The Borda count

A points, or positional, method. Its variants score sports awards and the Eurovision Song Contest.

The Borda count gives each option points for its place on every ballot. With three options, first place earns 2 points, second earns 1 and last earns 0. The option with the most points wins, and the totals put every option in order.

Nickname: Keeping score
Proposed: 1784, and by Nicholas of Cusa in the 1430s
Used for: Two seats in Slovenia's parliament, Nauru's parliament (a variant), baseball's MVP award, the Heisman Trophy, and Eurovision (a variant)
Best at: Rewarding options that are high on most ballots, and ranking every option
Watch out for: Near-identical options teaming up, and strategic voting

How it works

Everyone ranks the options. Best first.
Hand out points. With N options, each ballot gives N − 1 points to its first choice, N − 2 to its second, and so on down to 0 for last.
Add them up. The option with the most points wins, and the totals rank every other option too.

Example: nine friends pick dinner. Four rank tacos, pizza, sushi; three rank sushi, pizza, tacos; two rank pizza, sushi, tacos. First place is worth 2 points, second 1.

Ballots	🌮 Tacos	🍕 Pizza	🍣 Sushi
4 taco fans	8	4	0
3 sushi fans	0	3	6
2 pizza fans	0	4	2
Total	8	11, wins	8

Pizza wins with 11 points. It was only two friends' favorite, but it was second for the other seven. See the points add up in the story.

Variations

Top lists: Only the top few places score. The Heisman Trophy gives 3, 2 and 1 points for a voter's top three players; Eurovision gives 12, 10, then 8 down to 1 for a country's top ten songs. A group's top 10 list works the same way.
The Dowdall system: Used in Nauru: first place gets 1 point, second 1/2, third 1/3, and so on, which rewards first choices more.
Finding the worst: When everyone ranks every option, the lowest total is the option the group likes least, which makes points handy for tastings and contests.

A short history

1283
The Catalan philosopher Ramon Llull describes a points-like election in his novel Blanquerna.
1430s
The German philosopher Nicholas of Cusa, later a cardinal, proposes nearly the same method for electing the Holy Roman Emperor. It isn't adopted.
1784
Jean-Charles de Borda, a French naval engineer and mathematician, presents his method to the French Academy of Sciences. He said he'd first described it as early as 1770.
1795
The Academy starts electing its own members with the Borda count.
1800
The Academy stops relying on it, partly because members learned to game it, with Napoleon pushing for change. Borda's reply to complaints about strategic voting: “My scheme is intended only for honest men.”
1971
The Pacific island nation of Nauru starts electing its parliament with the Dowdall variant.
Today
Slovenia elects two members of parliament this way, for its Italian and Hungarian minorities. Sports awards such as baseball's Most Valuable Player and the Heisman Trophy use points, as does Eurovision.

Strengths and weaknesses

Strengths

Uses the whole ballot. Every place counts, so an option that's high on lots of ballots does well even if it's few people's favorite.
Finds broadly acceptable winners. Points reward the option the group as a whole likes best, which suits groups that have to live with the result together.
Ranks everything. The totals put every option in order, not just the winner, so the same count makes top 10 lists and finds the least favorite.
Simple arithmetic. It's just addition. Each polling place can total its own points, and the results add up.
More support always helps. Ranking an option higher can only raise its score.

Weaknesses

Teams of clones. Adding near-copies of an option can push it to the top, because the copies take high places on ballots that would have gone to rivals.
Strategic voting. Voters can bury a strong rival by ranking it last, even if they don't dislike it. Borda counts are unusually easy to game this way.
A majority's favorite can lose. If three of five voters rank A first and B second, but the other two rank B first and A last, B wins on points (7 to 6), even though most voters put A first.
Ranking a backup can hurt your favorite. Every point you give a second choice helps it catch up to your first.
Depends on what's on the ballot. Adding or removing an option that can't win changes the points, and can change who wins.

How it does on the fairness tests

Passes: more support never makes an option lose.
Passes: votes can be counted in separate places and added up.
Fails: if more than half the voters rank an option first, it can still lose.
Fails: an option that beats every other one-on-one can still lose.
Fails: ranking a backup can hurt your favorite.
Fails: similar options can team up and change the result.

Compare every method's results

Questions

How many points is each place worth?: In the classic Borda count, with N options, first place gets N − 1 points, second N − 2, down to 0 for last. Many variants use other scales, such as 3-2-1 or Eurovision's 12-10-8.
What if I don't rank every option?: It depends on the rules. Usually, options you leave off get no points, the same as last place.
Is the Borda count easy to game?: More than most methods, if voters coordinate. Borda himself knew it, and answered that his method was meant for honest voters.
Where have I seen it?: Sports awards like the Heisman Trophy and baseball's MVP, music competitions like Eurovision, and any ranking where places earn points.