[0:00]Is there a perfect voting system? One that truly captures the will of the people? From elections to awards, voting is everywhere. Today, we dive into the math of various voting methods, how they work and whether a truly ideal system exists?
[0:20]What is voting really? It's a way to give everyone a voice in making shared decisions, like choosing a leader. Everyone picks their favorite option and the one with the most votes wins. This method, called plurality voting is widely used in countries like India, the US, the UK, and Canada. It sounds simple, but has a flaw. For example, with this vote share, a candidate could win with just 30% of the votes, even though 70% did not prefer them. Or in a race with seven candidates, the winner might have only 20%. This is a serious drawback and it happens quite often. For example, Bill Clinton won the 1992 US elections with 43% of the votes. And in the 2015 UK election, the Conservative Party formed a government with just 36.9% votes. Both winning despite most voters not choosing them. So, how can we improve this? One way is to set a majority rule, where a candidate needs more than half the votes to win. For example, if A has 12 out of 20 votes, B has five and C has three. A wins with a clear majority. This majority rule is used in India's Lok Sabha, where a party needs more than 272 of the total 545 seats to form a government. But what if no one has a clear majority? For example, A has eight, B seven and C five. What's the solution? In such cases, coalitions can form, as often seen in India, but this isn't a mathematical fix. So one option is to hold a re-election. But with the same candidates and voters, the outcome may not change. But what if we eliminate the least preferred candidate C and then rerun the election? Now, voters choose between A and B, giving one of them a majority. However, with more candidates, eliminating the lowest one and holding a re-election multiple times can be impractical. A better approach is ranked voting, where voters rank candidates by preference. It's like holding multiple elections, but all at one. For example, with three candidates and 20 voters, each ranks their choices. Here, A and B have seven first choice votes, and C has six. With no clear majority, C with the fewest first choice votes is eliminated. C's votes are then reallocated based on the voter's next choices. After reallocation, A becomes the top choice for 13 voters, while B still for seven. So A wins, reflecting the majority's preference once the least popular candidate is removed. This approach, called Instant Runoff Voting, is used in Australia's House elections, Ireland's Presidential elections, local elections in New Zealand, and parts of the US. This method seems fair but has a very surprising flaw. Let's tweak our example a bit. We strengthen A's position by changing the preferences for two voters. A is now the top pick for nine people, B five and C six. Now B is eliminated and their votes are reallocated. Strangely, C wins this time with 11 votes compared to A's nine. Surprising, isn't it? While A was winning earlier, strengthening their position led to their loss. An odd outcome for a system aiming for fairness. Such scenarios can actually happen. In the 2009 Burlington Mayoral election, a couple of candidates were eliminated in the first round. Finally, Montroll was eliminated and most of his votes went to Bob, making Bob the winner. But if Bob had gained more first choice votes from Kurt's supporters, Kurt would have been eliminated instead. And with his remaining votes favoring Montroll, Montroll could be the winner. In this twist, Bob would lose despite being in a stronger position. To avoid such situations, a mathematician, Marquis de Condorcet, proposed a new idea. Elect a candidate who would win in a head-to-head election against every other candidate. For example, in this case, if you compare B and A, B wins. In a contest between B and C, B wins again, making B the overall winner. But in our original example, if we hold head-to-head contests, A beats B, B beats C and C beats A. This creates a cycle, known as the Condorcet loop, where there is no clear winner. In mathematical terms, this outcome is intransitive. Ideally, elections should be transitive, meaning if voters prefer A over B and B over C, they should logically prefer A over C as well. This was one of the five conditions for a fair voting system laid down by the mathematician Kenneth Arrow. But later on, his own Nobel Prize winning Impossibility Theorem showed that no voting system can meet all five criteria perfectly. So, is the search for an ideal voting system over? Not quite. New methods keep emerging like range-based voting, where voters score each candidate and the one with the highest total wins. Variations of this method are often used to rate products online, movies on IMDb, and in judging competitions. Another new approach is proportional representation, where each party gets seats based on their vote share. For example, in a district with 10 seats, a party with 50% of the vote gets five seats, one with 30% gets three seats and one with 20% gets two seats. Okay. So with all these methods to choose from, is there a perfect one that works in every situation? From national elections to simple group decisions, the short answer is no. Each system has its own strengths and flaws. But does that make the choice of voting system trivial? Far from it. In fact, it's the most powerful decision in any election. Let's see why with one last example involving three candidates and 17 voters. Using plurality voting, A wins with seven votes. With Instant Runoff, B is eliminated and C wins with nine votes. And in a Condorcet system, B wins against both A and C. The outcome changes entirely just based on which system is used. So where does the real power lie? With the voters or with those who choose the voting method?
