ACYCLICITY AND THE DISPERSION OF THE VETO POWER

Blair and Pollak (Econometrica (1982) 50: 931-943) prove that, if ther e are more alternatives than individuals, then, for every arrovian bin ary decision rule that is acyclic, there is at least one individual wh o 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 e ndowed with a similar power. Kelsey (Soc Choice Welfare (1985) 2: 131- 137) states precise results in this respect. In this paper, we first g ive a new and much simpler proof of the main result of Blair and Polla k 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 an d the minimum number of pairs of alternatives on which these coalition s may exercise their power. We also show that, if the veto power of th e coalitions of the minimal size attainable under the last objective i s limited to the minimum number of pairs of alternatives, then all lar ger coalitions have a veto power on all pairs. All the results are obt ained by appealing to an acyclicity condition found by Ferejohn and Fi shburn (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.