By Mark Colyvan & Brian Hedden

Most of us have great faith in democracy. Such is this faith that when democratic processes deliver poor results (as many believe occurred in the 2016 US election and the Brexit referendum) we wonder what went wrong and look for answers, rather than questioning democracy itself.

Why this faith in democracy? A popular thought is that democratic governance delivers power to the people and thus issues policies and decisions that are “the will of the people”. But this thought faces serious challenges from the mathematics of voting. The notion of the will of the people makes sense when the people agree, but what happens when there is disagreement? Disagreements are typically settled through voting. But there are many very different voting systems, which can, and do, lead to different results. Moreover, there is no one voting system that can be singled out as the right one.

Let’s start with majority rule, where a candidate or policy wins if a majority of voters pick that candidate or policy. This voting system works fine when there are just two options. But if there are more than two, it might not give any outcome at all, since it’s possible that no option wins a majority of votes. Worse still, sometimes the majority view (when it exists) is not the right or fair outcome, since it doesn’t take into account the strength of people’s preferences. Consider three friends deciding on a movie at a film festival. Two have a very slight preference for George Romero’s Dawn of the Dead but would also quite like to see the Ridley Scott film Blade Runner. The third friend has a strong preference for Blade Runner and cannot stomach zombie movies such as Dawn of the Dead. Clearly the fairest outcome is to see Blade Runner, despite Dawn of the Dead being the first choice of the majority.

If we take full preferences into account—not just first preference—things do not get any better. Consider a case of seven people deciding between three options A, B and C. Three vote (from best to worst option) for A, B, C. Two vote for B, A, C and two for C, B, A. On one way of tallying the votes, called “plurality rule” or “first past the post”, an option wins if it’s ranked first by more voters than any other option. In our example, option A wins by plurality rule because it’s ranked first by three voters, while each of B and C are ranked first by only two. But a clear majority (four to three) prefer B to A. Which, then, is the will of the people?

Making use of voters’ full preferences, as in “preferential voting” systems, can yield further problems. For example, there’s the Condorcet paradox. Consider three voters who express their preferences over three options, A, B, and C. Voter 1 prefers A to B, and B to C. Voter 2 prefers B to C and C to A. Voter 3 prefers C to A and A to B. Here, the problem is that the group seems to have irrational preferences. After all, if A is preferred to B and B is preferred to C, it is only reasonable that A is preferred to C. But by a 3 to 2 majority in each case, the group prefers A to B and B to C, but also prefers C to A. The individual voters are rational but the electorate (under this voting rule, at least) is irrational.

We’ve looked at only a few simple voting systems, but there are hundreds more. At first glance they can seem equally reasonable, but they can give different results. How might we choose one?

Let’s put in place a few conditions on what we think a fair way of aggregating individual preferences should look like:

1. The unique social ranking should be possible for any set of individual preferences. (The “unrestricted domain” condition);

2. If every voter prefers A over B, then the social ranking prefers A over B. (The “unanimity” condition);

3. The social ranking with respect to A and B should depend only on the individual rankings of A and B, and not how individuals rank A and B relative to some third option C. (The “independence of irrelevant alternatives” condition);

4. There should not be a single voter such that the social ranking is always the same as that voter’s ranking. (The “no dictator” condition).

These conditions seem very reasonable. But surprisingly, it’s impossible for any voting system to satisfy all of them. This remarkable result was proven mathematically by Nobel Prize winner Kenneth Arrow (1921–2017) in 1951. While the philosophical significance of Arrow’s discovery is up for debate, it seems to suggest that there is no perfect voting system that truly reflects “the will of the people”. Indeed, on one reading, there’s no such thing as “the will of the people”.

Of course democracy has a great deal going for it. For instance, it may safeguard against rule by a corrupt handful of elites, yield better economic growth, prevent wars and famines, and make citizens feel engaged and part of a connected society. But the mathematics of voting suggests that democracy doesn’t mean that policies reflect the will of the people, at least not in any straightforward sense. And if there’s no sense to be made of the will of the people, why would we want power resting with the people?

Mark Colyvan is Professor of Philosophy at the University of Sydney and Humboldt Fellow, Munich Center for Mathematical Philosophy, Ludwig-Maximilians University, Munich. Brian Hedden is Senior Lecturer in Philosophy at the University of Sydney.