Let's look at a simple example. Suppose on a weekly article about ice cream. And people have different opinion on what is the best ice cream flavor. Is it chocolate, vanilla or strawberry. There are six possible, configurations of opinion. Let's say three of them received some votes. And there are nine votes all together. Four goes to chocolate better than vanilla better than strawberry. Three goes to strawberry better than vanilla better than chocolate. And two votes goes to vanilla better than strawberry better than chocolate. Alright, So let's see what would happen with plurality, border count and Condorcet voting, respectively. By plurality voting, it's quite clear, okay? We will get C as the, top one. Chocolate is the best and then, strawberry gets three votes, putting it on the top, so get strawberry and vanilla got two votes putting on the top, so get vanilla But as you can see that this somehow does not utilize the fact that chocolate also get five. Which is more than four votes. Placing it at the bottom cuz we'll only care about the majority count of the top position, not the rest of them. But, what about position of voting? Let's say border count so we have to give some points. There are three candidates and therefore the top candidate get two points, then get one point gain, then get zero points. So chocolates gets four votes in which it gets two points. So chocolate get eight points and all together. Strawberry get three votes where it does two points so that's six plus two more votes will get one point. So its also got eight points, It's a tie. And vanilla got four votes with one point, three votes with one point and two votes with two points. So you've got eleven points and therefore vanilla is better than. Strawberry equals chocolate at tied. What about condoshea vote. Let's look at pairwise comparison here. So suppose we look at a C versus the chocolate versus vanilla. Chocolate is ranked higher than vanilla by four people. But vanilla ranked higher than chocolate by five people so vanilla wins chocolate. Now, what about let's say, strawberry or vanilla now. So strawberry is voted better than vanilla by three votes. But vanilla voted better than strawberry by six votes. Okay. So, vanilla better than strawberry. Now what about the last pair-wise comparison we need, Chocolate or strawberry? So chocolate is voted better then strawberry by four votes but strawberries voted better than chocolate by five votes so strawberry wins chocolate. Well, turns out that in this case Condorcet voting doesn't lead to a transit a cyclic output. We do have logically consisten reserved which is vanilla is the best better than both. And, strawberry is then, in turn better than chocolate. So these are the three different results. First of all, they're all different and perhaps very disturbingly, majority vote and Condorcet vote, both are very intuitive systems, give us completely different results, that are diagonally opposite. Majority of vote says, chocolate better than strawberry, better than vanilla. Condorcet says vanilla better than strawberry, better than chocolate. And vote account then gives you something different. Shows that even when all three have some meaningful result the three results can be so different that we don't know which one to, it will pick. Then this is a very simple nine candidate, nine voters, three candidate system. Imagine what would be if there's even there are more choices. Now you might object, first this is a synthesized example, maybe in real world it wouldn't be like this. Well true this is clearly synthesized to be a small example yet illustrate potential users, but then again there are many such paradoxes this is not an isolated incident. And finally how would you describe a typical real world. Scenario? This is where we're enough. Well, maybe we can start with some simple statements that we all can, buy in. We all can convince ourselves they should be true. And then, we'll take the logical implications, we call that, the axiomatic construction. Axioms, is that, these propositions we take to be true in order to look at implication of logical conclusions coming out of it. There'll be a few axiomatic system we go through, including in today's advance material, the Nash axiomatic system for bargaining, which leads to a unique positive result, and also I'd like to continue axioms for fairness valuation. But before then, we look at, a very famous axiomatic construction by Arrow. So Ken Arrow in 1950, said that maybe we can all agree on the following five statements. And then we can see what implications come out of that for voting systems. The first statement says that each input list should be complete and transitive. Alright, fine, as soon complete, Even though incomplete input is often the reality in, in real systems but let's say it's complete. Of course, we want them to be transitive otherwise, it's logically inconsistent. Second, output list should be complete and transitive too, Fair enough. Third, output should not be just identical to one input list no matter what the other input list are, Again fair enough. Otherwise you have a dictator then there is no point in voting anymore. And then the fourth one says, so-called Pareto principle. If all the input list says A should be better than B then output must also say A better than B because there's no disagreement among inputs. Finally it's what's called IIA, independence of irrelevant alternatives. It says that between a pair of choices A and B. Each inputs preference between A and B remains the same. Some say A's better than B, some say B's better than A, doesn't matter. Okay as long as the preference relative between A and B a pair wise comparison remains the same, that even if their preference involving other candidates like C moves around from here, to here, to here, output preference between A and B remains the same. Because where C lies is irrelevant to AB's comparison, A is still better than B and B is still better than A in different people's mind. So the output. Decision between A relative to B should not depend on where C sits. Independence of irrelevant alternatives. All right. All five statements sounds fair enough. Okay, now we can say, all right, I can convince myself that all should be true. Then, we think what kind of systems will satisfy all five axioms. And the surprise is. None. Zero. No voting system can satisfy all five axioms, as soon as there are three or more candidates. Again if there are only two candidates, life is easy cuz comparing scalers on a real line is fully ordered. This is the famous Arrow's impossibility result that says, it doesn't matter how many voters there are. If there are three or more candidates, than no voting system can satisfy all five axioms that we just believed to be, reasonable. Some-, something's wrong, if surprise factor. Is used to judge the elegance of a result. And this is one of the most elegant results that we're going to see in this course. Somehow our intuition isn't quite right. Which axiom gave us the trouble. Not the first three cause they really define what a meaningful logical voting should be. So it must be either parietal or IA axiom. It turns out that it's the IA axiom. Usually, the axiom that takes the longest to describe is the first one you should suspect giving you any trouble. The fact that these five are logically inconsistent among themselves, that's what negative impossibility results of error says is due to the fact that these seemingly irrelevant alternatives are not irrelevant after all. Where other candidate sit relative to A and B actually should make a difference in the output. Without that, you actually can have transitive inputs and yet cyclic output. In other words, axiom five can preclude the existence of x and two. And now we won't have time to go into the details, proof of that. But we will go through, in the next video segment, another impossibility result by sum, And we will see that, indeed, the seemingly harmless irrelevant alternatives are not irrelevant in the final voting after all. In other words, by compressing many lists into one single list. We need to know not just the relative position peer wise. We actually need to keep track of the scale and the position of, of the candidates. So, sometimes people quote Arrow's impossibility result to say that voting is flawed, well that argument actually is logically nonsense. What it says that, is that our intuition, is flawed. We cannot assume that these alternatives positions are irrelevant. They are relevant. And indeed later, researchers such as Sari have developed the possibility results by modifying this IA. For example, the so called intensity form. That says not only I keep track of, relative comparison pairwise between a and b. But also, I write down the number of candidates that's in between a and b, which could be zero, or one, or some number, up to n-2. We call that number the intensity. Of A better than B. Then, is we replace IIA by this so called IIIA, three I's. In the sense that the ranking of a pair of candidates depends only on their relative position and the intensity value. It turns out, that suffices to lead us to a possibility result. We don't need to keep track of who, but need to keep track of how many candidates are there in between A and B. Well then if its possibility result then give me a voting system that is indeed going to satisfy all these axioms, it turns out position, voting, border count can do that. So, people often ask, what is the true intent of the voter? Are we capturing that? Well, the true intent of the voter is already captured, that is the entire set of input lists, the input profile. That is the true intent. If you want to condense all that into a single list, some information will be lost. What this possibility and the Arrow's impossibility results highlight is the need to count, not just to order. We need to know the scale, more than just pair-wise relative positions.