The physics and maths of keeping elections fair and representative | Explained Premium

2024 national elections analyzed through history, mathematics, and physics to understand and improve electoral systems worldwide.

There are about 60 national elections in 2024 involving two billion people, including the biggest of them — the national elections underway in India — and the election to the U.S. presidency. Across the world, elections are a volatile mixture of emotions, aspirations, competing ideologies, and sometimes even violence. It might then surprise many that, despite the cacophony, there is science behind the election’s processes.

About 2,500 years ago, the earliest form of elections in ancient Athens was a system that ultimately depended on the candidate’s luck. Among all the suitable candidates, one was randomly chosen. Since the winning criterion was based on random choice, campaigning or influence couldn’t help the candidate.

Tenth-century Chola inscriptions at Uthiramerur in Tamil Nadu reveal the practice of choosing village representatives through a ‘Kudavolai’ system. The final choice was made by randomly picking one among the candidates the people had voted for.

Today, social choice theorists and mathematicians who study elections call this the approval voting system followed by a random choice. As a means of electing candidates, this process fails to reflect the will of the people. If this is a flawed process, what would be the right way to elect candidates? Surprisingly, mathematics tells us that there is no simple answer to this question.

The first-past-the-post (FPTP) system followed in India, the U.S., the U.K., and several other countries has many drawbacks. Critics have pointed out the disproportionate difference between the popular vote share and the seat share in many Parliaments. For example, in the 2015 Delhi Assembly elections, the Aam Aadmi Party received 54% of the popular vote but won 96% of the seats, whereas the Bharatiya Janata Party won 32% and 4%, respectively.

Second, winners in the FPTP system often secure far less than 50% of the vote share. No government in India, irrespective of its strength in the Lok Sabha (i.e. number of seats), has ever surpassed 50% vote share. Since 1918, only once, in 1931 in the U.K., did a government command more than 50%. So by the vote-share metric, rather than parliamentary seats, India and the U.K. were always ruled by “minority” governments. Expectedly, social choice theorists disfavour the FPTP system, though it continues to find wide use for its simplicity.

Are there better alternatives? Mathematical analysis to design better electoral systems dates back to the 13th century in the works of Ramon Llull, a missionary and theologian. His book ‘De Arte Eleccionis’, in the Catalan language, gives a detailed algorithm for a two-stage election process for church officials. It ensures that the winner, when pitted against each of the other contenders, receives more than 50% votes and is the most preferred candidate. This work was lost for centuries until it was discovered in the late 1980s.