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SPEAKER: This is a production of Cornell University.
LOU: So I'd like to welcome you to the second of this semester's Messenger Lectures. The third one, let me remind you at the beginning, will be tomorrow afternoon at 4:45. So as I said yesterday, I wanted to say a few words about the series itself. The series is supported by an endowment from the estate of HJ Messenger, Jr., who was a member of the Cornell class of 1880. The usual literature about him says he was a longtime teacher of mathematics-- a variety of reasons, I was skeptical about this.
His estate gave, in 1923, the equivalent of $880,000 today to support these lectures. It was a $74,000 bequest. So I went and looked a little bit about him and found that the university keeps files on all of its alumni. Be warned. And then once they die, these go to the deceased alumni files, which are down below the ground in Kroch library. And I got the deceased alumni file of Hiram John Messenger, Jr. And it's sort of an interesting story.
He got a B.Lit. In 1880. He came from Cortland, where-- right nearby-- is where his father was a banker. He founded the HJ Messenger Bank of Cortland and Marathon, which you might think would explain how he could give such a prize. But I think from what I could read, it looks like the bank didn't do so well in the years after the Civil War. But before that, Messenger, Sr. had a lot to do with the building of downtown Cortland, such as it is.
[LAUGHTER]
And so young Hiram John Messenger, after graduating from Cornell, returned to Cortland for one year and taught as a professor of mathematics at the State Normal School at Cortland, which is now SUNY Cortland. After this one year, he went west to Napa college in Napa, California, which eventually became the College of the Pacific in Stockton, which is now part of the University of the Pacific. And he taught there two years, also as a professor of mathematics.
Then he returned to Cornell as a graduate student in mathematics and in 1886, received a PhD in mathematics, which was the second PhD in mathematics given by this university. The first went to one Henry T. eddy who may or may not have his name on Eddy Street but was a very successful mathematician anyhow and went on. Eddy does have his name on a building at the University of Minnesota, where he ended his career.
Anyway, Messenger, after getting his PhD on more modern methods and conics, I think was what his thesis title was, moved to New York and became an acting associate professor of mathematics. See, a PhD got him a lower title. He became an acting associate professor of mathematics at the University of the City of New York, which was the early name for NYU. And he taught there for four years. So his math teaching career was not long-term. It was seven years.
In 1891, he became an actuarial clerk at the Metropolitan Life Insurance Company in New York and was in this position for seven years and then moved to Hartford and became an actuary with the Travelers Insurance Company, where he completed his career. During this time, he lectured extensively and wrote various things. Apparently he had worked for Travelers. He did work in finance through various ways-- wrote a pamphlet on different methods proposed for valuing the marketing marketable securities held by life insurance companies. And he went on to become a pioneer in the field of health insurance, of all things. He wrote a pamphlet on health insurance in the United States in 1903.
Anyway, he died in 1913. And his estate didn't pay Cornell-- I presumable that it was paying other people in the meantime-- until 1924. There's was an article in the-- or 1923-- in the October 12, 1923, Ithaca Journal. It was reported that at the trustees' meeting in Buffalo, the president reported this gift of $74,000. And it says the income of the fund to be used to provide a course or courses of lectures on the evolution of civilization for the special purpose of raising the moral standard of our political, business, and social life. These lectures are going to be known as the Messenger Lectures on the Evolution of Civilization.
So the tail end of the title seems to have been lost over the years. But I think it's pretty apt that the first of the things that he had chosen that lectures should be on was raising the moral standards of our political life. And this series of lectures is exactly about our political life and how it interacts with our mathematical way of reasoning about it.
Our speaker-- I'll introduce him again briefly-- is Michel Balinski from Ecole Polytechnique, who, as I mentioned, is an American mathematician so is qualified to complain about the American political system even though he has worked in France. He was born in Geneva, Switzerland, to Polish parents who were there because his maternal grandfather, Ludwik Rajchman was a Polish diplomat who headed the health section of the League of Nations. After the war, Rajchman came to New York and went to the United Nations. And he went on to be the founder of UNICEF.
Balinski moved to the US at a young age and received his entire education in this country and worked here for a few decades before moving to France in 1980. His lecture today is about voting systems and how to choose a winner from many candidates. It's based on his forthcoming book with Rida Laraki of a Ecole Polytechnique called One Value, One Vote, Measuring, Electing, and Ranking. In this book, they achieve a result which basically counters the-- well, he'll tell you-- counters the various paradoxes of social choice theory, this well-known Arrow's paradox which say you can't have a reasonable voting system. Apparently this is not so. This result of Balinski and Laraki has been hailed as a major breakthrough in the area of voting and social choice. And we'll soon hear about it. Michel.
MICHEL BALINSKI: Thank you very much, Lou. It's a pleasure as I said yesterday, to be here and to be able to talk about these problems. And of course, I feel very honored to have been invited to do so. And this talk, as Lou said, is entirely based on joint work with Rida Laraki. And what I'm going to do, of course, since we're trying to do this to a wide audience, is to emphasize the applied character of the thing. But of course, there are-- and I will point this out as we go-- through there are theorems and mathematics behind it which characterize the methods about which I'll be talking.
I'm going to start very quickly simply to put up three, four slides that I put up at the beginning yesterday, because this is a series of talks. So the first one, which I gave yesterday, had to do with apportioning seats. This is old work that was done with Peyton Young. The second now, which I will be talking with you about today, is why the first-past-the-post method of electing one among several candidates is seriously defective. And this somehow will get us into the subject matter and then see what we can do about it. And then tomorrow, I would like to talk with you about a scheme for circumventing gerrymandering.
Now again, just to put the context, this is in the context of saying we all claim to live in democracies. France is a democracy. England is a democracy. The United States is a democracy. And one asks the question, but exactly how democratic is this? And I think there are very serious problems in all of these countries and others, in fact, almost every place that I've been that I've looked at, where really one can say, my, god. This is not reasonable representation.
