Chapter 3: The tournament

Fay, a sushi fan, has been thinking.

“Instead of knocking places out, let's hold a tournament. Every place plays every other place, one on one. In each match, everybody sides with whichever of the two they ranked higher.”

First match: pizza against tacos. Forget sushi for a moment.

The four taco fans side with tacos, of course. But the sushi fans ranked pizza above tacos, and so did the pizza fans. Pizza wins, five to four.

Next: pizza against sushi. The taco fans ranked pizza second and sushi last, so they side with pizza. So do Hal and Ivy. Pizza wins again, six to three.

Last match: sushi against tacos. The sushi fans and the pizza fans all ranked sushi above tacos. Sushi wins, five to four.

That's the same matchup as the last round of the runoff, with the same result.

Add it up. Pizza won both of its matches. Sushi won one. Tacos won none.

When one option beats every other option head to head, it's called the Condorcet winner, after the Marquis de Condorcet, a French mathematician who described the idea in 1785. It's a hard winner to argue with: whatever you'd rather have instead, more of the friends prefer pizza to it.

Fay is a sushi fan, but she has to admit it. By her own rules, it's pizza.

Cal asks the obvious question.

“What if nobody wins every match?”

It can happen. Picture a different night with different ballots, where tacos beats pizza, pizza beats sushi, and sushi beats tacos. Like rock, paper, scissors, there's no undefeated champion. Mathematicians call it a Condorcet cycle. Everyone else calls it a loop.

The ballots that night: four friends rank tacos, pizza, sushi. Three rank pizza, sushi, tacos. Two rank sushi, tacos, pizza.

So a tournament needs a tiebreaker for loops. A popular one is the Schulze method, invented by Markus Schulze in 1997. With three options, it comes down to this: find the narrowest win in the loop, and set it aside.

Here, that's sushi's five-to-four win over tacos. Set it aside and what's left is simple. Tacos beats pizza, pizza beats sushi, and tacos wins.

Loops are rare, especially with lots of voters, but a good way of counting needs a plan for them anyway. The Debian project, which makes free software that runs on millions of computers, has elected its leaders with the Schulze method for years.

Back at dinner, there's no loop. Pizza beat everything.

But Dee, a taco fan, wants to try one more way... Chapter 4: Keeping score→