A note on binary strategy-proof social choice functions

Let 𝝫𝑛 be the set of the binary strategy-proof social choice functions referred to a group of n voters who are allowed to declare indifference between the alternatives. We provide a recursive way to obtain the set 𝝫𝑛+1 from the set 𝝫𝑛. Computing the cardinalities |𝝫𝑛| presents difficulties as the computation of the Dedekind numbers. The latter give the analogous number of social choice functions when only strict preferences are admitted. A comparison is given for the known values. Based on our results, we present a graphical description of the binary strategy-proof social choice functions in the case of three voters.