If a coalition D is decisive over a pair of social outcomes (x,y) under a social choice process that satisfies Arrow's axioms, then another coalition C (that differs from D) cannot be decisive over a different pair (a,b) of social outcomes.

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This statement is true according to Arrow's impossibility theorem, which states that no social choice process can satisfy all of Arrow's axioms simultaneously. In particular, the independence of irrelevant alternatives axiom means that if a certain outcome is chosen from a set of alternatives, the relative ranking of those alternatives should not depend on the presence or absence of other alternatives. Therefore, if coalition D is decisive over (x,y), the ranking of any other pair of outcomes (such as (a,b)) should not depend on the presence or absence of coalition C.

If a Coalition D Is Decisive Over a Pair of Social