On the structure of voting systems between two alternatives

Voting systems between two alternatives have been widely studied in the literature of Social Choice. One of the results given by Fishburn (The theory of social choice. Princeton University Press, Princeton, 1973) allows us to characterize anonymous, neutral and monotonic voting systems by means of functions satisfying adequate conditions. From among all kinds of functions, the class of affine functions is highly interesting because from them it is possible to obtain the voting systems most used in practice. In this paper we analyze the structure of the set of these functions and we show that this set is convex and its extreme points are the functions that generate the following voting systems: simple majority, absolute majority, unanimous majority and Pareto majority. Moreover, we suggest a simple method for choosing a voting system when two alternatives are under consideration.