And here, there is a presidency I don't need to talk about. I did this computation the other day. Senate is really just so lopsided that it's absolutely incredible from the point of view of representation. And you get this also from the point of view of political representation, as is this 51 Democratic senators representing 58% of the population, as versus 49 Republicans who represent 42%. And I will argue tomorrow with some facts that it is entirely possible-- and I suspect has often happened-- that a minority of the population has elected a majority of the House of Representatives. Well, that seems-- so how representative is all of this?
Now nothing different in the UK-- again, I'm sorry for those of you I'm repeating here. But you take the last six elections. You've never had a party who entered with a majority of the vote. They always had a minority, but a large majority in the seat, particularly in 2005, which was hailed as a wonderful, grand victory for Tony Blair. But he got a third of a vote-- you know, 35.2, as versus the conservatives, who got a third also, but just a little bit less.
In France, the situation is just as bad but in different ways. There, there is manipulation of the voting rules very consistently. There are changes in one aspect or another of the electoral code in all of the years that I've listed here. And this last table of four numbers is just to give you a sense of how different the congressional districts are in population. And they're meant to be defined in terms of population. So you see over the entire country, the second of Lozere is the smallest. The second of Val d'Oise is the largest. We're talking about a factor of five and a half to one, whereas-- and of course, that has two aspects. It's because of apportioning seats to each of the departments. That's the first decision. Then it's cut up.
So these two numbers on the left have two components of inequity, if you wish. One comes from the apportionment. The other comes from the districting, from the cutting up of the districts, whereas on the right, we're in the same department of Var. And there, you get a factor of 2.4 approximately. So you see, it's really, really quite bad.
OK. So the problem we're going to face today or to raise today is how to elect and I will put in the same boat, how to rank. Now incidentally, in the theory of social choice, these are two very, very different problems. And we have a theorem which I won't go into into these talks, which basically says the following thing. If you want certain properties where you have people saying, I have a rank ordering of competitors. So each of us, we're all in this room. We each have a-- there's some competition going on. And each of us makes a rank ordering of the competitors. And from that-- and this is the basic problem of the theory of social choice-- we want to get a common ranking.
Now you take a common set of axioms, of properties that you want this to satisfy. And you apply it to saying, I want to get a ranking overall of competitors. And you also apply it to I want to designate a winner. And the result basically is this, It says, well. In that case, you're going to have to be satisfied with saying, Mr. A is the winner-- sorry, Mr A is first. Mr. B is second. Mr. C is third. Mr. D-- this is a men's competition in this example. But the winner is Mr. C. That is, there is not an accord between the first in the ranking and a winner. There's an essential contradiction.
All right. So rhetorically, are elected presidents those wished by the voters? And I would say that-- and that's our problem today. So we want to choose one candidate-- or we want to get a ranking-- among several. And of course, what do we do? We have a mechanism for doing this. Mechanisms, the usual mechanisms, do they do the job well? I mean, you know what the mechanism is for the US presidency. It's an electoral college. And of course, as we all know, it may well elect somebody who does not have a majority of the votes. This happened in 1876, happened in 1888. As you know, it happened in 2000.
But we can look more closely and ask one of the reasons why it happened in 2000. And that is-- at least one reason can be given-- is that Ralph Nader was a candidate in Florida. He got 97,488 votes. Florida had 25 electoral votes. It's quite clear that had Nader not been there, most of those votes would have gone to Gore. He would have carried Florida. And therefore, he would have been president, without talking about all the chads and all the rest of that.
So this is precisely a violation of Kenneth Arrow's famous independence of irrelevant alternatives condition. What does it say? It says that we're interested in who's first between, say, two candidates, two major candidates-- in this case, Bush and Gore. And the presence of a third candidate can flip this decision one way or another. This is a condition that seems to be really very difficult to accept. And of course-- well, I will give you Arrow's theorem in a minute-- it is something that is violated by every method that has and is being used to elect people.
And I do everything in the content of the French presidential election, because the examples are extremely useful to be able to illustrate the things I would like to say. And the Constitution of the Fifth Republic says the president of the republic is elected by the absolute majority of votes. If it is not obtained in the first round of the election, then a second round is held two weeks later, always a Sunday. The only two candidates who may present themselves after the eventual withdrawal of more favored candidates-- that has never occurred-- are those who have the largest number of votes in the first round-- so now, no electoral college to mess things up.
And now let's first of all look at this famous collection of 2002, where you all know, the extreme right candidate, Jean-Marie Le Pen, survived into the second round. Here are the votes in the first round. So as you see I mean, it's very, very flat. Chirac, who is the standing president, only got just slightly-- didn't even get 20% of the vote. Jospin was the standing prime minister, socialist. Everybody expected the second round to be between Jospin and Chirac. But here's what happened. Of course, we have here 16 candidates. Now second round, overwhelmingly for Chirac, because overwhelmingly, to be more honest, against Le Pen, as you can see. Chirac barely got 20% of the votes, didn't even get the 20% in the first round.
Now let's look at this and ask several questions. Chevenement, who picked up 5.33%, he was a socialist and clearly took away some of the socialist votes. It seems quite reasonable that had he not been there, several percent of those votes at least would have gone to Jospin. It seems quite clear that Jospin would have survived to the second term. And the vote would have been between Chirac and Jospin.
Now less known perhaps, but there was a candidate by the name of Taubira. She also was a socialist. And she obtained-- now, wait a second. Where are we? Taubira, as you see, she's down at the last row here on the left. She obtained 2.32% of the vote-- very little. However, she offered to withdraw from the first turn. When she offered to withdraw on the condition that her expenses would be covered. And the socialists refused to do that, so she stated. I've learned subsequently that-- that is, I've learned-- how should I put this to make sure that I'm not, in some sense, lying? I have been assured by people who I think should know-- two people, in particular one who used to be the head of the second television station in France-- that, of course, it was well known that Chirac supported her candidacy.
At the same time, there is an ally, an ex-ally of Chirac by the name of Pasqua, who had been minister of the interior, who was very close ally of Chirac across the years. He had said he would run and then-- at any rate, he said he would. And then he didn't. Why, I'm unable to say. But now I postulate. Suppose he had and Taubira-- and he replaced Taubira and Taubira had withdrawn. Well, then what would have happened? It seems quite clear that most of Taubira's votes would have gone to Jospin. Jospin and Pasqua would have taken away some of the votes of Chirac. And it would be totally possible for this to have occurred, which would then mean that it would be Jospin against Le Pen at the second term. And therefore, quite clearly, Jospin would have won by a very large margin.
