 Okay, so today we are going to discuss social choice and voting theory, okay, so this is some material that you will find in the book of Masquerel and Winston and Green in chapter 21 or in Easley and Kleinberg, so in Masquerel you will find a more mathematical, so what is the problem? So the problem is that we want to go from the individual to a society and to see whether we can define the preferences of a society from the preferences of the individuals. So we have S alternatives, you can think of them as different candidates for president and individuals, each of which has a rational preference over these individuals, so which means it's complete and transitive. So and we define a social choice rule as a rule that is a function of all the preferences of individuals and that gives us a preference for society as a whole, okay. And the main question we are going to discuss is whether there is any social choice rule that satisfies some minimum requirement, okay, so let me make some examples. So there are indeed many social choice rules and these are essentially used in elections and so for example you have one is plurality voting, so when we go and vote then we just cast one vote for our preferred candidate, our top candidate, okay. So for example in this example, which is taken from this paper of the French election in 2002, I mean this is just made up and the idea was that there were three candidates and essentially 30% of the people, if 30% of the French people, French voters had these preference 36 and these other preference and 34 and these other preference, then plurality voting would have ended up by choosing Shirak over Lepen over Jospen, because just you sum up and you see that well Shirak will get 36% of the votes, Lepen 34% of the votes and Jospen 30% of the votes. However you can think of the different social choice rule, which is called board accounts and in board accounts you just give two votes to the best and one vote to the second and no vote for the third, okay. So if you do the calculation is a little bit more complicated here, but you see that well again in this case Shirak will have 36 times two plus 30 and he will get more votes than Jospen and Lepen, but you see that already using this rule the order between Jospen and Lepen has changed between the previous one. Yet another rule that you can consider is the one of majority voting, you pick two candidates and you make an election on just this pair of candidates, okay, and if you do so then you will find that well with respect to Jospen and Shirak, well there are 30% plus 34% of the people who prefer Jospen to Shirak and only 36% that prefer Shirak to Jospen. So Jospen in a pair wise majority voting would win against Shirak and now you can see also that Shirak will also win against Lepen because essentially these people prefer Shirak to Lepen, but also these people prefer Shirak to Lepen. And so you end up with a different preference ordering for the majority, okay. So there is another, so depending on which which voting rule you choose you end up with a different decision, so which voting rule should one choose. But there are more serious problems because essentially if you take for example pair wise majority problem rule then this is the problem that the majority may not be transitive. So this is another example of an election and you may see that in this election we have 44% of the people that prefer Gore to Bush to Nader, 46% that prefer Bush to Nader to Gore and 10% that prefer Nader to Gore to Bush. Then by pair wise majority you will see that Gore is better than Bush because 54% of the people prefer Gore to Bush. Bush is better than Nader because there are 44 plus 46%, 86% of the people prefer Bush to Nader. But at the same time also Nader is better than Gore because there are 46 plus 10, 56% of the people prefer Nader to Gore. And then Gore is better than Bush. So you see that the majority rule pair wise majority is leads to a non-transitive, non-rational outcome. So okay, so but if you take another rule then you can see that this also has other problems. So another problem in social choice rules is the one that you do not like that your outcome, social outcome be manipulable. Okay, so let us take an example. Imagine that a society has to choose whether to invest in solar energy or in gas. Okay, so society has to choose what is the energy source that is going to use. And then if 58 of the people have this preference and 42% of the people have this other preference then the society will choose solar to gas. Okay, but then you can say well wait, I mean there is also wind energy, wind turbines that can be used. Then if you include also this other choice then now you will see that the 58% of the people that had this preference over solar and gas split and 48% have this preference where wind is stirred and 10% have this preference when wind is the best choice, is better than solar and gas. And then in this case you see that by adding one choice you end up in the same situation as before, but not only you see that now depending on how you make your decision you may end up in a different conclusion. So imagine that you use pairwise you compare, you let society compare these two options in pairs. Then imagine that you first consider whether to go for solar or to go for wind. And then by the same argument as before you will find out that wind is preferred. And then you can consider whether the society will go like to go for wind or for gas. And then by this table you will see that the majority will prefer gas to wind. But now if you reverse the order of the choices if you ask