# 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.