So what's the point? The point is that we have here a vicious occurrence of the relevance of irrelevant alternatives. That is, here are clearly irrelevant alternatives. Taubira, Pasqua, Chevenement had absolutely no chance of being elected. But they could have a very strong influence on the election.
Now, of course these-- and I say now-- these examples prove that with the two round first-past-the post system, the presence or absence of unelectable candidates can determine the outcome. And of course, this is a violation of Arrow's condition. Now we're going to come to this last election. And of course, election of 2002 had a big impact in France, people very aware of what had happened. And obviously, the value of the votes given to Chirac, I mean, it was not as though 82% of the population wanted Chirac as president.
Here is the first round result of the 2007 election. And here is the second round. As you know, Sarkozy was elected. Now there are lots of very interesting things about this election. First of all, as I just suggested, the lessons of 2002 were very, very present in the spirits of people, therefore, very high participation and, I believe, more strategic voting, more realization that maybe you should not vote your conviction. Maybe you should vote otherwise. Because you don't want to have the kind of occurrence you had five years before.
Secondly, it seems very clear that there is one candidate who would have beaten every other one in a direct confrontation. All the polls show this. I'm showing you one poll. And that is Francois Bayrou. Bayrou, in this particular poll three weeks before the first round, was shown beating Le Pen by 84% of the vote, Segolene Royal by 57%, and Sarkozy by 54%. This was consistent for the last-- every time this kind of a poll was done, the same qualitative results occurred.
One of the reasons is that of those who were for Royal, 72% they would prefer Bayrou to Sarkozy. And of those who said they were voting for Sarkozy, 75% said they would vote for Bayrou against Royal. So he was picking up all of those from both sides.
Of course, this shows what, if we believe this? The French electoral system, in fact, any first-past-the-post system, can fail to elect the so-called Condorcet winner, Condorcet winner being one who beats every other candidate. Now of course, I haven't done the table for the other candidates. There's not much point-- again, who beats each one in a head-to-head confrontation. So you're not being confused by other candidacies.
Now let's look at the traditional view of the theory of social choice. Traditionally-- commonly, I would say, and in theory-- a voter is assumed to have a list of candidates in his or her mind, ordered from best to worse. Now, we say-- and when I say we, it's not the royal we. It's Rida Laraki and I-- we say, that is nonsense. And I can give you two instances of why. I can give you two proofs of the nonsense.
First, the old system for judging figure skating-- I know probably many of you watched it on television. And in some sense, it misled people as to thinking about how the system worked. Judges announced scores-- 9.19, 9.8, or whatever it might be. And then immediately, the television people would take averages and say who's ahead of whom and so on and so forth. But that's not the way the numbers were used. If a judge gave a 9.8, that number was remembered till the end of everybody's performance. And it was used only for one purpose. It was to order the candidates according to him. So his grades were not used as grades. They were simply used as an input of a rank ordering of the performers. OK?
So now what does that say? Well, it's just too damn difficult to write down a rank ordering of the competitors. There might be 20 competitors. How are you going to keep track of who's going to be where and where you're going to slip them in and all that? Much easier to give scores-- that's first. So it's not practical.
Secondly, let's take politics-- 12 candidates this year in France, five years ago, as you saw, 16. No one had a rank ordering on the 12 or 16 candidates. What do you think? Well, there are a bunch of candidates. I would never consider them. The hell with them. I'd forget them. Then there's some candidates. Maybe I have a preference. Maybe I have a second preference. But of course, I have intensities of preferences. But certainly, it's not a rank ordering.
And thirdly, I will tell you about an experiment shortly, where we tried a method and were implicitly-- what happened? We asked people to grade. I'll come to that in a moment. That's what our approach does. And it was extremely telling, because a third, a little over a third of the voters in our experiment, gave the highest grade possible to at least two candidates. So they did not even want to make a choice, a single choice. They preferred saying it's two candidates. And three are my top choice. So I say that is nonsense.
Now secondly, I would like to give a little bit of the background, a flavor of social choice in order to be able to explain what Arrow's theorem says. And the first, of course, the parting point is that there may be no Condorcet winner. And here is an example. And what these numbers say here is this, that if I take A against B-- that is, how many votes-- let's see. Do I do that? All right.
A against B, he gets 29% from the first list, right, and plus 38 from the second list. That's 67%. So A beats B with 67% of the vote. B beats C with 61%. And C beats A with 70%. And of course, this is the famous Condorcet paradox. In fact, Condorcet was preceded by Ramon Llull, who proposed a more refined idea in 1299, where though he does not say in what he writes that he has recognized that there could be a Condorcet paradox, but his method was to do the following thing-- give to every candidate a point for each candidate he defeats in a head-to-head race. If they're tied, a point to both. Then elect the candidate with the most points.
That, if there is one candidate who beats all the others head-to-head-- that is, a Condorcet winner-- obviously, he will have the most points. But maybe we don't have that. So this is a more general rule. This is today known as Copeland's method, attributed to Copeland. This is 1951.
Now a lot of objections we can make about this approach, and I'm not going to try to make a whole big story. But I would simply like to take two concepts and show that there is an essential contradiction. Suppose that we have three n voters, 300 voters, who have the following-- whoops-- list. That is, one hundred of them prefer A to B to C. Then 100 prefer B to C to A, and Another 100, C to A to B. And you see this as cyclic. So A is in first place. Each of them is in the first place, second place, and third place as often as the others. So there's a perfect symmetry between them.
And what can we conclude from this? Well, quite an obvious thing, that collectively, what do these three end voters say-- that the three candidates are tied. Or another way of saying it, their opinions cancel each other out. So we can forget them. The others will decide.
So now look at this example. Suppose the electorate's lists are as follows. OK, so here are the six possible orderings of preferences. And notice then that on the left, I can take out-- I can let my n be 13%. I can take them out. And on the right, I can relate my n be 4%. I can take them out Sorry-- I'm getting ahead of my story. I can do that. But we'll do that in a second.
