 One problem with many voting systems is the existence of what's called strategic voting. In a strategic vote, a voter casts a ballot that doesn't reflect their true preferences. We say they have cast an insincere vote. We can look at the problem as follows. Suppose a voter could change their vote after seeing how everyone else voted. If they do so, could they obtain a better result? Now, why would anybody submit a vote other than what they really wanted? Well, let's take a look. Suppose the preference schedule for an election is as shown, and if the election were to be decided by plurality, how should voters vote? To answer that question, we need to know how the election turned out, and we see that using plurality, T wins. So now the question to ask is, if you could change your vote after, would you do so? So let's consider. To begin with, consider the 3 plus 2, these 5 voters who supported T. In plurality, T wins, so these voters have no reason to vote other than how they did, they got the outcome they wanted, and hopefully they'll get what they deserve. Now, consider the 3 plus 1, the 4 voters who supported C. Again, in plurality, T wins, and so the question is, if they could have voted differently, would they have done so? And that depends. So let's consider this one voter who ranked C over T over S. Now, while their favorite candidate, C, didn't win, their second favorite, T, did actually win, and so T winning, well, that's a reasonably good result for them. So while their first choice didn't win, their second place did, and their least favorite, lost, and so they probably don't want to change their vote. But consider the voters who ranked the choices C over S over T. Now, since this is plurality, this means they voted for their first choice, so they actually voted C and lost, and since T won, their least favorite won. So the question is, if they voted differently, could they have gotten a better outcome? So if they ranked the candidates S over C over T, and voted for S, then the new preference schedule would be, and now, S has 5 plus 2, 7 votes, and S would win the election. For these voters, it's a better outcome. So if the three voters who really preferred C over S over T submitted an insincere ballot ranking S over C over T, the outcome, S wins, is better for them. Now, this is actually a major problem with plurality, and it's a very well-known problem. Because voters can only support one candidate, they often cast a ballot for who they think can win, and not necessarily who they like most. Likewise, third-party candidates can split the vote, leading to the victory of a candidate acceptable to the fewest people. Now, when we talk about voting theory, we say that a social choice function satisfies independence of irrelevant alternatives, IIA, if the relative ranking of two candidates depends only on the individual rankings of the two candidates. So we saw that in the original preference schedule, T wins a plurality vote, and there are 1 plus 3 plus 2, 6 voters who feel that T is better than S. Now, if the three voters who preferred C over S over T voted insincerely as S over C over T, the new preference would be, and the thing to realize is there are still 1 plus 3 plus 2, 6 voters who feel T is better than S. But now S wins even though no voter changed their minds about S or T. The only difference is how these voters felt about C, a third choice that should be irrelevant to the race between S and T. And this says that plurality fails this independence of irrelevant alternatives. Does the board account satisfy independent of irrelevant alternatives? Well, suppose our preference schedule for an election is as follows. Now, if we use a board account because there's three choices, then a first place vote gets two points, a second place vote gets one point, and so using a board account, A gets 24 points, B gets, and C gets, and then C has the most points, C wins. Now, consider the voters who ranked B above A above C. They don't like the fact that C wins, and if they submitted insincere balance A over B over C, the new preference schedule would be, and again using a board account, we find, and now A wins even though no one changed their relative ranking of A and C. These voters began ranking A over C, and they ended ranking A over C. And with a little effort, we can also find a preference schedule that shows instant runoff also fails independence of irrelevant alternatives. We'll let you do your own homework and come up with an example. So the question is, can we create a voting system that satisfies independence of irrelevant alternatives? And the answer is, yes, but let's consider some other ideals.