 Arrow's theorem proves that no voting system can satisfy independent of irrelevant alternatives, monotonicity, non-imposition, neutrality, and non-dictatorship. However, there is a very important assumption in the proof. Arrow's theorem relies on the existence of a preference schedule. This gives us a few ways we can avoid arrows. In approval voting, described by political scientist Stephen Brams and mathematician Peter Fishburne in 1978, voters approve of all choices they would be satisfied with. They can approve of as many or as few choices as they want, and the winner will be the choice with the most approvals. For example, suppose in an election the following approval ballots are returned. Voter 1 votes A and B. Voter 2 approves of A and C. Voter 3 approves of B and C, and so on. And we can try to determine who won the election and maybe make a few other observations. So let's collect the approval votes for each of the candidates. So A was approved of by these five voters. B was approved of by these five voters. And C was approved by six voters. So C wins the approval vote. Now what's important to notice here is that C won the approval vote, but C was supported by six of nine-two-thirds of the voters. And so whoever wins an approval vote knows how much of the electorate supported them. And conversely, the electorate knows how much of the voting population supported a winning candidate. So let's think about why it's better. Since voters can approve of as many of the choices they want, there's no reason why they wouldn't include all the choices they'd approve of. Now it's still possible to vote strategically, but there's no point. Also, since voters only vote for someone they actively approve of, the winner knows their true level of support among the electorate. Now approval voting has the following problems. First, it wasn't used in Colonial America. And second, it gives people too many choices. And finally, it tends to select candidates that appeal to everyone. So if you've been winning elections because you've limited your voters to those who could vote in Colonial America and by limiting their choices and by running candidates that only appeal to small segments of the population, approval voting is terrible. Majority grade is a relatively new voting system, first suggested in 2007 by Michel Balinski and Rita Laracchi. In majority grade, every voter grades the candidate. So, for example, they might grade the candidate's excellent, good, fair, poor, reject. Grades can repeat, and it's not necessary that all grades be used. The winner is the candidate with the highest median grade. For example, suppose the ballots for an election are... Now this may look like a preference schedule, but it actually isn't. And that's because the candidates are not ranked, but they're graded. And that distinction is important. So to find the median grade, we'll put the rankings of each candidate in order. So candidate A got these grades, good, reject, good, fair, reject, good, fair. And if we put those in order, similarly for candidates B and C. The median grade is the middle grade. So we see that candidate A's, the median grade is this one, fair. Candidate B has a median grade of good. And candidate C has a median grade of fair. And since good is better than fair, then that means that candidate B wins the election. Majority grades satisfies modificity, since raising the grade of a winner makes them win more, neutrality, non-imposition, non-dictatorship, and it's the only system we've seen that satisfies independence of irrelevant alternatives. Another advantage is that it actually better reflects how people make choices. When you look at a restaurant menu, you don't rank every choice on the menu, but you typically have a few choices at the top, a few in the middle, and most in the don't care for category. Finally, the winner knows the true sentiment of the voters. So a candidate could win with a high median grade, which means the voters viewed them as better than the alternatives. But a candidate could also win with a low median grade, which means the voters viewed them as the least bad among the alternatives. Majority grade seems to be an ideal voting system. Unfortunately, it's subject to two rather bizarre paradoxes. We'll take a look at those next.