  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 $\gamma$.