But if I look at this particular problem, then the Condorcet winner is A. A gets 51% against B. And he gets 70% against C. But if I take out the 39, the n equal 13 on the left and the n equal 4 on the right, I'm left with the problem in red. And I ask you to look at the problem in red. Who's the obvious winner? Well, C is dead last. And B certainly beats A, whereas what is the Condorcet winner? He's A.
So by the presence or absence of quote, unquote, "preferences"-- that is, and I don't like that word, but ordered lists that are canceling each other out-- that determines the Condorcet winner one way or the other. That says this notion is not necessarily the right one. Of course, could be the right candidate. But in the literature and in the concept of people, the notion of the Condorcet winner is the strongest one that has survived. And it continues to be talked about. And yet, I think it is a perfectly flawed idea.
OK. Now in 1433-- and incidentally, one of the reasons that I want to emphasize this age is that people have been thinking about this problem in the same way since 1299. That is, rank orderings-- it's always making comparisons with no weights involved, no nothing. It's just a rank ordering. Nicolaus Cusanus proposed then what is known as Borda's method. That is, it works as follows. A voter-- again, come back to the situation. We have these lists. A voter contributes k Borda points to a candidate if he. Prefers that candidate to k others. And so, we look at everybody's lists. And each candidate gets a certain number of points by this process. The one with the most points wins.
A candidate's Borda count is the sum of the Borda points, as I've said. Sorry, I'm getting ahead of myself. And the Borda ranking ranks the candidates according to their Borda counts. Now obviously, this method, the cyclical things, it washes them out, right? Because everybody appears in that, say, in the first position the same number of times and so on. So they're going to get the same number of points. So it will not change when you add one of these Condorcet components or not. It clearly cancels properly.
On the other hand, look at this example. Suppose the electorate's lists are this. Now we'll see that Borda's winners are not necessarily legitimate. The Borda winner here is A, because the Borda ranking is as I've given it. For example, A's Borda count is what? Well, he's on the left. I've put him in red. So A beats to people in the first 33%. So that gives him two times 33. Then he gets 2 times 16. And then he's in second place for 30%. So his Borda count is 128. B's is 117. And C's is 55.
But if C withdraws-- and C is just the last candidate, right-- what happens? Well, things turn around. B becomes the Borda winner. So the fact of C being last, if you drop him, then we change the result. That again, it's a violation of Arrow's independence of irrelevant alternatives. The presence or absence of C determines the election.
This is unavoidable in the traditional model. If voters truly have lists in their minds and they can do whatever they like, if when one candidate is the first on everyone's list, then that candidate must be elected. And finally, if winners do not change because of some irrelevant candidacy, then Arrow's impossibility. There isn't any method of election that does this.
Now this has usually been interpreted as meaning there can be no satisfactory mechanism for amalgamating the wants, wishes, or opinions of an electorate. And politicians are very pleased with this result. Because that lets them manipulate as much as they want and try to change things in order to suit themselves. Nevertheless, today, the debate goes on in voting theory. It gives the Condorcet winner a place of honor. Others are arguing for Borda as the method to be used. Sometimes the two are convoluted into a Baroque marriage of incompatible ideas. For example, this suggestion has been repeatedly made. If there is no Condorcet winner, then you should use Borda's method.
A matter of fact, Borda's method was used to elect members of the French Academy of Sciences from about 1794 until 1800, when a newly elected member, Napoleon Bonaparte, insisted it be abandoned because of its manipulability. And here, Laplace is the man who really pointed this out very distinctly. He had imagined that voters, that they give number grades to candidates. And then he did a probabilistic analysis to justify Borda's method, but afterwards, rejected it. And here's what he said. Borda's method would be, without a doubt, the best if considerations alien to the merit of a candidate did not influence the choice of the electors, even the most honest ones, and did not determine them to rank last the most dangerous opponents to their favorites.
Indeed, in the traditional model, where again, it is assumed the voters have lists, manipulability is unavoidable. This is the famous theorem of Gibbard-Satterthwaite, which says there is no method for which a voter's best strategy is to vote his convictions. And of course, especially in a popular election, that's what people want to do. That seems very clear. And so our conclusion is, well, look, thinking about voters or judges and thinking that they rank order, it's just a bad model.
First of all, in practice-- and we've looked at practice. We've looked at piano competitions. We've looked at wine competitions. I'll come to wine a little later. We've looked at skating, gymnastics, diving. That's not what they do. What do they all do? They give scores.
And so now I will describe to you the translation of this into an experiment that we conducted in the first round of the French presidential election this year. Now we begin-- this is exactly the ballot that people had to vote in. I will give you the French translations in a second. But I mean, for those of you who read French, we thought about this statement a very long time. And I would like to say right away that it was quite influenced-- we think it's very important to address the question. And this was very much initially motivated by Ramon Llull's charge in 1299.
"It is necessary," he said, "to ascertain that in the election, three things should be concerned, of which the first is honesty and holiness of life. The second is knowledge and wisdom. And the third is a suitable disposition of the heart. Each person having a vote in the chapter should take an oath by the holy gospels of God to consider these three things and to always select the person in whom they are best embodied." You know, the thing that struck me in reading this was I wondered, what would a modern voter do were he confronted with such a statement, translated, of course, into terms that would be more acceptable. Maybe people really vote differently.
Well, here's the translation of the end of our ballot. To be president of France, having taken into account all considerations, I judge in conscience that this candidate would be-- and now you had to say either excellent, very good, good, acceptable, poor, or to be rejected. Now I'm not sure these are the good words in English. They are excellent for French. Because excellent, very good, good-- the exact terms we had in French-- excellent, very good, good, acceptable, poor, are the grades given to all French school children. Everybody knows what these means. These constitute a common language of evaluation. The one thing that was added was to be rejected. So candidates were asked to check for each candidate, say, give a grade. And it was explicitly said that if there was no check, then that meant the candidate is to be rejected.
