Blair and Pollak (Econometrica (1982)
50: 931-943) prove that, if there are more alternatives than individuals, then, for every arrovian binary decision rule that is acyclic, there is at least one individual who has a veto power over a critical number of pairs of alternatives. If the number of individuals is larger than the number of alternatives, there need not be single vetoers but there could be small coalitions endowed with a similar power. Kelsey (Soc Choice Welfare (1985) 2: 131-137) states precise results in this respect. In this paper, we first give a new and much simpler proof of the main result of Blair and Pollak and complete proofs of the generalization of this result by Kelsey. Then we give a precise answer as to the minimum size of the coalitions that must have a veto power under any acyclic binary decision rule and the minimum number of pairs of alternatives on which these coalitions may exercise their power. We also show that, if the veto power of the coalitions of the minimal size attainable under the last objective is limited to the minimum number of pairs of alternatives, then all larger coalitions have a veto power on all pairs. All the results are obtained by appealing to an acyclicity condition found by Ferejohn and Fishburn (J Econ Theory (1979) 21: 28-45). In the case of symmetric and monotonic binary decision rules, proofs are even easier and illustrate clearly the reasons for the veto power. This possibility has been alluded to by some of the aforementioned authors. But Kelsey (1985) is the first, if not the only one, to state precise results in this respect. He cleverly extends the result of Blau and Deb (1977) to coalitions of an arbitrary partition of the voters and derives interesting corollaries from this extension. His proof consists in giving the same preference to all members of each coalition in a partition of voters. Then coalitions may be interpreted as individuals and the usual proofs invoked. Kelsey also "converts" or "translates" the main result of Blair and Pollak "into a result on the power of groups of individuals" but the proof of this more general result has not been published.
In this paper, we first give a new and much simpler proof of the main result of Blair and PoUak. Using the same technique, we next present complete proofs of the generalization of this result by Kelsey (1985), namely his Theorems 7, 8-9. We also extend these theorems to the case where the number of coalitions in a partition is equal to the number of alternatives.
Given the objectionable nature of the veto power, a social objective in designing binary decision rules could be to restrict the veto power to coalitions as large as possible. The third and main contribution of this paper is a precise answer as to the minimum size of the coalitions that must have a veto power under any acyclic binary decision rule and the minimum number of pairs of alternatives on which these coalitions may exercise their power. We also show that, if the veto power of the coalitions of the minimal size attainable under the last objective is limited to the minimum number of pairs of alternatives, then all larger coalitions have a veto power on all pairs. Our paper also innovates from a methodological point of view. Indeed, all the results are obtained without constructing a single profile of preferences, contrary to the practice in this literature. This is made possible by appealing to an acyclicity condition found by Ferejohn and Fishburn (1979). In the case of symmetric and monotonic binary decision rules, proofs are even easier and illustrate clearly the reasons for the veto power.