first whether the society will prefer solar to gas instead of solar to wind, then you will find that they prefer solar. And then if after this you ask whether they prefer solar to wind you will end up in wind. So you see the outcome of the choice of the society here depends on the order in which you pose the questions, the binary questions. Okay so are there any questions about this? Is everything clear what we are talking about? Professor I have a question. It's regarding the function of the different binary relations that we defined as preferences. How could we define such a function? So how we can define this function F here? Yes. Okay so I mean these are three particular ways of defining this function. So plurality voting. Plurality voting means that you just let every individual just express one vote for their top candidate and then you rank the votes. Okay so this is a social preference. Okay and this border count is another one and majority votes is another one. And there are many many different alternatives. Okay. Okay. Okay thank you. Okay so if there are no other questions so I see that. Professor. Professor. First slide yes. I have a question. Yes. It means that they organized the vote three times. In this example you mean no? In the previous one. In the previous one? Yes. Yeah for instance this one. Since we have three people it means that we organized the the vote three times to make the final choice. No well it means that say you could think like the following. So instead when you go and vote instead of expressing only what is your best preference best candidate you just submit the whole preference relation. And then the mechanism the social choice mechanism compares each of these each pair of different alternatives. Okay and what the Cronus paradox implies is that the problem with this is that you may end up with a preference relation that is not transitive. In this case it means that people have to fill the form of preference. In this case yes in this case I mean this is not usually used pair-wise majority is not a system that is typically used. But this could be a way of also running the elections. Okay. Thank you. Could you explain how this Excuse me. Yeah. Yeah so could you explain how this 44 46 and 10 came like let's say even I mean there are only three alternatives right. So how with these numbers with this table. Yeah in this table how did you come up with x-axis I understand the y-axis there is a like I mean when we vote we'll be voting between one two and three. Yeah. So how did you get like we get this x-axis in the sense 44 46 and 10. So these these I made it up. It's not does not reflect any real situation. I'm just saying let's imagine a society where 44 percent of the people have this preference relation so they prefer this gore to bush to another 46 percent have these other preference relation and 10 percent have these other preference relation. Okay. I made all these up. Okay. Okay. Okay. Thanks. Thanks. Professor I have another question. So does that mean that the function that we use as social choice should conserve rationality of the of the output preference relation? That would be yes. I'll come to that. This would be a very nice requirement. No. Yes. Okay. Okay. Thank you. Okay. So let me go to the main question that we are going to one of the main questions that we are going to discuss today is it how so is there a social choice rule that is that is the best one. Okay. And again we assume that we have an individual size each alternative and we ask whether there is a way to aggregate individual preferences into a social choice function that satisfies some reasonable requirement. What are these reasonable requirements? Well, these are these requirements. So the first is that we want that these prefer this social choice function works whatever are the preferences of the individuals. Okay. So this is called a restricted domain. Okay. So whatever preference individual have. So this we could be able to extract a social choice preference. So the second thing is rationality. So we want the society to be rational. So we want the social choice rule to be complete and to be transitive. Okay. This also looks like a minimum requirement. But another requirement that we want is what is called unanimity. And this means that if everybody prefers A to B, then the society should prefer A to B. Okay. So this is a minimal requirement that we should also ask. And the other requirement is what is called independence of irrelevant alternatives. Which means that the social choice between A and B should only depend on individual preferences between A and B. Okay. It cannot depend on what are the preferences between A and another choice C. Okay. Because if it is so, then the social choice can be manipulated. Okay. One can fictitiously introduce another option like a wind, as we saw, that will turn the choice between A and B the other way around. Okay. Okay. So the main result in this is what is called arrows in possibility theorem. And what arrows in possibility theorem is that if you look at, if you want a social choice that satisfies all these requirements, well, there is one. And it is essentially dictatorship. Dictatorship is a social choice rule such that you pick one individual and the preference of the society is equal to the preference of this individual. It is clearly a preference social choice rule that is artistic domain. It is, it has unanimity. It is rational and it's independent of irrelevant alternatives. Okay. Of course, this is not a very nice result because, well, we do not think this is a good choice for a democratic