All right. So what is this point of view? Voters here do not vote. They evaluate. And I underline they evaluate with a common language of grades, as Laplace had originally imagined. A candidate's set of grades determines that candidate's what we call majority grade. I'll come to it in a moment as to how it's defined. And it is, if you wish, the final grade that is conferred upon the candidate by the electoral. The candidates are ranked according to their majority grades. And the first among them is the winner.
Finally, when two candidates have the same majority grade, then there is a procedure which will always rank one ahead of the other, unless both have an exactly identical set of grades. Consequence-- this method, we call this the majority judgment, satisfies Arrow's independence of irrelevant alternatives. Why? Because the grades belong to the candidate. And whether there's another candidate who withdraws or does not withdraw, it doesn't change anything. So it automatically satisfies that.
As a practical matter, we believe it is very important to pose a clear question, that by and large, people try to answer the question, that this common language, again, is absolutely essential. And from this point of view of this language, in the new theory, Arrow's impossibility says something very important. It says without a common language, there can be no consistent collective decision. So it plays a very important role here.
Now a lot of people say, yeah, but of course, these common languages don't exist. You know, I have one professor here who is very harsh on another who's not and so on. You never get anything together. Well, as a matter of practice, we have observed that this is not true. Common languages do exist. When you look at diving, when you look at diving, you look at skating, you look at a lot of professional-- excuse me-- sporting competitions, they all use scores. Piano competitions do, too. They always use scores. And you can observe that when somebody says this is a performance that's worth 9.2, everybody knows what that means-- don't necessarily agree, but they know what it means.
Now look. It's not perfect. Lou will tell me that's blue. And I don't see the blue, that the color is blue. I have something in my mind when he says that. It'll be blue. But it could be somewhat different. So it's not perfect. Incidentally, a centimeter isn't perfect, either, when you use it. There's a little bit of margin of error and so on. So it is a measurement. And as I said before, we believe that the words were used in the French [INAUDIBLE] do really constitute for France a common language.
All right. Now what is the majority grade? It is the middle most of the grades, or the median. A majority of voters have assigned the candidate, at least this grade and a majority at most this grade. So for example, if we got this distribution of grades-- 8% excellent, 23% very good, 27% good, et cetera, then this candidate's grade would be good. Because 8 plus 23 plus 27-- so 58% of voters assigned at least a good and 69 at most a good. So the candidate's grade is good.
And let me say this right here. Voters love the idea that candidates would get grades. So they know you're just acceptable. Or you're just good. No. It doesn't go beyond that. Don't forget. OK. Now why do this? Well, it combats strategic voting for the following reason.
Suppose a voter-- you're a voter. You voted, but you voted very good in this list. And the candidate got good. So with the usual way of adding, what would you have imagined? You'd say, oh, gee, I made a mistake. I should have given an excellent because that would have helped move the score up. But with this, you can move from very good to excellent and won't change anything. You know, the 58%, at least by going from very good-- sorry-- by going from very good up to excellent, it certainly doesn't change where the middle is. And obviously, this remark goes the other way.
Suppose you thought the candidate was acceptable, in the ordinary voting, you would think, oh, boy. [INAUDIBLE] giving points. You think, I should have moved down. But it doesn't do any good, doesn't change anything. So it is what we call strategy-proof-in-grading. This is quite important for wines.
Now a second property which is immediately obvious is that the majority grade respects a majority opinion. That is, if a majority gives a grade to a candidate and the whole majority says this candidate is acceptable, then that candidate is great. Now it turns out that these properties uniquely define the method. There are many other properties that do also, which I'm not going to go into here today. But there are many characterizing properties. In fact, almost everything that we could think of that seemed reasonable, you could then use to characterize the method that we've proposed.
Now let's look at breaking ties and voting situation. And then I'll show you how it works out in the wine situation. So now let's assume two candidates have a good. So who's better between the two with the grade of good? Well, if a candidate's percentage better than the majority grade is greater than his or her percentage that is worse than the majority grade, then we say this is a majority grade plus. So let's say it's good.
If a candidate's percentage of grades that is better than good is greater than that are worse than good, then it's a good plus. And otherwise, if it's the other way around, it's a good minus. And now what do we say? A good plus is obviously better than a good minus and same thing for the others. And if we have two good pluses, then the one with the higher percentage of better than good is ranked ahead. And if there are two good minuses, the one with the higher number of worse than good is ranked behind. So this is presumably, with many voters, plenty enough to distinguish between everybody.
All right. Here is that the description of the experiment. It was conducted in three of 12 voting bureaus of a small town called Orsay, outside of Paris. I wish to emphasize this is not representative of France. Participants were well-informed about this ahead of time in many different ways. And they cast their ballots in exactly the same conditions that they cast their official ballots. They first cast their official ballots in one part of the room. And then they came in another. And we tried to persuade them to participate. And 74% of them did participate. So we had quite a large rate of participation.
Yes. A lot of this was interviewed by a television person who was present. And so we have testimony that people were really very pleased about this way of voting. And here are the results. Now I know of one person here who followed these elections closely. But every French person to whom we've shown this, when we take out the names of Bayrou, Sarkozy, and Royal. We'll look at the distribution of these grades and be able to identify them. Does anybody care to try to do that?
Now let me quickly say that these are the rejected. We have recorded the blanks. But the blanks are interpreted as reject. And voters were well-informed about that. So if you wish-- now of course, one of the wonderful things here is Le Pen, you see that he was rejected by 74% of the vote. He's dead last with this system, whereas in the actual system, he was fourth. Anybody want to hazard a guess?
SPEAKER 2: Well, I think the first one is [INAUDIBLE].
MICHEL BALINSKI: Yes.
AUDIENCE: I think the second one [INAUDIBLE]. The second is [INAUDIBLE].
MICHEL BALINSKI: Well, it's the-- almost. You reversed Royal and Sarkozy. You see, Sarkozy got the largest number of excellents but also a very large number of-- lots of reject. He was very controversial. And these numbers say a lot. In red are their grades. So you see there were three goods. One was the green candidate who ended up eighth officially. With us, she was a strong fourth. And I think that shows exactly what the electorate wanted. They are very conscious of ecological problems. But they didn't want her as president. So you get a lot of information out of this.
