This lecture will look at voting schemes a little more closely, and the main message of this segment will be that even very reasonable sounding voting schemes can run into problems. And we'll do it by a series of examples, we'll look at situation, I'll ask you to pause the video and think about the situation and then we'll and to continue when you're ready. So let's get started. Let's start with the Condorcet condition, which on the face of it is incontrovertible. And consider the following examples. So here we have a thousand agents and here are the preferences. So for example, 499 of them prefer A to B to C, and so on for the others. First question is, not every voting situation has a Condorcet winner does this one have. This is a good time for you to pause the video and think about it. Well the answer is yes. There is a Condorcet winner and it's B. And why is that? Well let's look at the relative preference of B to A and the relevant preference of B to C. We have here that 501 of the agents prefer B to A. And 502 of the agents prefer B to C. So clearly, B is the Condorcet winner. So where's the problem? Well, now, think about the simplest sort of voting we're familiar with. Plurality voting, everybody votes for their top candidate. Who would win the plurality voting here? Well, again, you can pause the video or in this case, it's very straightforward, right? Clearly A would win it because 499 agents would vote for A. And the next highest number would be C with 498. So plurality voting doesn't give you the candidate that on the face of it should be the clear winner. What about voting plurality with elimination? So this might take a little more time to think about. So you might want to pause the video here just for a second. And now when you think about it you see that C would be the winner under plurality within a nation and why is that? Well you'd first run a plurality and you'll see that B is the loser. So B would leave the competition, if you wish, and now it will be head to head between C and A and in this case C would be the winner because 501 out of the 1000 agents prefer C to A. So C would be the winner. And so two voting teams, both of them on the face of it reasonable, would give you different answers, and both answers different from the criterion that on the face of it seems quite a safe criterion, namely the Condorcet condition. So here's another example and let's think about what would happen in this case. What would happen under plurality voting? Well clearly under plurality voting A would win since A would get 35 votes and the second highest would be B with 33. What would happen under the Borda voting? This takes a little more thinking, and you might want to pause, just for a second, the video. But when you continue, by that time, you'll quickly realize that again, A would be the winner under Borda, and clearly, you have A, B and C appear in each of the places one, two and three, with A appearing with the largest number of agents and the higher locations. So A would have the highest Borda count and would be the winner too. So this looks very good. But now what happens if C drops out. So, C realizes that he has no chance of winning the election and drops out. Now what would happen under the both plurality and Borda? Just, you might want to pause for a second the video and think about it. And when you do you realize that in both cases B would win. And so, should you have a candidate that has no chance of winning and his sole role, if you wish, is to change what otherwise would be the outcome of the elections. Here is another peculiarity of voting schemes, and imagine that we're doing pairwise elimination. That is we're going to take one candidate compare him to another, take the winner compare him to a third and so on and so forth. So the order in which we compare the candidates we call that the agenda. So somebody need to set the agenda. We call that person the agenda setter. So imagine pairwise elimination with the order of comparison A, B, C. In other words, A will be compared to B and then B will be compared to C. So who will be the winner of this election? A good time to pause the video and think about it. And once you do, you realize that C would be the winner because when A is compared to B, well, we have B prefered to A by a majority of the agents. So A would be eliminated. And then when the winner, namely B, is compared to C you'll see that C is preferred to B by the majority of the agent, and therefore C will be the winner of this election with this ordering. What happens with another ordering like A, C, B? Again you might want to pause the video, and by the time we resume we'll realize that this case B would be the winner. And perhaps not surprisingly when you ask about the third ordering B, C, A. You'll see that A would be the winner there. And so it's a little perhaps disconcerting that the same voting scheme, merely by deciding on the order in which you run it, will lead to very different results. And here is another example. There are three agents and four candidates and the preferences are as written up there. Now consider again pairwise elimination with the A, B, C, D ordering and what would you get? Again, pause the video for a second, think about it and realize that the winner would be D. What about this case? We're now not talking about different orderings. We're talking about a given ordering and an outcome. But there's something a little troubling about this outcome. What is it? Pause the video. Think about it. And realize that the problem is that everyone prefers B to D here. Right? So B is preferred to D here. B is preferred to D here and B is preferred to D here. And yet D this Pareto-dominated candidate wins. So something is wrong in this picture. Well the goal here was not to give us the final answer of what is the right way to vote. In fact, that is not well defined. The goal was to alert us to the fact that a lot of reasonable sounding voting schemes can be problematic. And so with this note of caution, we'll finish the segment.
