On Locally Rationalizable Social Choice Functions

We consider a notion of rationalizability, where the rationalizing relation may depend on the set of feasible alternatives. More precisely, we say that a choice function is locally rationalizable if it is rationalized by a family of rationalizing relations such that a strict preference between two alternatives in some feasible set is preserved when removing other alternatives. Tyson (2008) has shown that a choice function is locally rationalizable if and only if it satisfies Sen's $\gamma$. We expand the theory of local rationalizability by proposing a natural strengthening of $\gamma$ that precisely characterizes local rationalizability via PIP-transitive relations and by introducing the $\gamma$-hull of a choice function as its finest coarsening that satisfies $\gamma$. Local rationalizability permits a unified perspective on social choice functions that satisfy $\gamma$, including classic ones such as the top cycle and the uncovered set as well as new ones such as two-stage majoritarian choice and split cycle. We give simple axiomatic characterizations of some of these using local rationalizability and propose systematic procedures to define social choice functions that satisfy $\gamma$.