The paradox of voting
Going to the cinema alone
On a walk by yourself, you stop at the cinema. The three-screen cinema is showing Robot Laser Wars, Lawyer 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 Lawyer 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 Lawyer 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 Lawyer 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 Lawyers.
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 > Lawyers > Giggles|
|Spouse:||Lawyers > Giggles > Robots|
|Kid:||Giggles > Robots > Lawyers|
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 Lawyers, Robots win.||(You and your kid will vote that way.)|
|If the contest is between Lawyers and Giggles, Lawyers win.||(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 Lawyers, and Lawyers 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. Definitionally, it is better than second-best and 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.
See also: Arrow's impossibility theorem
The irrationality of voting
Here's a separate question. This is not "the paradox of voting." It is perhaps related, though.
Is it rational to vote?
"...the entire problem of collective action is that it’s rational to act collectively where it’s not to act alone." - Alyssa Battistoni, "Spadework: On political organizing." n+1. Issue 34: Head Case. Spring 2019.
Ryan Brumberg's article "The Democratic Religion" in the Fall 2006 issue of The Dissident explored why we vote. "It's not logical to vote," he began. After all, the likelihood that a large election will be able to be flipped by a single voter is about 0.0000001 (as calculated by William Riker and Peter Ordeshook, probably in their 1968 article "A Theory of the Calculus of Voting"). There's a greater chance that your vote will be subject to a counting error. You are also far more likely to die on the way to the polls (by accident or murder) than to cast a tiebreaking vote. (Then, assuming you have a moral imperative to keep yourself safe, might you not be morally obligated to stay home despite your desire to influence the election? What obligation would pull you into the relatively dangerous streets?) Even if you want to believe — with faith, contrary to logic — that those odds are somehow in your favor and that you are that one person who will flip the election, how can you justify it? Why do you bother to go to the polls? And why do so many people make the same choice, despite the investment of time and risk? Why do people vote?
Some make the Kantian argument that there is a "categorical imperative" to vote, regardless of the consequences of voting or not voting. The same obligation applies to everyone. Everyone is supposed to vote. That's that.
If consequences matter to you, you ought to consider (as Brumberg put it) that "any benefits derived from an election are available to everyone," regardless of who they voted for or whether they voted at all. Voting is not like paying taxes. Whereas an individual's tax payment has "a real (however minor) impact" on the total available funding, "in winner-take-all elections such as those we hold in the United States, it does not matter whether a candidate wins by 5 votes or by 5 million; a win is a win." The public outcome is exactly the same; adding your vote to the final total doesn't affect the elected leader's mandate.
Brumberg references ideas of Melissa Acevedo and Joachim Krueger (perhaps in this article?) regarding psychology. The "voter's illusion" is that everyone else (or, especially like-minded people, that is, people who would vote for the same candidates) will tend to make the same decision they will. If they vote, so will everyone else; if they don't vote, neither will anyone else. This may either be a predictive expectation, or it may be magical thinking that their behavior somehow influences others. Either way, people vote to maintain their optimism. Then there's the "delusion of personal relevance," the belief that their personal vote might tip the election. "It may be that people are used to participating in small-scale cooperative situations where their actions have a readily perceivable effect on the overall outcome," Brumberg paraphrased. For voting, which is "a much larger-scale cooperative problem, they are "inappropriately transferring a belief in personal relevance to the national scale." Brumberg believes that both of these psychological mechanisms — the "illusion" and the "delusion" — "seem to be fallback defenses that people use when they're challenged about the rationality of voting."
Brumberg believes that voting is a social norm, and so people vote without "think[ing] much about whether voting itself makes sense." He is interested in the research in reciprocal altruism, especially when the reciprocity is indirect. Here, he cites the work of mathematicians Martin Nowak and Karl Sigmund (probably their 1998 paper "Evolution of Indirect Reciprocity"). They point to the fact that "discerning cooperators—who, in the simplest case, are charitable to fellow cooperators, but punish those who do not cooperate" tend to win in contests that demonstrate how they "reap the benefits of repeated cooperation without being taken advantage of." They build their reputation in the community.
This can't be the whole story, he says, because the anonymous act of voting doesn't deliver benefits or build one's reputation. What, in addition to reciprocal altruism, could explain it? Brumberg notes that people "have been taught that voting is an obligation of citizenship" and are therefore "impervious to any evidence or reasoning about the inconsequentiality of their votes. Faith in democracy, like faith in a supernatural being, is outside the scope of reason, oblivious to its power — and this faith is, therefore, an effective enforcer of norms. By contrast, any norm that is subject to shifting calculations of the odds will not be very reliable." With this attitude, people "police themselves to perform the normative rituals of democracy — such as voting — even though these rituals are of no help to themselves or to anyone else." Of course, "some of us are believers and others are cynics," and why people might possess either fundamental attitude is itself a mystery, he says. He concludes:
...the gene for social cooperation is seized upon by every kind of culturally transmitted religion, which can induce human beings to take individually irrational actions precisely because they are individually irrational—but collectively promulgated. That is the essence of faith; its ritual acts are those of collective endeavor.
To demand justification for religious rituals will prompt people to produce rationalizations, not reasons. And in the democratic religion, the ultimate ritual is the act of voting.