On locally rationalizable social choice functions

Abstract 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. It is known that a choice function is locally rationalizable if and only if it satisfies Sen’s $$\gamma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> . We expand the theory of local rationalizability by proposing a natural strengthening of $$\gamma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> that precisely characterizes local rationalizability via PIP-transitive relations. Local rationalizability permits a unified perspective on social choice functions that satisfy $$\gamma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> , 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 $$\gamma$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> .