society. But essentially, so this is why this is called an impossibility theorem. And because if you want to rule out dictatorship, so if you want to impose also another requirement that is called anonymity, which means that the social choice rule should be invariant under any permutation of the individuals. Then arrows in possibility theorem tells you that there is no such a rule. And of course, if you have only two alternatives, then say pairwise majority works satisfies all these requirements. Okay. So the problem arises only when you have more than two alternatives. And this is indeed why many political systems have evolved into systems where essentially you have only two alternatives between, say, conservatives and liberals left and right, only two main parties. Of course, the other choice is to have just a strict alternative to just one choice, but that is not very democratic. Okay. So now, well, if you want to check your understanding you should be able to understand what is wrong, what of this desiderata or this requirement fails in plurality voting. And what is wrong with deciding on a complex issue by dividing into binary issues. Okay. So questions. So let me... I have a question. Please. Are you suggesting that in choosing a candidate you should use the binary choice over other kind of choices? No, no, no, no. What I'm saying, and this is a mathematical theorem. It is a theorem that there is no social choice rule that satisfies all these reasonable requirements. So what I'm saying is that if you want to design a society, a democratic society that takes a decision in a say optimal rational way, then this is just impossible. Okay. So you have to give up one of these reasonable requirements. Okay, thank you. Sorry. Yeah. Can I ask you what's the relation in between a social rule and a binary rule? So a binary rule. I don't know what you mean by a binary rule, but say you mean in this example here I was saying in this problem here, say, I mean, what I mean is you can think of a social choice rule that works like a tournament. In a tournament, what you do is you pick say one candidate, one alternative, and you compare it to another one, you take the winning of these two candidates, all these two alternatives, and you compare it to a third one, and you go on like this. Okay. Yes. So the social rule to which Arrows theorem refers to is a binary. Any social choice. Of any such choice. Okay, thank you. So Arrows theorem tells you that there is no way in which you can get a say a rational social choice out of the rational aggregating the preferences of individuals. Okay, again, let me stress that what I'm doing here is just to summarize what is explained in the lectures that are pre-recorded and you find on the website. Then I also give, you can also find an idea of how you prove this theorem. Okay, other questions. Okay, so if not, let me continue and say, well, this Arrow impossibility theorem is really a worst case result. So it tells you that say, however you find a social choice rule, I can find a set of preferences of individuals such that the social choice rule will violate at least one of these of these actions, of these desiderata, unanimity, rationality, etc. Okay, however, you can ask how bad is the situation typically? And you can ask, say, what is the social choice rule, which is least bad, essentially. And there is this result of this group that I'm asking that show that essentially majority rule is the least bad. And the idea is that whenever majority, let's say, whenever some other rule works, then also majority rule works. But let's say the majority rule in some sense is the one that has the largest domain of applicability. Okay, then you can also ask typically, so what is the probability if you imagine that you have a random society, so where essentially every individual has a preference between alternatives, which is just a random permutation of the possible alternatives. You can ask yourself, how bad is this? Okay, what is the probability that they pick a set of preferences of individuals, such that the majority rule is not transitive? And then you can see that, well, as the number of alternatives grows large, the probability that you end up in a non-transitive majority also becomes larger and larger. Okay, and if you ask a simpler question, so what is the probability that in the in pairwise majority, there is one alternative that is better than all the other alternatives, this is called a condorsae winner. Okay, then this probability, you can compute it in this limit of a large random society, and it falls off very, very slowly. So as a square root of log s divided by s. So essentially, this tells you that even if the majority rule is not perfect, well, in practical cases, it may be that there's a large chance that it works. Okay, then there is another argument which is suggested by Amartya Sen and says that what this result by Canetaro is not really a negative result. What this tells you is that if a society has to make a choice between a certain number of alternatives, then the individual preferences do not provide enough information to make this choice. Okay, and the idea is that, say, if it is really a choice for the society as a whole, then the individuals should maybe say interact and share ideas and come to a consensus on what is the best choice for society. And this is what politics is, I mean, is building a consensus, which is essentially means is equivalent to aligning preferences, making the preferences of different people closer to each other. And you can see that, well, this is a paper we did, say, 15 years ago, you