Now here's the ranking. I actually had them in the ranking that we had. So you see, Bayrou was a good plus, Royal, a good minus, Sarkozy a good minus, and on down. On the left is the national order. So Sarkozy was first, Royal second, Bayrou, Le Pen fourth, and so on, these numbers. These give the actual order finished nationally.
And you see-- well, all right, what else do I want to point out? This is very significant. The last is very significant. It shows really that France really did not want Le Pen, no question about this. And well, I guess I don't. Now, I would like to make another remark, which--
AUDIENCE: What's the [INAUDIBLE] to reject rather than just having poor as the lowest?
MICHEL BALINSKI: Well, we felt that in our estimation, voters want to be able to express themselves, OK? Now there's a problem of giving too big a language because it becomes too fine. But there seems to be, we felt ahead of time, a lot of people saying, look, this candidate, never. I really just know, and in particular Le Pen-- that in particular Le Pen, that's clear, and so that seemed to respond to a very strong wish.
Incidentally, i don't give you the statistic. But it was very interesting. The least used mark was excellent. The next least was very good. I mean, it just went up to reject. The distribution of the marks was very, very clear. Incidentally, I'm sure this would happen anywhere. I mean, that's how our politicians are looked at. And it would be very salutary for them to know it.
Now look, I'm going to look here in our experiment-- I mean, there are so many things I could tell you. But let me just, again, say in our experiment, Bayrou was first, Royal was second. So I'm sticking to this experiment. And I asked the question, could strategic voting, could voters have manipulated who would have preferred to have Royal winner, could they have manipulated and made her the winner rather than Bayrou? And the answer is yes, of course. If there were many of them, they could. But now if on the other hand, you look at those voters who graded Royal above Bayrou and if they postulate, if they did as I will show you in a second, then they would have failed.
So here's the way to read this slide. There are 2.8% of all the voters who gave an acceptable to Royal and either a poor or to reject to Bayrou. OK? And let's say here, there is 6.9 who gave a very good to Royal and a good to Bayrou, and so on. OK?
Now the first point is that, of course, here we assume, OK, they preferred Royal to Bayrou. But I mean, they didn't think much of either one. Would they be tempted to cheat and to exaggerate? You know we say, no. None of them did that. Now please look at this sentence here. Polls estimated at 20% the number of people who gave strategic votes different than what their convictions were.
OK, so now I will assume that 33% of those who said Royal was excellent and Bayrou was very good would vote strategically. How can they vote strategically-- by lowering Bayrou below good. But it doesn't serve them to go any further. Because as soon as you go down, that's the only that will change. You go any further, it's not going to change things, right? So you can cheat. But there's no point to going very far. And everything is in this slide.
Now one of the things to see right away is that the voter who can raise one candidate's grade cannot lower the others and vice versa. So you can cheat one direction but you can't in both. Now incidentally, when you're using points and summing, none of this is true. Any one person changes, he has an effect on the result. And so if you do as we have postulated-- and of course, this 9.2% here, if they either put Royal excellent or very good and Bayrou one of these, they can't do anything to change. There's no way they can influence matters at all. So this is resistant to change. And we have several theorems having to do with characterizing the method and showing why it is the least manipulable possible.
OK. Do I have another few minutes? Well, this incidentally, what got this started was wine. Because I wrote a series of popular articles in the French Scientific American five years ago. And as a result, a very well-known oenologist called me and said, look, I think your techniques would have something to do and could really help in our procedures for grading wines for competitions. I don't think that these methods are very good, and so on and so forth.
And he came. And we spoke. And then finally, he and I decided, well, let's look at this problem. And this was very, very important, I think, because it gave us another mindset. Everybody since 1299 has thought that the only way you can think about ranking candidates is what-- listing them one after the other as an input. But when you have wines, you don't. Somehow, it liberated us. And so we started looking at that problem. And of course, very soon, we saw what we felt had to be done for that, and then, of course, that it goes on and is applicable elsewhere.
Now it has been used, this majority judgment has been used in this French wine competition known as Les Citadelles du Vin in June, 2006 and 2007, which is in the Bordeaux region. Both times, we're talking about 1,250 different wines. It was used in parallel with the usual system. The usual system, I will show you the rating form that is used. But the argument of this oenologist was to say that the sum of points-- they use a point system, which you'll see in the second-- misses the point because it has difficulty in detecting exceptional wines by overly favoring those that are taste-wise correct.
Secondly, he said that, moreover, of course, he felt that the quality of a wine is not the sum of individual qualities. It's something that goes well beyond, and that in effect, really respectable judges do not judge in that way. What they do is taste, decide the final grade they want, and then attribute the points so that they add up correctly.
Well, here is the form that they use. I'm sorry. I've had to-- just to get things together on one slide. So the first column says excellent, very good, good, passable, inadequate, mediocre, bad. And for each of these-- I think there are 14 or 15 constituents-- there are points associated with this. And what a judge is supposed to do is check-- well, in this case, as you see, I've checked very good here for limpidity, excellent for nuance, a very good for intensity, and so on. And that gives you altogether a score of 90. The sum of all the first column is 100. The last sum is obviously zero. So you get a score of zero to 100.
And in our experiment, it goes like this. For you, this wine is excellent, very good, good, passable, inadequate, mediocre, or bad. In wine, there are typically five judges. And for any wine, the grades may be, of course, listed from best to worst. Here's an example-- Bordeaux, very good, good, good, good, bad. And so, of course, median is good. So it's grade is good.
However, sometimes there may be an even number of judges. So there isn't any middle. And in that case, we say you must give the lower of the two middle. So if we had here six judges, the grade for this Arbois would be passable. Now why? Because I will compare these two for the following reason. We say that in this example, the Arbois should rank ahead of the Beaune. Because all of the Arboi's grades belong to the middle interval of Beaune's grades.
So look-- here's the Beaune. And here's the Arbois. Take the middle interval of the Beaune's one and the Arbois-- I'm sorry. Is that way I'm saying it? Which is better? And Arbois is right smack in the middle. What is that telling us? There's a much greater consensus around the grade given to Beaune than there is about Arbois-- sorry, to Arbois than there is about Beaune. Excuse me. I've got it here.
