The independence axiom derives most of it intuitive strength from the fact that if you violate it, you can be money pumped when presented with a sequence of decisions.

If your preferences aren't transitive, then even your one-shot decision making system is completely broken, since it can't even yield an action that is "preferred" in a meaningful sense. Vulnerability to money pumping would be the last of your concerns in this case.

Money pumping is an issue in sequential decision making with time-discounting and/or time horizons: any method to aggregate future utilities other than exponential discounting ( * ) over an infinite time horizon yields dynamic inconsistency which could, in principle, be exploited for money pumping.

The intuitive justification for the independence axiom is the following:

• What would you like for dessert, sir? Ice cream or cake?
• Ice cream.
• Oh sorry, I forgot! We also have fruit.
• Then cake.

This decision making example looks intuitively irrational. If you prefer ice cream to cake when they are the only two alternatives, then why would you prefer cake to ice cream when a third, inferior, alternative is included? The independence axiom formalizes this intuition about rational behavior.

( * with no discounting being a special case of exponential discounting)