can see that if indeed the preferences of individuals are not really independent, but they depend on each other, they are correlated, then also, in that case, the probability to have a pairwise majority is can be close to one. Okay, and well, and then as I told you, well, if you have just two candidates, if you stick the choice to just two alternatives, then there is no problem. Okay, there are other situations where the other impossibility theorem does not apply. Okay, so the idea is that we may give up the requirement of a restricted domain. Okay, and one idea is that, well, in many cases, say, the preferences of individuals are, say, can be ranked on a spectrum. Okay, so if you think about political opinion, then people will have a certain preferred location on the political spectrum between left and right, and their preferences will decrease from the maximum, both towards the left and towards the right. Okay, so in other words, if you restrict attention to a single-picked preference, what are called single-picked preferences such as these ones, so which are preferences where if you put the rank of the alternatives against the alternatives sorted in a particular way, each individual will have just a single pick, and then its preferences will decrease both on the left and on the right. So if all individuals have these type of preferences, then what you can show is a result called the median voter theorem that tells you in this case, otherwise majority voting will always be transited. And the main idea of this result is that, say, if you pick the median voter, so the one whose top choice is in the middle, then it is clear that his top preference, if you think a little bit about it, so his top preference will be preferred with respect to any other preference, because, say, if you take a preference which is on the right, then there will be more people that prefer this preference here, these alternatives here to these other alternatives, than people that prefer the other way around. Okay, so there are several other results of this type that tell you that, say, under some circumstances, you can figure out that some social choice rule may work. And in particular, if, in particular, the majority, fair-wise majority rule is the one that will work more often. Okay, so it is another good time to stop for questions, if there are. Are you completely lost? Hello? I have a question, sorry. In reality, how often is there an association of a single big preference system in social choices? Okay, so remember that we are talking about preferences of individuals which are not observable, which are preferences are not observable features of individuals. And so on one hand, you see a lot of, so you see two things. So one thing is, you see that in politics set in the agenda, what is the order? What is the priority in which you discuss issues is something that is very important. So this means that politicians know about these results very well. And the other thing, I mean, you also see that there is this tendency of, say, polarization of opinions, okay, so that everything is, say, projected on the same axis, okay, whether you are in favor of something, whether you are against something, whether you are somewhere in between, okay, on a political spectrum. So this is part of the dynamics of politics. I mean, I find it really interesting that, say, these mathematical results or these mathematical, say, theorems have a consequence in the working of our society and of politics. Okay, and if you think about it, they are really very deep results. I don't know if I can answer your question though. Yes, thank you. Okay, so, okay, can you explain our restricted domain again? So our restricted domain means that you want a social choice rule that works whatever are the preferences of individuals. Okay, what is wrong with Borda count? It's also to Borda, this seems the most logical choice. This is a very good question that you can, so you can see that in many situations, in some situations, Borda count may be manipulated. So imagine a situation where I imagine this situation here, okay, and I use Borda count, and the choice is Shirak, Jospen, Le Pen, okay. Now the people who voted for Jospen here, who prefer Jospen to Shirak to Le Pen, they may think, well, okay, so if I really give these votes according to Borda count, I will give one vote to Shirak, and this will mean that Shirak will win. So maybe it is better that we misrepresent our preferences and give two votes to Jospen, one vote to Le Pen, and zero to Shirak. In this case, if individuals, so if the individuals with this preference misrepresent their preferences, then Jospen will get more votes than Shirak, okay. So it is in the interest of these people here not to vote according to their preferences. And this is what is called incentive compatibility. So in a society, you want a mechanism for society to work that should not encourage people to cheat or to misrepresent their preferences. By the way, this is what happens all the time in voting. For example, in many cases, people do not vote the candidates they prefer, because they think the candidate they prefer has no chances of being elected, okay. And maybe they vote for the second best or for the third best, okay. But this distorts the preferences of society, okay. So did I explain you why board account does not work? Okay. Okay, so let me go to another set of problems, which is the one of the wisdom of the crowd, okay. So the wisdom of the crowd is a general So Professor, I have a question. Yes, it's about the single-pict preferences. So said that the preferences