Voting definitions and examples

Approval Voting

Approval Voting is an alternative voting system for single-winner races.  Each voter simply votes for (approves) one or more candidates; the winner is the candidate with the most approval votes.  Ballot formats and voting equipment don't have to be changed for Approval Voting and runoffs are never necessary.  In fact, in a sense, Approval is even simpler than the current plurality system, since the vote-for-only-one restriction is removed.  No ballots are thrown out for overvotes; the votes are simply added up.  Unlike more complicated alternative single-winner systems like IRV, Approval Voting completely eliminates the spoiler and lesser-of-two-evils problems of plurality.

consistency

A voting system satisfies consistency when combining groups of votes that individually give the same winner never results in a larger group of votes with a different winner.  For example, say a town election uses two precincts of equal size, and they cast the IRV ballots

In precinct 1, Clark is eliminated first and Brown wins; in precinct 2, Allen is eliminated first and Brown wins.  So Brown wins each individual precinct under IRV.  But when the votes are combined for the town-wide result, giving

IRV eliminates Brown first and gives the victory to Allen.  Therefore, like most ranked-ballot voting systems, IRV is inconsistent.  By contrast, it's easy to see that Approval Voting is consistent:  Combining results from different precincts entails simply totaling each candidate's approval votes, and if the same candidate wins in each precinct, that candidate will necessarily win the combined election as well.

Instant Runoff Voting

Instant Runoff Voting, or IRV, is the name given to a single-winner ranked-ballot voting system by the Center for Voting and Democracy for promotional purposes in the United States.  Outside the U.S. it's known as Hare, preferential voting and the alternative vote.  IRV requires ballots that allow ranking all the candidates in each race.  IRV can be thought of as simulating repeated runoff elections.  It looks only at the top choices of the ballots and eliminates the one candidate with the fewest first-place votes, then redistributes the votes for the eliminated candidate according to the ranked votes and continues eliminating candidates until only one is left.  To illustrate, consider an election with five candidates arranged along a left-right spectrum:

So 19% of the voters rank FarLeft first, Left second, Center third, Right fourth and FarRight last; 21% of the voters rank Left first, FarLeft second, and so on.  FarRight has only 16% of the first-place votes and so is eliminated by IRV first.

Now FarLeft has the fewest first-place votes with 19% and is eliminated.

Center is eliminated next with 26%.

Right beats Left in the final simulated runoff 51% to 49% and wins.

How fair is this result?  Well, 66% of the voters preferred Center to Right, the IRV winner.  In fact, Center would have won a final runoff against any one other candidate, and more voters had Center as a favorite than any other candidate!  It's hard to justify IRV's choice of Right in this election.  Anomalies like this aren't uncommon under IRV, and since IRV requires more complicated ballots and fails at eliminating the spoiler and lesser-of-two-evils problems of plurality, we consider it a poor single-winner voting reform.

lesser-of-two-evils dilemma

Also known as the wasted-vote syndrome, the lesser-of-two-evils dilemma is one of the two principal problems of plurality voting that alternative systems such as Approval Voting and IRV try to solve (the spoiler effect is the other).  A voter is faced with the dilemma when it's smart to vote a compromise candidate over a favorite.  For example, Nader supporters in Florida faced the lesser-of-two-evils dilemma when deciding whether to vote sincerely for Nader, who didn't have a reasonable chance to win, or insincerely for Gore, whom they preferred to Bush, the other frontrunner.  If more Nader supporters had made the smart choice of voting insincerely for Gore as a hedge against Bush, Gore would certainly have won Florida and thus the presidency.  Most voters have at some time had to choose between "throwing their vote away" and voting for a "lesser of two evils."

IRV tries to get rid of the lesser-of-two-evils dilemma by tranferring votes for eliminated candidates to compromise candidates.  Unfortunately, doing so only sometimes solves the problem.  IRV can often eliminate a voter's compromise candidate before the favorite and allow the least favorite candidate to win, giving that voter incentive to vote the compromise insincerely first.  For example, given the votes

IRV would eliminate McCain first and Bush would beat Gore in the simulated runoff.  (Note that McCain would have beaten either Bush or Gore if he had made it into the runoff.)  The Gore voters let their least favorite candidate win since their preference for McCain over Bush is ignored.  But what if they moved their lesser evil up to first place and dropped their favorite?

