Electoral Systems

Electoral systems are used for a variety of purposes:

• Electing single office-holders, such as mayors or presidents
• Electing groups such as councils and parliaments
• Selecting between two or more policies, especially on a public scale

They are also used by different groups and in different contexts:

• Small groups
• Companies and other organisations
• Localities and states

No electoral system is perfect: selecting one always involves trade-offs. Some of these are mathematical in nature while others are political. I think that that is what I find fascinating about them: comparing different systems and the ways that they go wrong. This page will provide an overview of electoral systems and of the properties that they may fulfil. This, and my other pages, will eschew formal notation and rigour in favour of being somewhat comprehensible to a layperson.

Components of electoral systems

Electoral systems are comprised of rules that prescribe:

• Who can vote
• Where, when and how they vote
• The structure of the ballot and what information the voter puts on it
• How votes are counted
• Who can run for election or what the options to choose from are
• The method used to determine who or what the winners are
• If an assembly is being elected, how the districts are delimited

I will focus on the method of determining the winner as that is what interests me the most. The other components are important, perhaps moreso than the specific method used, but they aren't as interesting to me.

Types of electoral methods

There are multiple ways that systems are classified. The most important are whether they elect one member or multiple and how format of the ballot. Single-winner systems are used for electing individual office-holders or selecting between alternatives. Multi-winner systems are used for electing assemblies. In general, multi-winner systems are definied for an arbitrary number of winners; single-winner systems are a special case of multi-winner systems. Conversely, single-winner systems can be converted into multi-winner systems by either dividing the electorate into groups, each of which elect one member, or by removing the first winner and repeating the system with the remaining candidates. The two main ways a ballot is filled out are by selecting a single option, ranking the options or by giving each option a score. First-past-the-post and most party-list systems require a single mark. Ranked systems are the most diverse and include instant-runoff voting and points systems, among others. Rated systems may limit how many points each voter can give out or allow them to rank each option independently.

Properties of electoral methods

Electoral systems can meet certain properties. Some of these represent what we want in electoral systems, but are hard to exactly define:

• How fair they are to voters and candidates
• How well the results represent voters' interests
• How difficult they are to game
• How easy it is to vote

These often involve factors beyond the electoral method, such as voter registration requirements, the availability of alternatives to in-person voting on the day and how districts are drawn. As such, there can be quite a deep discussion of these properties without involving much maths at all.

Other properties are more rigourously defined, but focus only on the electoral method alone and are only an operationalisation of the looser properties described above. Some of the ones that I feel are the most important are:

• The sincere favourite criterion: Does it always make sense to give full, though not neccesarily exclusive, support to your preferred option?
• The participation criterion: Is an honest vote always benign or can it backfire and lead to a less-preferred option being chosen?
• The monotonicity criterion: Can giving a candidate a better score or rank cause them to lose?
• The later-no-help and later-no-harm criteria: Can how a voter ranks or rates less-preffered options influence the fortunes of more-preferred options? If so, how?
• The majority, mutual majority and Condorcet criteria: How well does a single-winner electoral system reflect majority preferences?
• Proportionality for solid coalitions (ranked) and various proportionality criteria (rated): Is a multi-winner system propotional?

That being said, these don't always correspond to the looser properties that I am actually interested in.

Specific electoral systems

Here, I'll just list specific systems. In time, each will have their own page that describes them.

Single-winner systems

• Single-member first-past-the-post
• Approval voting
• Points systems
• Score voting
• Bucklin voting
• Highest-median systems
• Instant-runoff voting
• STAR voting
• Schulze system
• Condorcet/instant-runoff hybrids

Multi-winner systems

• Single-winner systems in multi-member contexts
• Plurality-at-large (single non-transferable vote and multiple non-transferable votes)
• List systems
• Single transferable vote
• Approval-based systems
• Mixed-member systems