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 \(W+1\) 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 \(\gamma \)) 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.

我优化了多用途模型，并确定了足以使路径独立选择函数合理化的总订单的最小数量W。该标识引用了众所周知的信鸽原理：通过W排名合理化的大小为\（W + 1 \）的任何数据集都必须包含至少两个不同的观察，其中相同的排名被最大化。通常，即使对于合理的选择功能（例如前十个规则），索引W也会很大。如果W受约束，则可以通过显式焦点算法在多项式时间内找到最小化合理化。扩展公理（Sen的\（\ gamma \））描述了一种特殊情况，其中索引W必须等于容量-可以在菜单中一起选择的最大元素数。