Minimal rationalizations

I refine the multi-utility model and identify the minimal number W of total orders that is sufficient to rationalize a path independent choice function. This identification invokes the well-known pigeonhole principle: any dataset of size $$W+1$$ W + 1 that is rationalized by W rankings must contain at least two distinct observations where the same ranking is maximized. In general, the index W can be huge even for reasonable choice functions, such as top-ten rules. If W is constrained, then minimal rationalizations can be found in polynomial time via an explicit focal algorithm. The axiom of Expansion (Sen’s $$\gamma $$ γ ) describes a special case where the index W must equal the capacity—the largest number of elements that may be selected together in a menu.