should have certain order. Yes, exactly. So is that order can be played by the utility function of the preferences that we have? Yes, it exists. Yes, yes, yes. So it can be, yes, it can be represented as in terms of utility functions. Yes, yes. Okay, thank you. Okay, let me go back to the wisdom of the crowd. So the idea is that this way of the social choice rules, these voting schemes, what they also allow to do is to aggregate information from different people. And so the aggregated information is wiser than the information that the individuals have. So imagine a situation where society has to make a decision and there is an objective truth. Okay, so think about whether say climate change is a serious threat or not. Okay, so this is an objective thing. And if it is a serious threat, then we should do something. If it is not, then we can continue business as usual. Okay, imagine a situation where the different individuals on this binary choice, whether climate change is true or not, may have different opinions or preferences because they have different information. Okay, now if you let them vote, then this voting will aggregate all this information. And what you can see is that if the information that each individual has is noisy but is informative in the sense that there are more people who are likely to have the right information, then if you look at the majority, how the majority will behave, then the majority will also reflect what the true information is. Okay, because by the law of large numbers. Okay. And so this is why this and we use it many times also when you go to on the internet or say if we see that something is popular, then it must also be good. Okay, so we associate popularity with the sort of truth of or good. Okay, now there are some situations in which this, the cloud may not be that wise just because individuals are rational. Okay, not just because I mean, of course, if individuals have say misbehaved or have different preferences, do not behave rationally, then you can have whatever happens. But even if they are rational, then you can get into irrational outcome. There are a couple of situations that are discussed. So this is one situation that is discussed in the book of Klember. So imagine that you have these three people and they have to, so the game is that each of them picks a bowl from an urn, which can be either this one or this one. And what they have to do is they get this bowl, they look at the color, they have this information, what is the color of the bowl that they picked. And on the basis of this, what they have to vote is whether the true urn is this one or this one. Okay. Now you would say, well, okay, if I pick a black bowl, then I should vote for a black bowl, I should vote for this urn. And if I pick a white bowl, then I should vote for this urn. Okay. But if you think a little bit more, then you can think, when is it that my vote will really be decisive? Because if the urn is really this one, then no matter what is the bowl that I put, I mean, what is the vote? What is my vote? Well, the others also will have white bowls. And so they will report a white, the true urn is this one. Okay. But the only situation in which my vote is decisive is when I pick a white bowl and the true urn is this one. And some other guy also picks the white bowl. Okay. So this is a very low probability, but only in this case is my vote decisive. Because if I misreport, then the society will take the wrong decision. Okay. So if you work a little bit about this, then you figure out that then, even if I pick the white bowl in this case, it is, and even in this case, if I pick a white bowl, I better report that it is a black bowl. And as a result of this, then society as a whole ends up taking the wrong decision. Okay. So, and there are a number of cases where these type of situations occur. So one interesting one is in court cases in the United States, where in order to plead guilty a person, you should have essentially the majority of the people that are convinced that he's guilty. But then if there is just a, it is just a sufficient that one person does not find that person guilty, and he will be discharged from everything. So then this in the end puts a pressure on people to, even if they think that that particular person is not guilty, then they better report he is guilty, because that is the only situation which they will be decisive. I mean, if all the others are convinced that he is guilty and he is not. Excuse me, Professor. Yeah. So this kind of reasoning seems a bit glitchy to me. I mean, it seems, I can understand why one would make such an argument. But I don't see how one can say that this is a rational argument. At least it sounds very arrogant. No, it's a rational argument from the individual point of view. Okay. So if you think, say, so imagine that, say, you think a person is not guilty. Okay. And you say he's not guilty. Then the only situation in which your vote is decisive is everybody else is convinced that he's guilty. And your vote that is not guilty. Okay. So because of you, this person will go to prison. Okay. Now, but if he's really guilty, and being guilty is an objective thing, then there will be someone else that sorry, Diago, it was a little bit different. Yeah. So if he's not guilty, then if I report that he's guilty, then it will not make any difference because there will be other people in the jury that will report that he is not guilty. Yeah, but it kind of puts an irrational, I mean, it puts, I think excessive faith in other people. Because even in the first case with the ball, what the person who's