Or: Why Tyranny of Majority only supports questions with simple yes or no answer.
Voting between two choices is simple: each voter votes one or the other and the one with the most votes wins. Easy enough.
However when there are three or more options things get more complicated. And not just a little bit but a lot. The kind of complicated that there has been scientific research on it and the result is a theorem that there does not exist a completely problem-free method.
Instead of one best method we have several different methods, each with it's own strengths and weaknesses, though some methods are still clearly better than others.
Sceptical? Can it really be that difficult to make a choice between 3 options instead of 2? Let's have a simple example: A group of 15 people need to make a decision. There are three options: A, B and C. Only one of them can be chosen. The group can't come to a consensus about the best choice, so they decide to choose the best option for the group by a democratic vote. With only 3 choices, how hard can it be?
Example 1: Plurality - the simple way
Let's just ignore the complications and use the same system we used for two options: everyone gets to vote the option they think is the best and the option with most votes wins. This is the most obvious, simple and common solution, but also the most problematic.
The three options receive votes from 15 voters like this:
- A gets 6 votes.
- B gets 5 votes.
- C gets 4 votes.
=> A gets most votes, therefore A wins.
Choice A has most supporters and wins, which seems fair. However, what if those who voted for B and C consider A to be the worst possible choice? In that case the election ends up with a choice that a majority of voters (9 versus 6) consider to be the worst possible outcome.
Point of the voting system is to reflect people's will, and if people end up with a choice that majority consider the worst alternative, their will does not seem to be reflected very well. Yet this is the most commonly used voting system around the world.
Another thing: what if choice C were dropped from the election? Then suddenly B would win instead of A. That means that the choice between A and B depends on whether a third option C is available or not. This violates Independence of irrelevant alternatives criteria which states that introducing a third option should not affect the choice between A and B. That is, adding choice C should affect the election result only if C ends up winning.
This can sometimes discourage candidates running in elections: if B and C happened to be candidates who belong to the same or similar party, then if both run, it would result in opposing party candidate A winning. But if either B or C decided to drop out before election, then the remaining candidate would win over A. Therefore it would be rational to one of them to voluntarily drop out of the election.
Plurality causes problems also for the voter: let's say a voter prefers option C, but could live also with B and loathes A. If he suspects that B might be more popular than C, he might figure that it is better to vote for B instead of his preferred choice C, just to prevent A from winning. And that would be correct: if all supporters of C voted B instead (or vice versa), then B (or C) would win instead of A. So instead of voting honestly for the best choice it can be rational to tactically vote for another candidate instead. Probably needless to say, a voting system which discourages honest voting is not a good one. Also simply a perceived weakness of C can cause voters to flock over to B, making it a self-fulfilling prophecy, even if in reality C would have been the more popular candidate.
Whether done by candidates by themselves or by the voters, the plurality voting tends to limit voter options to the (assumed) strongest two choices. This kind of solves the problem of multiple-choice elections by simply giving voters only two choices to choose from. However from the perspective of democracy it would be better to have more, not less, options to choose from. Plurality tends to lead to polarization and two-party systems where instead of the best candidate people vote for lesser of the two evils just to keep the bigger evil from winning.
In this sense even random ballot would be a better method: at least in it a voter can vote honestly because his choice between B or C does not affect in any way the chances of A winning.
Example 2: Multiple rounds
One way to fix this is to have the election in two rounds: on the first round the candidate with the least votes is eliminated and the final round is between two choices, which is again simple. So in our example C would be eliminated in the first round and on the final round those who voted for C vote instead for B:
- A gets 6 votes
- B gets 9 votes
=> B gets most votes on the second round, therefore B wins.
B beats A 9 - 6 and B is declared the winner. So while A has more direct supporters than B, majority of voters prefer B to A. Much better result, right?
Yet what if all of the A supporters would have preferred C over B? That is, if there were an election between B and C, C would win? This would mean that election was won by a candidate who would lose a one-on-one election to another candidate, which does not seem fair. Only reason this didn't occur is that C happened to be eliminated in the first round.
Example 3: One-on-one between each candidate
A way to fix this would be to have one-on-one elections between each of the candidates:
- A vs. B: B wins 9 to 6
- A vs. C: C wins 9 to 6
- B vs. C: C wins 10 to 5
=> C beats every other candidate on one-on-one elections, therefore C wins.
A loses to both B and C, since most voters prefer any other candidate than A.
Since A supporters like C more than B, C wins B in one-on-one and therefore the whole election, even though it was the first one eliminated in multiple round voting.
Conclusion
So there we have it: three different voting methods, three different results, yet each can be said to be chosen democratically. This shows that voting between three or more options is a very different thing from voting between two options and the method of determining the winner clearly matters.
Recommendations:
- Stick to selections with two choices as much as possible, just to keep things simple. If you have a question with multiple choices, try to split it to several two-choice questions if possible.
- If you really need to have multiple choices (say election with multiple candidates), choose your election method carefully.
- Avoid plurality as much as possible. If simplicity is important, use approval voting. Or even random ballot.
- If a bit more complexity is acceptable, use multiple voting rounds. If the voters jot down their order of preference in a single ballot, no need to even have voting between rounds.
- If complexity is not a problem (either can use computers or have plenty of time for manual counting) one of the Condorcet methods is a good choice. Schulze method seems to be popular.
