 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 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 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 their 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 Contorcet winner. Does this one have? So it's a good time for you to pause the video and think about it. Well the answer is yes. There is a Contorcet winner and it's B. And why is that? Well let's look at the relative preference of B to A and the relative preference of B to C. Well 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 a Contorcet winner. Well 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 could pause the video or in this case it's fairly 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 is the clear, 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 with elimination 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 would be head-to-head between C and A and in this case C would be the winner because 501 out of the 1,000 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 condorsate 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 border 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 border and clearly you have A, B and C each appearing in each of the places one two and three with A appearing with the largest number of agents in the higher locations. So A would have the highest board account 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 no chance of winning the election and drops out. Now what would happen under the both plurality and border? 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 here 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 to compare him to another take the winner comparing 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 needs 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 would be the winner of this election? I got 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 preferred 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 ACB? Again you might want to pause the video and when we read by the time we resume you'll realize that in this case B would be the winner and perhaps not surprisingly when you ask about the third ordering BCA you'll see that A would be the winner then 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 written up there. Now consider again pairwise elimination with the ABCD 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 what about this case? We're not 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. Something is wrong in this picture. Well the goal here was not to give us the final answer what is the right way to vote in fact that is not well defined. The goal was to alert us to that the fact that a lot of reasonable sounding voting schemes can can be problematic and so with this note of caution we'll finish the segment.