Aizerman and Malishevsky (1981) show that any path-independent choice function on a finite domain Z can be rationalized by some set T of total orders. Kopylov construct’s a minimal rationalization T such that any set T* of smaller size is inconsistent with the choice data. Given that T has at most k orders, it can be found in polynomial time O(Z^k) and from choices in menus that have at most k+1 elements. Selections in larger menus can be distorted by any behavioral effects or not observed at all.

Any binary rationalization |T| = 2 is minimal. For at least 95% of choice functions that have binary rationalizations T, such T is determined uniquely. In the general case, uniqueness holds asymptotically for the restriction of minimal rationalizations T to any fixed menu B in a large Z. Moreover, minimal rationalizations identify some robust variations of the random utility model.