Of course, the price of $1 could be too steep, then we could look for something less valuable to exchange.
However, payments of arbitrarily low value need not exist! Sure, you can propose a lottery, where the agent only has to pay 1$ with probability p. But arbitrarily low probabilities need not exist. To offer a lottery, you need to have a physical method to generate events with that probability. If p equals 1 divided by Grahm’s number, how many coins would you have to flip to run this lottery? Even if you said “the agent will pay $1 when a lump of solid gold materializes out of thin air”, there are numbers lower than that probability. I hate to be an ultrafinitist, but it’s true, extremely small (or large) numbers are not physically meaningful.
Note, here I am assuming the axiom of continuity, i.e. that if B<A<B+$1, then for some p, we must have B+p·$1<A. However, we should be able to violate this axiom as well.
What does this imply? Probably nothing. This perverse agent is almost indistinguishable from an agent which sets U(A)=U(B)=U(C). The rule is that you can violate any axioms, but only when it doesn’t matter.
I believe for practical purposes, “I (or you) buy a cheap lottery ticket, and if it’s the winning ticket, then you pay me $1” is low enough.
That’s a measly one in a billion. Why would you believe that this is enough? Enough for what? I’m talking about the preferences of a foreign agent. We don’t get to make our own rules about what the agent prefers, only the agent can decide that.
Regarding practical purposes, sure you could treat the agent as if it was indifferent between A, B and C. However, given the binary choice, it will choose A over B, every time. And if you offered to trade C to B, B to A and A to C, at no cost, then the agent would gladly walk the cycle any number of times (if we can ignore the inherent costs of trading).