Let me propose a situation to you: you, and eight friends of yours, all live on the same street in some city or large town. As a crazy random happenstance, you are all precisely equally distant from your nearest neighbors: two blocks away. The nine of you decide that you're going to go out for a big dinner somewhere along your street, and there are two more-or-less identically good options: you could go to ApricotBug's, over on the west end of town, or you could go to Beryl Thursday's, closer to the east end.
How do you decide? Well, let's suppose you decide to put it to a vote, since that seems most democratic. Let's further suppose that, since the restaurants are more-or-less identical, everyone simply votes for the one closest to their house. Three of you live west of ApricotBug's, four of you live east of Beryl Thursday's, and two of you live in between. Let's vote!
You can easily see that everyone west of ApricotBug's votes for it, everyone east of Beryl Thursday's votes for it, and the two in between split their votes one and one. Beryl Thursday's wins, 5 to 4.
Notice that, even though each of you used a selfish algorithm (or to put a positive spin on it, an honest, true-to-yourself algorithm) to decide how to vote (the option that minimizes your own travel time), that as a group you also voted for the globally-minimum amount of travel; it would take a total of 45 blocks traveled for everyone to get to ApricotBug's, but only 41 blocks for everyone to get to Beryl Thursday's.
Continuing with the story: You have so much fun, you decide to do it again next month. However, a new restaurant has opened down the street, CBIM (Can't Believe It's (Already) Monday).
How does CBIM change the voting?
If we used the well-known method of plurality voting, we can see that the vote for ApricotBug's stays at 4, but the 5 votes that originally all went to Beryl Thursday's are now split 3 for it and 2 for CGIM. Which means that, now, ApricotBug's wins.
This is no longer the globally-optimal solution, and if you think about it, it's really kind of stupid that adding more options, particularly an obviously bad choice like CBIM, would ruin the vote like this.
What to Do, What to Do...
After much existential gnashing of teeth, you decide to use "instant runoff voting" (because one of you heard about it on the internet and, at first glance, it sounds pretty good!) With IRV, you can pick additional options beyond just your first choice, and if your first choice doesn't win, your vote will be moved over to the next in line.
Everyone west of ApricotBug's and your friend who lives right next to it will vote A > B > C. Also, the two friends closest to CBIM will vote C > B > A. For the three in the middle, two vote B > A > C and one votes B > C > A. (Count the distances yourself if this isn't clear to you.) So which restaurant wins now? IRV tells us to eliminate the option with the fewest first-place votes. That's CBIM. We then move those votes to the voter's second choice, which in this case is Beryl Thursday's for both voters.
Those two votes puts Beryl Thursday's back over the top, which is good, since we know that it's the globally-optimal solution, with respect to total distance traveled. So it sure seems like IRV has improved things for us. (If you're curious, CBIM would have taken 81 blocks; almost twice as bad as ApricotBug's!)
Next month rolls by, and gosh darnit, you've all gotten really attached to this big night out, so you're going to do it again! One minor change though: CBIM had a little problem at their original location (it was too popular, so it wasn't large enough) and so the owner relocated four blocks over (a total distance of 53 blocks for everyone to reach.)
No problem; we'll let IRV handle this! It's still 4 for A > B > C, and 1 for B > C > A, but now we're up to 3 for C > B > A and down to 1 for B > A > C. That's 4 first-place votes for A, 3 for C, and two for B. Uh oh...
Now we're back to the exact same problem we had before under plurality. None of you or your friends have moved. ApricotBug's and Beryl Thursday's haven't moved. The only thing that changed is a that a third fringe option has moved to a slightly-more-acceptable position. The mere presence of CBIM spoiled the vote under plurality, and now its mere presence has spoiled the election in the exact same way under IRV.
Perhaps you're thinking to yourself that, no, some of the CBIM fans will see what's going to happen, and they'll instead vote for Beryl Thursday's first. But they could have just as easily done that under plurality voting; why would IRV be different? And even if it is, for some unexplained reason, different, why should they have to choose among the lesser of two evils, between ApricotBug's and Beryl Thursday's, when in their heart-of-hearts, they want CBIM? Wasn't IRV supposed to fix that problem?
IRV Isn't a Good Idea
If you think IRV is a good idea because it will allow third parties to win elections, you are wrong; it will be just like under plurality. If you think IRV is a good idea because it fixes the spoiler problem, you are wrong; it will be just like under plurality.
In the real-world, we don't have access to simple metrics like "fewest total blocks traveled" to guide us in choosing our political leaders; we only have everyone's biased, greedy, short-sighted, and misinformed opinions. I believe that, deep down, we all want what's best for everyone, but that, obviously, opinions differ on what that is. If IRV can't properly decide which of three kitsch restaurants to hang out at in a simple, one-dimensional scenario like this, I guarantee that it won't do any better in the real world.
Next up, I'll have a dinner-vote using score voting, and then maybe we'll try a two-dimensional example, which will show the problems with Condorcet methods.