voting black, even though he picked white, is thinking is, well, probably other people will make the more straightforward choice because that's just what people are. But I myself, as the more intelligent and rational individual, will pick the opposite of what I should pick, because in the case my vote is decisive, it won't matter because the other people will make the the correct decision. Yes, exactly. So you should not have too many rational and smart people in a committee. And that's, that's, that's really the situation. Okay. Yeah, okay. I mean, but if you want, I mean, these cases discussed together with several other cases is discussing a very convincing manner in the book of Kleinberg and easily. So I really recommend you have a look at this. Okay, no, I mean, I can see why it is rational from a very subjective point of view. Your lecture is described as well. So, okay, so let me consider another situation where the wisdom, the crowd may not be so wise. And this is goes under the name of a rational herding. Okay, or yes, rational herding. So, so imagine that we have a situation where I have to choose between two restaurants. And one restaurant is better than the other. Okay. And I have some information. And this information is a noisy signal. Okay, it's a noisy signal with that with the probability larger than one half tells me that indeed, the A is better than B. And with one minus B tells me the wrong information. Okay. And then what should I do when I should behave according to my information? If, because I know that the probability of this information is right, is larger than one half, then I should follow the advice of my signal. Okay. And, but if I do so, if I follow the, this recommendation of this private signal, then what I do will reveal my private information. Then now imagine the situation where there are many people making the same choice. They have to decide whether they want to go in restaurant A or in restaurant B. Now, Mr. K, the gay number K may have his own personal information and he may decide to act upon this information, but he may also observe what other people have chosen. And if he observed that other people have chosen a particular restaurant, then he may consider that they did so that he can observe their private information. Okay. And this information may be of better quality with respect to his own private signal. So essentially, because of this, what this guy may do is just to follow the action of others inferring the signals they received. But if this guy does so, then his action will no longer reveal his own information. So is the information that he received gets lost. Okay, because he is not going to act according to his information, but he's going to act according to the information of the crowd. Okay. And essentially you can show that say it is enough that the first two guys get the wrong information, then everybody else will end up in the wrong restaurant, in the worst restaurant. Okay. So this is called an information cascade because it's like a avalanche where essentially all these people behave in a way that completely neglects their own information. Okay. And so this set of issues which are related to how we interact with information, as you can imagine, is a very interesting subject. And because as we know, there are a lot of reasons why all these information systems and social networks that we have are essentially failing and giving problems. And this is a subject that is really very much studied. And there is a lot of other, say, ideas in game theory and in economics of how you can describe situations where you have, say, this collapse of trust, this type of phenomena occurring. Okay. So with this, I think if there are some quick questions, we can take them. Otherwise, you have a lot of material on the website. So what if the individual recognizes that a collective view is rational? I don't understand this question very rich one. Can you please explain a little bit better this question? Maybe, let me, okay. So in the domain restriction, you can order the alternative, not along an oriented line, but in a circular space, where the extreme of the political spectrum are close to each other. Well, then in this case, again, I think that the result will work because essentially, you will have just one maximum on this circle. And then the same example will hold. Then there is a question on easy models. So yes. So there are many people who have worked with spin models who describe social choice and voting and opinion dynamics and the idea is that this temperature is something that is related to how rational people are. But you can also interpret it in a different way, temperature. So this is also explained on the website. So for each one, Mike is not working. What if the definition of an individual's optimal decision may not be as straightforward as in most simple cases? Factoring complexity of individual thought when subjected to a spoon. So yes. So the, in many cases, the social choice, so individual choice itself may be complex and individuals may not have a transitive rational preference themselves. This we discussed. And of course, all these problems will be even more, the problems of social choice will be even more serious when individual rationality is not there, is not assumed. But what we have shown is that even if you have rational individuals, then society may behave in a very irrational manner. Or the crowd may behave in a very irrational manner. Okay. So I think if there are no other questions, then I think we can stop here and maybe take 10 minutes of break and reconvene in half past four.