Independence axiom
Also known as: Substitution axiom, Cancellation
A component shared by two gambles should not affect which one you prefer.
What it means
The independence axiom is the requirement that if you prefer lottery A to lottery B, you must still prefer a mixture of A with some third lottery C over the same mixture of B with C, in the same proportions. The logic is that the common C component is irrelevant: either it happens, in which case your choice made no difference, or it does not, in which case you simply face A versus B. It is the load-bearing axiom of expected utility theory, the one that forces utility to be linear in probabilities, and so underwrites most normative models of risky choice in economics and finance. People violate it systematically: the Allais paradox and the certainty effect show that a shared component is not treated as irrelevant when its presence makes an option certain. It matters because most alternatives to expected utility, prospect theory included, are built by weakening it.
Examples
Two raffle tickets carry the same consolation prize. Strip that shared prize from both and people's preference between the tickets flips, though the axiom says the common part should cancel out.
A patient ranks treatment A above B, then reverses once both options carry an identical added complication risk. The shared risk should have cancelled, but it changes how each option feels.
Two pension plans both include the same guaranteed state top-up. Removing it from both descriptions shifts savers toward the riskier plan, though the shared component is irrelevant to the comparison.
First described in von Neumann & Morgenstern (1944); Samuelson (1952).