Now McCain wins, and the Gore supporters succeeded in preventing Bush's win.  As in this example, IRV often rewards voting a lesser evil over a favorite candidate, but Approval Voting never does.  Approval may sometimes reward voting for compromise candidates in addition to a favorite, but it never encourages dropping the favorite like plurality and IRV do.

monotonicity

A voting system satisfies monotonicity when improving the position of a given candidate on some voted ballots can never cause that candidate to lose, and dropping a candidate on some voted ballots can never cause that candidate to win.  A voting system that can cause that counterintuitive behavior is said to be nonmonotonic.  For example, if the votes in an Approval election are

then Evans wins with 65% approval.  If some of the 10% started approving Evans, it could never turn Evans from a winner to a loser, and if some of the 35% stopped approving Davis, it could never turn Davis from a loser to a winner.  It's easy to see that Approval Voting is monotonic.  On the other hand, consider the following IRV example:

Imran is eliminated first and Grace wins the simulated runoff against Hayes.  But if the 10% voters decide that Grace is better than Hayes after all and uprank Grace to first place, the votes become

and then Hayes is eliminated first and Imran wins the simulated runoff against Grace.  So Grace went from a winner to a loser when some voters started voting him higher.  Not even plurality is capable of such ridiculous behavior.  Voting systems that use an elimination scheme, like IRV, are often nonmonotonic.

plurality voting

Also known as first-past-the-post, plurality is by far the most common voting system for single-winner races.  Each voter is allowed to vote for only one candidate, and the candidate with the most votes wins.  Plurality is simple and easy to count, but its well-known pathologies include the spoiler effect and the lesser-of-two-evils dilemma.  According to Duverger's Law, the result of these problems of plurality is the two-party system.

reverse-symmetry

A voting system satisfies reverse-symmetry when exactly reversing each ballot never results in electing the same winner as before.  In Approval Voting, reversing each ballot approves each candidate previously unapproved and vice versa, so the previous last-place candidate would then win the reversed election—a common-sense result.  IRV, however, is non-reverse-symmetric; if the ballots are

IRV eliminates Laker first and gives the victory to Knott.  But reversing the ballots gives

and Jones is eliminated first, allowing Knott to win again.  IRV doesn't change the winner when all voters vote their opposite preferences—it contradicts itself!

spoiled ballot

Not to be confused with the spoiler effect, a spoiled ballot is simply a ballot that has been cast incorrectly.  For example, a plurality ballot that has votes for two candidates in the same race is considered spoiled.  Similarly, an IRV ballot that gives the same rank to two candidates in the same race is spoiled.  Approval Voting minimizes spoiled ballots; there's no such thing as an overvote in Approval.

spoiler effect

Also known as the vote-splitting effect, the spoiler effect is one of the two principal problems of plurality voting that alternative systems such as Approval Voting and IRV try to solve (the lesser-of-two-evils dilemma is the other).  A spoiler is a candidate that turns another candidate from a winner to a loser simply by running.  For example, if the sincere preferences in a plurality election are

and everyone votes sincerely, Gore will win with 48% of the vote.  But Bush and McCain split the Republican vote; if McCain hadn't run, Bush would have won with 52% of the vote.  McCain spoiled the election for Bush (or, depending on your point of view, Bush spoiled it for McCain).  This spoiler effect is the main reason that parties have primaries and run only one candidate in each race of a general plurality election.

IRV tries to get rid of the spoiler effect by tranferring votes for eliminated candidates to lesser-evil candidates.  IRV would succeed in keeping McCain from spoiling the above election for Bush, but in doing so it would make Gore a spoiler for McCain!  Gore would be tempted to drop out of the race to keep Bush from winning, and if he didn't, his supporters could do it for him; they'd be tempted to vote McCain insincerely in first place, dropping Gore to second and effectively out of the race.  So IRV's clumsy attempt to avoid spoilers fails and causes strategy problems.  Approval Voting is immune to the spoiler effect since the choice of voting for one candidate is independent of voting for other candidates in the same race.