Suppose you and a pair of friends are looking to order a pizza. You, and one friend, really like mushrooms, and prefer them over all other vegetable options, but you both also really, really like pepperoni. Your other friend also really likes mushrooms, and prefers them over all other options, but they're also vegetarian. What one topping should you get?
Clearly the answer is mushrooms, and there is no group of friends worth calling themselves such who would conclude otherwise. It's so obvious that it hardly seems worth calling attention to. So why is it, that if we put this decision up to a vote, do so many election methods, which are otherwise seen as perfectly reasonable methods, fail? Plurality, top-two runoffs, instant runoff voting, all variations of Condorcet's method, even Bucklin voting; all of them, incorrectly, choose pepperoni.
The Tyranny of the Majority
The answer is simple: All of these methods obey the majority criterion. This criterion seems so reasonable: If a majority of voters prefer one choice over all others, then that choice should be elected. And that's the case here. A majority, 2 out of 3, prefer pepperoni to all other options. Therefore, pepperoni must win. It's completely reasonable. And it means you're willing to send your friend home hungry.
If we examine the ballots these three friends would submit to the pizza-topping election, we can see the problem. Even though you and your one friend both really like mushrooms, the ballots don't show that. They only show that you like them less than pepperoni. There's no way for you to indicate the importance of that preference, and no way to mitigate it. You can't say, as you would if you and two friend were actually face-to-face and making such a decision "I like pepperoni better, but, dude, since you couldn't eat it, I'd totally be alright with eating mushrooms." Well, almost no way: You could choose to rank mushrooms above pepperoni, making a deliberate action to, after considering the thoughtful dialogue of the other voters, submerge your own narrow self-interest by voting for the choice you have determine is the one that would deliver the best result for society at-large, despite your own personal preference....
Or we could use a voting method that already knows how to do that, and lets you express your personal preferences in the same natural way that you would among your friends.
The Tyranny of Weak Preferences
The problem here is that these majoritarian methods have no way to indicate strength of preferences. Your weak preference between pepperoni and mushrooms is given as much importance as a your friend's overwhelming preference between pepperoni and mushrooms. And yet, the ability of approval voting and score voting to cleverly subvert the majority criterion in this way has been used as an argument against it. The detractors argue that it is horrible, absolutely horrible, that a candidate could be preferred by the majority of voters but that the election method would pick someone else. But is it really so horrible? After all, didn't you say you really liked mushrooms?
The truth is, under the voting methods for which I argue, voters have the power to express a willingness to compromise, and if they choose to express that desire, then these voting methods have the ability to honor it. And conversely, if the voters do not have an interest in compromising, then they can express that too.
Now, I know that, in the voting booth, we're not all friends. But that only makes it more important that we use election methods which strive for consensus, instead of divisiveness. You don't have to vote on pizza toppings with your friends, because it's a mostly unimportant decision with only a few people affected, none of which are especially invested in the outcome. You trust each other, and can quickly talk out all possible outcomes and come to a mutually-acceptable agreement. That's harder to do with 100 million angry people deciding on their government. But you still want—no, you still need—a consensus result. The majority criterion is detrimental to that goal.
Arguing Both Sides
What I find especially funny is that the other common argument used against approval and score voting is that they will degenerate to plurality voting, because voters will only vote for their single favorite choice. We know that's not true, but it's also contradicted by their arguments favoring the majority criterion. Because, for an approval or score voting election to actually exhibit majority failure, a substantial number of the voters would have to vote for multiple candidates; while, for accusations of bullet-voting to carry any weight, nearly all voters would have to vote for just one candidate. And yet, detractors argue that both of these things will happen, which is logically inconsistent.
The truth is, sometimes voters will bullet-vote, but only when it suits them to do so. And sometimes (rarely) a majority candidate won't win, but only when the voters who prefer that candidate have made the decision to express that preference as a weak preference, and defer to the whims of the minority. In other words, only when they have said "I like this other candidate better, but dude, if you really hate him, I'd totally be alright with one of these other guys."
It's called a compromise. It's how friends stay friendly, and how civilization stays civilized.