The subtle mathematics of voting

That’s Maths: Although well-defined, the process of Irish-style PR is complicated and gives rise to debate and disputes

The selection of leaders by voting has a history reaching back to the Athenian democracy. Elections are essentially arithmetical exercises, but they involve more than simple counting and they have some subtle mathematical aspects. The scientific study of voting and elections, which began around the time of the French revolution, is called psephology, from the Greek word psephos, a pebble: pebbles were used as counting tallies in ancient times.

The Marquis de Condorcet (1743-1794), a French philosopher, mathematician and political scientist, was one of the founders of the mathematical theory of voting. He had studied under the renowned mathematician d’Alembert and he wrote several books on mathematics. He discovered a counterintuitive result now called Condorcet’s paradox.

Suppose we have three candidates. If a majority of voters prefer A to B and a majority prefer B to C, then it would appear obvious that there must be a majority who prefer A to C. This combination of two results is known as transitivity, but Condorcet showed that it can fail to hold. It is possible that more prefer C to A, resulting in a cycle of preferences: A before B before C before A. This is similar to rock- paper-scissors, where each choice wins over a second but loses to the third option.

Fancy omelette

During the French revolution, Condorcet fled Paris to avoid capture and possible execution. Stopping at an inn, he betrayed his aristocratic status by ordering a 12-egg omelette. He was arrested and imprisoned, and died soon afterwards.


Condorcet’s studies were considered to be a key contribution to the French Enlightenment. After his death, his widow, Sophie, undertook to publish all his writings. This work was continued by their daughter Eliza, who had married Arthur O’Connor, a United Irishman.

In the 1950s, the mathematical theory of games, devised by John von Neumann, was used to analyse voting systems. Later, Kenneth Arrow used mathematical arguments to show that certain desirable properties of voting systems were mutually exclusive; all systems are inherently limited and compromises are unavoidable. Arrow's impossibility theorem is the most frequently quoted and applied result in voting theory.

In the single transferable vote system used in Ireland, voters rank the candidates in order of preference. The idea of this proportional representation system is that the number of seats won by each party or group of candidates should be proportional to the number of votes cast for them. PR systems tend to result in several political parties, whereas single-vote or “first past the post” systems – as used in the UK and US – usually result in dominance by just two parties.

In the Irish system, the method of counting is set out in minute detail. The method is algorithmic in nature; that is, it may be implemented as a series of clearly identifiable steps, under conditions that are explicit and unequivocal. Although well-defined, the process is complicated and gives rise to debate and occasional disputes.

Nowadays the benefits and weaknesses of a voting system are expected to be demonstrable in mathematical terms. Modern research focuses on devising new criteria and new methods of fulfilling them. With high-power computers, it is feasible to simulate elections and to study the practical implications of modifications in voting and counting procedures. Large ensembles of simulations yield statistically robust conclusions.