Choice functions: Rationality re-examined (Q1588300)

Let \(X\) be a finite set and \(P(X)\) the collection of non-empty subsets of \(X\). A choice function is a map \(F:P(X) \to P(X)\) such that \(F(A)\) is a nonempty subset of \(A\) for all \(A \in P(X)\). Let \(R\) be a binary relation on \(X\) and define the set \(M(A,R) = \{ x\in A\mid xRy\;\forall y\in A\}\) for each \(A \in P(X)\). In terms of the preference relation induced from \(R\), this means that the elements of \(M(A,R)\) are the maximally-preferred elements of \(A\). A choice function \(F\) is called rational if there exists some binary relation \(R\) on \(X\) such that \(F(A) = M(A,R)\) for all \(A\in P(X)\). However, some reasonable choice functions are not rational in this sense. The authors prove the general result that for any choice function, there is some binary relation \(R\) such that \(F(A) \subseteq M(A,R)\) for all \(A\in P(X)\). The bulk of the paper is then concerned with analyzing such a relation \(R\) when the choice function satisfies certain additional properties. The analysis is divided into sections depending on whether there are multiple maximals or no maximals. The authors give a number of clear characterizations of choice functions satisfying conditions that are weaker than classical rationality.