The Arbois' grades are either good and passable. They're all good or passable. And the middle interval here is very good to inadequate. So everything in Arbois is in the middle interval of Beaune. Conclusion-- Arbois is better. Why? Greater consensus-- consensus is what majority decision is about. We're looking for a consensus on which is the best. And we say that is the basic principle. That gives us the lower middle.
Therefore, if we want to rank three wines. Let's say there's an Anjou, a Bourgogne, and a Chablis. Suppose they're all good. Well, what do we do? Well, look-- now we know they each have a grade of good. But we want to differentiate between the three wines. So what do we do?
Well, we know they're already good. So the grade of good has been given all the information it can give. We drop it and apply the same procedure.
Now the secondary grade is good for Anjou, good for Bourgogne, passable for Chablis. Chablis is last. But now we have to do the same. Again, we'll drop the goods for Anjou and Bourgogne. We find both are now as a tertiary grade very good. So we do it again. And what do we find? Anjou is-- now it's passable mediocre. And so Anjou is better than Bourgogne, which of course, we know already is better than Chablis.
Now this idea and this procedure, it is from this that we deduced what you do for the voting when there are many, many, many judges, where you can do it in terms of percentage. And you don't have to do this painstaking, tie-breaking rule. Now from that, as you can see, any two competitors can be ranked unless their grades are exactly the same. Otherwise, you always get a difference.
And I will finish by saying that as usual in life, you think you've invented something different. But somebody had the idea essentially before. And that was Sir Francis Galton. Galton had the germ of that idea in 1907. And he applied it in the context of what we call the budgeting problem. He said, "A certain class of problems do not yet appear to be solved according to scientific rules, though they are of much importance and of frequent occurrence. One, a jury has to assess damages. Second, the council of a society has to fix on a sum of money suitable for some purpose. Each voter has equal authority. How can the right conclusion be reached, considering that there may be as many different estimates as there are members?
That conclusion is clearly not the average of the estimates, which would give a voting power to cranks in proportion to their crankiness. I wish to point out that the estimate to which least objection can be raised is the middle most estimate. Every other estimate is condemned by a majority of voters as being either too high or too low." Thank you.
[APPLAUSE]
LOU: There's time for a question or two. Are there any? Yeah? Yeah?
AUDIENCE: Do you think that voters will react negatively to the fact that at its base, only one vote really counts in the system?
MICHEL BALINSKI: I didn't-- repeat, please.
AUDIENCE: Do you that voters will react negatively to the fact that at its base, only one vote really counts in the system?
MICHEL BALINSKI: Well, I still didn't understand-- out of what?
AUDIENCE: In your system, only the median vote matters. So do you think that voters will react negatively to the fact that only one vote matters in the system?
MICHEL BALINSKI: Oh, I'm sorry. One vote does not matter in the system, does it? I mean, after all, it's got to be the middle one. And everybody somehow-- that I don't think so, no. I mean, look, what is being used today-- the average, right? And now you're substituting the median for the average, if you wish, except that it's a finite language, whereas if you're doing sums, you're getting something different.
AUDIENCE: I don't know. I guess in average, each voter has defined [INAUDIBLE] in the sense that they can't move the average, however [INAUDIBLE] it may be. But in this sense, only one vote actually decides.
MICHEL BALINSKI: Well, look, I mean, you can't say that one vote decides. Because how do I put this? If you drop other voters or if you drop him or if you drop other voters, I mean, you're going to change the outcome. No, I mean, all voters have exactly the same influence on the final outcome. Yes?
AUDIENCE: So what you've done is-- one of the most surprising things to me here that you've done is that you were able to get this implemented even in a test basis in that one place in [INAUDIBLE]. Do you have a strategy, or can you imagine it expanding? In other words, it's a wonderful theoretical enterprise. Most of us here [INAUDIBLE] see it implemented on a grander scale. Because it has many properties. And can you imagine a scenario in which this would be adopted?
MICHEL BALINSKI: Yeah. Obviously, we think about this. Our feeling is that what we would like, what we're going to try to do, and we've started to try to do-- well, first of all, I'll give you one example. There's a thing called the [SPEAKING FRENCH] in France. And a person-- this is a competitive exam. You study for a year. You go up before a jury. The jury is forced to rank you one, two, three, four, five. OK? Everybody's got to be ranked until a certain cut off and that's finished. And the ranking is announced. We are now in the process of trying to persuade the [SPEAKING FRENCH] in public law to adopt this method. And he is very interested.
So secondly, I think of departments. You're going to appoint one person, let's say, or maybe two, or maybe three. It doesn't matter. But you're going to appoint somebody. And if you cannot reach a consensus and you need to go to a vote, then I say use this. Because you're going to be combating strategic behavior. And you know well there is strategic behavior. It's well in your experience. So I say, do that.
Then the wine people-- I mean, this oenologist, he's already put in into his book. He feels this is the right way to do it. Then we're going to try to sell this to the skating community. And the hope is that as this gets-- and to the musical community and so on. And as this gets slowly used here and there and proves its value, then hopefully, we can get to politics. Now the problem with politics is, of course, that the actors are the referees. And the ones who are in power are those who won with the old rules. It's very difficult to get them to be interested.
Now the ones who are interested, they're the ones who have lost. Now Bayrou, Bayrou is very pleased by this result. I mean, I know him. He's called me in for consultations and things of this sort. So this result, he thinks is absolutely wonderful. But Sarkozy, I suspect is not very happy with this result at all.
So I think you have to start with the little places and places that are willing to experiment. And as I say, I make a could plea to you. I hope that in your departments, you'll try this. And I think the procedure would be to say, OK, we must decide on a common language. What is our common language? That's the debate. And once you've got the common language, try it. And this would be a way of--
AUDIENCE: When you say that this [INAUDIBLE] strategic [INAUDIBLE], it seems to me that you're saying it [INAUDIBLE] strategic voting by one individual. Because that one person wouldn't change the grade. But it seems like if every Republican said, I always vote excellent for the Republican and rejected the Democrat, they could make a difference in where the median would end up.
