Choice functions over a finite set: A summary
Let A be a finite set of outcomes and B a nonempty subset of A. Let S be a choice function, such that S(B) is nonempty for all S. The following conditions on S are crucial: Chernoff: \(B\subset B'\to S(B')\cap B\subset S(B)\). Expansion: S(B)\(\cap S(B')\subset S(B\cup B')\). Aizerman: [S(B')\(\subset B\subset B']\to [S(B)\subset S(B')].\) The main results are the following. S is rationalizable if and only if it satisfies the Chernoff and Expansion conditions. S is Pareto- rationalizable if and only if it satisfies the Aizerman and Expansion conditions. Finally, S is pseudo-rationalizable if and only if it satisfies the Chernoff and Aizerman conditions. Although these results have existed in Russian for some time, this is the first time that they have been readily available in the West.
scientific article; zbMATH DE number 3922310