Let's look at a simple example. Suppose on Wiki article about ice cream, and people have different opinions on what the best ice cream flavor. Is it chocolate, vanilla, or strawberry? There are six possible configuration of opinions. Let's say three of them received some votes. And there are 9 votes all together. 4 goes to chocolate better than vanilla, better than strawberry. 3 goes to strawberry better than vanilla, better than chocolate. And 2 votes goes to vanilla better than strawberry better than chocolate, all right? So let's see what will happen with Plurality, Borda count, and Condorcet voting respectively. By plurality voting is quite clear, okay? We will get C as the top one, chocolate is the best. And then strawberry gets 3 votes putting it on the top, so it gets strawberry. And vanilla got 2 votes putting it on the top, so we get vanilla. But as you can see that this somehow does not utilize the fact that chocolate also get 5, which is more than 4 votes, placing it at the bottom. Because we only care about the majority count of the top position, not the rest of them. Well, what about position of voting? Let's say Borda count. So we'll have to give some points. There's three candidates and therefore the top can get 2 points and then get 1 point then get 0 points. So chocolate gets 4 votes, in which it gets 2 points. So chocolate get 8 points all together. Strawberry get 3 votes where it got 2 points, so that's 6 plus 2 more votes, we get 1 point. So it also got 8 points, its a tie. And vanilla got 4 votes with 1 point, 3 votes with 1 point, and 2 votes with 2 points, so you've got 11 points. And therefore, vanilla is better than strawberry equals chocolate, they are tied. What about Condorcet vote? Let's look at a pair wise comparison here. So suppose we look at C versus V, chocolate versus vanilla. Chocolate is ranked higher than vanilla by 4 people but vanilla ranked higher than chocolate by 5 people. So, vanilla with chocolate. Now, what about let's say strawberry or vanilla. So strawberry is voted better than vanilla by 3 votes, but vanilla voted better than strawberry by 6 votes. So, vanilla better than strawberry. Now what about the last pairwise comparison we need, chocolate or strawberry? So chocolate is voted better than strawberry by 4 votes. But strawberry is voted better than chocolate by 5 votes. So strawberry wins chocolate. Well, it turns out that in this case, Condorcet voting doesn't lead to a cyclic output. We do have a largely consistent result, 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 vote says chocolate better than strawberry better than vanilla. Condorcet says vanilla better than strawberry better than chocolate. And Borda count 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 pick. Now this is a very simple nine voters, three candidate system. Imagine what would be if there are more choices. Now, you might object. First, this is a synthesized example. Maybe in the real world, it wouldn't be like this. Well, true, this is clearly synthesized to be a small example yet illustrate potential issues. 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 real world enough. While maybe we can start with some simple statements that we all can bind, we all can convince ourselves they should be true. And then look at the logical implications, we call that the Axiomatic Construction. So axioms means that these are proportions we take to be true in order to look at implication of logical conclusions coming out of them. There will be a few axiomatic system we go through including in today's advanced material an nash axiomatic system for bargaining which leads to a unique positive result. And also in lecture 20, axioms for fairness evaluation. 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. All right, fine, this one is complete, even though incomplete input is often the reality 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. Output should not just identical to one input list no matter what the other input lists 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 a Pareto principle. If all the input list, says A should be better than B, then the output must also say A better than B. because there's no disagreement about the inputs. Finally is what’s called IIA, independence of irrelevant alternatives. It says that between a pair of choices A and B. Okay, each input preference between A and B remains the same. Some say A is better than B, some say B is better than A, it doesn't matter. Okay, as along the preference relative between A and B, a pairwise comparison remains the same. That even if their preference involving other candidates, like C, moves around from here to here to here, the output preference between A and B remains the same. Because where C lies is irrelevant to ABs comparison 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 sound fair enough. Okay, and 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 actions? And B, 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, because 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, then no voting system can satisfy all five axioms that we just believed to be all reasonable. So some things wrong. If a surprise factor is used it to judge the the elegance of a result then, 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, because they are really defined what meaningful, logical voting should be. So it must be either Pareto or IIA axiom. It turns out that it's the IIA 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 by Arrow says, is due to the fact that these seemingly irrelevant alternatives are not irrelevant after all. Where other candidates sit relative to A and B actually should make a difference in the output. Without that, you actually can have transitive inputs and get cyclic output. In other words, axiom five can preclude the existence of axiom two. And now, we won't have time to go into the details, the proof of that, but we will go through it in the next video segment. Another impossibility result by sum, and we'll 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 pairwise. We actually need to keep track of the scale and the position of all the candidates. So sometimes, people quote Arrow's impossibility result to say that voting is flawed. Well, that argument actually is logically nonsense. What he says is that our intuition is flawed, okay? We cannot assume that these alternatives, positions are irrelevant, they are relevant. And indeed, later researchers such as Asari have developed the possibility results by modifying this IIA, 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 0 or 1 or some number after n minus 2. We call that number the intensity of A better than B. Then is would replace IIA by the so called IIIA, three I's. And that says the ranking of a pair of candidates depends only on the 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, [COUGH] we don't need to keep track of how many candidates are there in between A and B. Well then, if it's possibility result, then give me a voting system that is indeed going to satisfy all these axioms. It turns out position voting Borda 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 voter is already captured, the entire set of input profile. That is the true intent. If you want to condense all that into a single list some information will be lost. With this possibility and the errors end possibility results highlighted is the need to count, not to just order. We need to know the scale more than just pairwise relative positions.