Going to the cinema alone
The three-screen cinema is showing Robot Laser Wars, Legal Drama, and Giggles the Bear. You’re hoping for Robot Laser Wars; it’s something you can only see without your spouse, who wouldn’t appreciate it. When you arrive, there is a confusing sign saying “Next showing sold out.” You’re unsure which of the three films is sold out. Before you approach the ticket window, you privately rank your preferences so you’ll have a second choice ready to go in case it’s your first choice that’s sold out. It’s not hard. You’ll settle for Legal Drama. You feel much too old for Giggles the Bear.
If your spouse were in the same situation alone at the cinema, you’re sure that Legal Drama would be his first choice, and furthermore that he’d rather see Giggles the Bear before setting foot in Robot Laser Wars.
And your child? She’s still young enough to prefer Giggles the Bear. She might be entertained by Robot Laser Wars as a second choice, but she’d have no comprehension of or attention for Legal Drama.
The purpose of a “first, second, third” preference ranking is obvious. Regardless of which film is sold out, you know which of the two available films you want. If your first choice is sold out, you want your second choice. If your second or third choice is sold out, it’s no problem—you still want your first choice.
It barely rates mentioning but, for the record, and because it will be relevant later: Since you place your first choice over your second choice, and your second choice over your third choice, it stands to reason that you place your first choice over your third choice. For you, personally, Robots still beats Giggles regardless of your middle preference for Legal.
Going to the cinema with your family
Now look at what happens when the family goes to the theater together. You retain your individual preferences:
|You:||Robots > Legal > Giggles|
|Spouse:||Legal > Giggles > Robots|
|Kid:||Giggles > Robots > Legal|
You first need to find out which film is sold out so you can rule it out. Given the two remaining films, you’ll take a majority-wins vote. If everyone votes their own interest, the vote will come down 2 against 1.
|If the contest is between Robots and Legal, Robots win.||(You and your kid will vote that way.)|
|If the contest is between Legal and Giggles, Legal wins.||(You and your spouse will vote that way.)|
So, if the contest is between Robots and Giggles, won’t Robots win? After all, if the group prefers Robots over Legal, and Legal over Giggles, doesn’t it stand to reason that the group prefers Robots over Giggles? That’s how it works for you as an individual when you rank your first, second, and third choices. It’s just what it means to rank your individual preference. To you, Robots is the best, so it remains the best. It is better than second-best and definitely better than third-best. It seems as if that's what the group believes, too. And yet...that’s not always how it works for a group. In this scenario:
|If the contest is between Robots and Giggles, Giggles wins.||(Your spouse and your kid will vote that way.)|
Effect on elections
Individual preferences seem to operate by slightly different rules when they are aggregated into a “group preference.” This is called the paradox of voting. As Pierre Lemieux puts it, in the situation above: "The electorate is irrational even if each voter is rational....'We as a society' is more a casino roulette than a rational actor."
When a society doesn't have a clear preference about which direction is best, narrowing the available choices down to two and then holding a majority-wins election is one way to reach a short-term solution, but it won't resolve the long-term question about which direction really is best. The society will keep cycling the question and rehashing the debate. They really don't agree and the only way to pretend there's a majority census is to artificially narrow the options or change the framing. If there is a way to bring people to consensus, it may require a novel approach (such as teaching people to empathize with each other's wants and needs); holding yet another election probably won't do it.