If you're building a system that incorporates voting, or helping determine the election rules for an institution, there are a lot of potential voting systems for you to choose. This article explores why you might get better outcomes by choosing a voting system that is more "expressive" than common systems like plurality vote and ranked choice voting.
Intuitively, we consider a voting system to be more "expressive" the more detail it allows voters to provide about their preferences. Slightly more formally, we consider voting system A to be “at least as expressive” as another voting system B if we can define a function that translates any set of voter preferences in B into a set of preferences in A that would generate the same election outcome. If B is not also at least as expressive as A, then we say A is “more expressive” than B.
As an example, consider the common "Borda Count" ranked voting system in comparison to the “Range Voting” system which allows every voter to rank every option on a continuous scale (say, anywhere between 0% and 100%). Range voting is at least as expressive as Borda count: it is trivial to turn Borda ranks into range voting scores that will lead to the same election outcome. For instance, in an election with three options, you could assign the top-ranked Borda option a range-vote score of 100%, the middle-ranked option a score of 50%, and the bottom-ranked option a score of 0%. However, Borda count is not as expressive as range voting. To see why, just consider that range voting allows voters to give two options the same score -- how would you express a tie when you are forced to rank each option?
If you are willing to assume that voters are honest (that is, they won’t misrepresent their true preferences in order to get a better outcome), then a more expressive voting system will collect more information about the overall utility of each option and will thus be able to identify superior choices for the group of voters as a whole. In fact, even with dishonest voters, expressive voting systems can still capture more information.
Consider a case where two voters are considering three options, but one has much stronger opinions than the other. If Sue and Al use a Borda count to decide what to do, they’ll end up watching TV:
|Strong Sue||Apathetic Al|
|First Choice||Go to the Beach||Watch TV|
|Second Choice||Watch TV||Go Hiking|
|Third Choice||Go Hiking||Go to the Beach|
Now consider the same two voters using range voting:
|Strong Sue||Apathetic Al|
The ranking is the same as before, but Al just doesn’t feel nearly as strong as Sue about the issue. With range voting, they’ll end up going to the beach. You might say it’s unfair that they chose Al’s least favorite option, but collectively they should be happier.
A theoretically more expressive voting system is also more likely to satisfy voters’ emotional desire for expression. Whatever other motives we have for participating in elections, we almost always want to feel that "our voice has been heard". In some contexts, that feeling of “being heard” is more important to the voter than the actual outcome of the election.
Often it is provably impossible for any voting system to simultaneously satisfy even a small number of reasonable-sounding requirements -- for an example see Kenneth Arrow’s famous "Impossibility Theorem". Because of these surprising constraints, choosing a voting system always involves making trade-offs regarding which requirements are most important.
Let’s consider the range voting method we’ve been using in the examples so far. It’s a good candidate as an "expressive" voting system for allowing a group to choose or rank a finite set of options. Specifically, in 2011 Marcus Pivato proved:
Let X be a ﬁnite set. Range voting is the most expressive X -valued voting rule which satisﬁes reinforcement, neutrality, overwhelming majority, and does not admit minority overrides.
While range voting satisfies a number of reasonable requirements, there are also lots of requirements it doesn’t satisfy. Perhaps one of the most obvious requirements it doesn’t satisfy is the "Majority Criterion": an absolute majority of voters may rank one option as their top choice, but a vocal minority could cause a different option to be chosen. To see why, just imagine a case where four voters rate option A slightly above option B, while the fifth voter rates option B “100” and option A “0”. The majority prefers A, but the fifth voter will be able to tilt the scales to B. If you’re wondering how the majority criterion is different from “overwhelming majority”, check out this summarization of Pivato’s characterization result.
In fact, range voting fails to satisfy several criteria that might seem a necessary part of a "fair" voting system. To get an idea of what criteria are satisfied by various popular voting systems, check out this great chart from the Wikipedia article on voting systems.
There’s rarely one right answer when it comes to voting systems. If you’re planning an election, you need to carefully consider who your voters are and what they hope to get out of the election.
When we were building VoteUp, we decided that we wanted to focus on elections among groups of friends, where opinions mattered but the stakes were low and trust was high. Given those conditions, range voting fit well with our goals. Even so, some of our users have requested we switch to some version of the very similar Score Voting -- it’s just like range voting except the scores are discrete instead of spanning a continuous range. Score voting is still a very expressive voting system and might in some ways be easier for users to think about, but we decided that the extra emotional expressiveness of range voting was worth the trade-off.
People’s opinions are messy, but as a matter of faith we believe that anything we can do to help people express themselves more fully will eventually help the world.