Explorations on X-self selectivity

X-self selectivity of a social choice function is de ned as being self selective relative to the set of test functions X and all of its subsets. We explore the self-selectivity of social choice functions which satisfy independence of irrelevant alternatives, against di erent kinds of sets of test functions. We observe that testing against a smaller set can be su cient to deduce that a given social choice function is also self-selective relative to a larger set, under certain conditions. Moreover, we show that X-self selectivity is closed under set intersection and union. This leads to the notion of degree of self-selectivity, which allows us to compare the self-selectivities of two social choice functions under certain conditions.