Look, if everybody does that, we're just back to usual voting, right? OK. So [INAUDIBLE] case. But here's an example. Sorry, that's not the one, you know, the example I showed you before here, where you're getting-- I don't know-- what the total number of people, about 15%, are cheating as much as they can in order to-- or 20% of all the voters. And they couldn't upset the result. So it's not one.
Now technically speaking, it is strategy proof in grading if you believe the following thing. You're a judge. And you say, this is worth a-- let's say it's a point system-- nine. And if it is true that the further away from nine it is, the less you like it, and we're doing wines, where I have it under authority that nobody knows what they're tasting. So there aren't these things, then you can't cheat. OK?
Now in politics, you can't do that because there are these other things going on. However, again, I think-- and now we're talking in not mathematics. We're talking about this nebulous business of modeling, mathematics, and psychology, and other things and how do people behave? I think people want to answer the question they're posed. They don't like to cheat. Of course, there are few with like to cheat, obviously, those who like to play around. But the vast majority I don't think are-- they feel better in being able to say what they really think.
So yes, I think it should work quite well because of that. And I think the experiment is a step in that direction. Clearly, it was the first time they used it. So maybe in time, they're going to have different behaviors. But of course in time, if this is used, then this will change the behavior of the politicians. Because today, what is the object? Today, the object is to have a bit over 50%.
And I was very struck-- I was saying this at dinner when we were talking the other night-- that in the French presidential election this year, there were various television appearances of the major candidates, where the major candidates appeared in a forum like this with supposedly randomly picked or whatever citizen, which was of course, was doubtful. But the point is that the questions were of the nature, I have a broken leg. What are you to do for me? And that was the nature of the answer, too. I'm going to do this for you.
And I say, no. I think it's possible to address the [INAUDIBLE] the fine question, what's good for the country, and address the question directly on paper. This is where I think Lou made a very deep impression on me. And I think that having a statement which people are asked to respond to, yes, many in great numbers will respond to that. But of course, now we're not talking about-- we're talking about something else. I don't know exactly what. But I do feel that this is true.
LOU: Are there any others? Yeah, in the back.
AUDIENCE: Is it the weakness of the-- is it a weakness of the [INAUDIBLE]. For example, politicians get very low grades and wines high grades. Is it a weakness that [INAUDIBLE] the objective [INAUDIBLE]?
MICHEL BALINSKI: Well, I cannot tell you that I have experimental evidence to say that. You're saying wines get high grades and politicians get low grades.
AUDIENCE: Well, uniform grades of, say, [INAUDIBLE].
MICHEL BALINSKI: But look, if it's in common, it doesn't matter. Do you see what I mean? That is if there is a sense of-- again, I come back. Very essential to our notions is that you have a common language. And we do develop this idea in somewhat more detail than what we're writing. But this has to do with the very ancient problem that we all know about, where you have to have the professors who give very harsh grades and other who are very generous. And so you get these imbalances if you want to try to get to evaluate a student over the course of his entire or her entire career, or over a year or whatever. And that, of course, the extent to which people grade have a different interpretation of the language, that will be a weakness. I don't know if I've answered your question, but I--
LOU: One here, and then one more up there. And then-- so you're up.
AUDIENCE: Yes. This sort of strikes me as a refinement on approval votes.
MICHEL BALINSKI: Well, approval of voting is a particular case in which you have a language of two words. Now the problem with approval voting-- first of all, I don't think that's enough. Secondly, approval voting, you're not giving voters very much with which to express themselves, right? It's very meager what you're giving them.
Secondly, I would say-- we maintain-- I mean, I'm sorry. I hope you don't think this will be arrogant in some sense, but the trouble with all of the analyses for and against approval voting is that they've been done in the context of the traditional model. It's not viewed as a grading system. It's viewed as a voting system. OK?
And so what are their claims? Well, they make claims like approval voting has a tendency to choose the Condorcet candidate. Well, of course, that's false to begin with. Or another claim is made-- approval voting is completely indeterminate, very nice mathematically. What it says is that you can have, in approval voting-- but again, you see it departs from the notion that you have a rank order in your mind. Everybody is a rank order in your mind. And so what they will do if they're approval voting is they will choose a cutoff point somewhere and give ones. They'll vote for all those above and not for those below.
Now it's been shown that you could take one profile and produce any order whatsoever because of these cutoffs. But that's ridiculous. I mean, it's a very nice mathematical thing. It's lovely. But it is ridiculous in any true sense of voting. There are some candidates you really like and don't like the rest. You're never going to just place it anywhere. On the contrary, where you place the cutoff is the whole business. And that amounts to a great deal. So that's nonsense as a practical matter. It's a lovely-- otherwise, it's a lovely observation. But I think as a practical model, it just doesn't mean a thing.
LOU: [INAUDIBLE] one more.
MICHEL BALINSKI: Just the last thing-- and I think that, frankly, I've always thought of approval voting, there's not one theorem in approval voting which gives you any sense that this is a reasonable way to vote. All of it is very nice seat of the pants arguments, reasonable arguments, but nothing really definite. Finally, there is an [? analysis, ?] except in the case of so-called dichotomous preferences. That, of course-- but that's the only exception.
LOU: OK. Last one.
AUDIENCE: Do you think that--
LOU: No. There.
AUDIENCE: [INAUDIBLE] in order to avoid the assumption of the common language, [INAUDIBLE].
MICHEL BALINSKI: You want to rescale afterwards? Is that what you're saying?
AUDIENCE: Yeah. [INAUDIBLE].
MICHEL BALINSKI: I know this has been suggested often. Of course, I think it's tampering in a way that I find-- it's very arbitrary. And I dislike the idea.
LOU: OK. So I'll remind you, the last talk in the series is tomorrow at 4:45 in this room. And let's thank the speaker again.
[APPLAUSE]
Why the "first-past-the-post" method of electing one among several candidates—the most used method of all—is seriously defective, often elects the "wrong" candidate... and what to do about it.
The second in a series of three Messenger Lectures on representing, electing and ranking by Michel Balinski, Professor Emeritus, École Polytechnique, Paris.
