Operations research and voting theory

One main concern of voting theory is to determine a procedure for choosing a winner from among a set of candidates, based on the preferences of the voters or, more ambitiously, for ranking all the candidates or a part of them. In this presentation, we pay attention to some contributions of operations research to the design and the study of some voting procedures. First, we show through an easy example that the voting procedure plays an important role in the determination of the winner: for an election with four candidates, the choice of the voting procedure allows electing anyone of the four candidates with the same individual preferences of the voters. This provides also the opportunity to recall some main procedures, including Condorcet’s procedure, and leads to the statement of Arrow’s theorem. In a second step, more devoted to a mathematical approach, we detail a voting procedure based on the concept of Condorcet winner, namely the so-called median procedure. In this procedure, the aim is to rank the candidates in order to minimize the number of disagreements with respect to the voters’ preferences. Thus we obtain a combinatorial optimization problem. We show how to state it as a linear programming problem with binary variables. We specify the complexity of this median procedure. Last, we show, once again through easy examples, that the lack of some desirable properties for the considered voting procedure may involve some “paradoxes”.