SPEAKER 1: This is a production of Cornell University.
LOUIS BILLERA: My name is Louis Billera. I'm in the mathematics department. And I'm happy to welcome you to the fall term Messenger Lecture Series being given now by Michel Balinski of Ecole Polytechnique in Paris. I just want to say a few words of introduction about Michel. I'll say something about the lecture series itself tomorrow.
Michel is an American mathematician who moved to France in 1980 and has worked there since then, retiring as director of the Econometrics Laboratory at Ecole Polytechnique in 1999. He now has Emeritus status from that university. His formal academic training started with a Bachelor of Arts in Mathematics from Williams College in 1954, an MS in Economics from MIT in 1956, and a PhD in Math from Princeton in 1959.
He's held academic positions in the US at variety of institutions, including Princeton, Penn, both Wharton and the Economics Department, the City University of New York Graduate Center, the Yale School of Organization and Management, and SUNY Stony Brook, where he, in 1986, founded the Institute for Decision Sciences, which he directed for a few years.
He's held visiting research positions-- and I'll probably miss some-- in Austria, , Chile, France, and Switzerland. And he holds honorary degrees from Yale and the University of Augsburg in Germany. His awards include two for writing, the Lanchester Prize from the Operations Research Society of America and the Lester R. Ford Award from the Mathematical Association of America. In 1969-70 he was an IBM World Trade Corporation Fellow.
He's been on numerous editorial boards and the founding editor of at least one. And his research over the years has been in theory and algorithms for optimization problems. But over the past 40 years, he's had an interest in the use of mathematical methods on problems that arise in political environments.
In 1982, he published what is considered the definitive account of the history and theory of apportionment, which is going to be the topic of today's lecture. The title of that book was Fair Representation: Meeting the Ideal of One Man, One Vote. It was written with H. Peyton Young, now at the Brookings Institution, or maybe not now at the Brookings Institution. It was published originally by the Yale University Press. It was reissued by the Brookings Institution in 2001. It was translated into Japanese in 1989.
This topic, the topic of this lecture in particular, is an interesting mix of American history and mathematics. It's a fascinating topic, and I think you will enjoy it. Let's welcome Michel Balinski.
MICHEL BALINSKI: Well, thank you. Thank you very much, Lou. I feel quite honored to be invited to give these lectures and very happy, too, because it's very hard to get oneself heard when it comes to doing anything about electoral reform. But I hope to convince you that there is a lot of reform necessary, not just in the United States.
Nevertheless, I'd like to begin by saying quickly what I plan to do in these three lectures, to which I've given the title "Representing, Electing, and Ranking". I'd like to add another point quickly. I'm not going to be giving references throughout because some of you don't give a damn, I suspect. Others who are in the field do. But there is a site that Lou has arranged in the math department where I put down the references that I've used for these letters in case anybody wants to check. At any rate, the work on apportionment, of course, was done with Peyton Young.
So the first lecture will concern apportionment. And what I would like to show you is that the present method is not an equitable method, and of course, as usual, was chosen for political reasons. This is always the case in almost everything that we'll be talking about. And then what can we do about it?
The second lecture, I'll first talk about the first past the post system, if you wish, the method that is currently used both in the United States, England, many countries, of electing one among several candidates, which, of course, is the most used method throughout the world. It's so used that people cannot imagine you could do anything else. But I will show you why it is seriously defective, often elects the wrong candidate, and what we can do about that. That work has all been done with a colleague just very recently whose name is Rida Laraki.
The third lecture concerns the blatant gerrymandering that goes on in the United States. Incidentally, I am American, and my French colleagues are saying, oh boy, you better let it be known because they're going to say, here you come, and you criticize a country. But I do the same thing in France. There it's a little bit less appreciated. Why it is, in some sense today, unavoidable, and what we can do about that.
Well, first of all just a few facts. Observe. A minority of voters can, and of course have, elected a president of the United States. I don't know if you ever did this-- 50 senators represent 16% of the population. Today, if you look at the breakdown, 51 democratic Senators for 58% of the population, 49% of the Senators, 49 senators for 42% of the population. In fact, I will also argue that it is entirely possible that voters in the United States, a minority of voters in the United States, has, in fact, elected a majority of the House.
Now this is not unique to the United States. I think England, we've always talked about Tony Blair and his great victories. His last great victory he had 35.2% of the vote. That is his party. In fact, if you look at the winners across the last, what is it, one, two, three, six elections in England, always-- as you see, it's never above 45% gives a whopping majority.
In the last election, I would say Labor got 1/3 of the vote and the Conservatives got 1/3 of the vote. But look at what the figures are from the point of view of representation. Incidentally, this has impact. Tony Blair went into Iraq against the will of his nation. And yet he had this majority which enabled him to do this.
Now France is pretty bad, too. First of all, France, there are changes in the electoral law in each of these years. '80 through '85, '91, '98, '99, 2002, '03. And you cannot pretend that this was to improve matters. It was to arrange matters. Today's Assemblee Nationale, districts were drawn in 1986 on the basis of the census of 1982. The law says that after every second census there should be a reapportionment and a redistricting. There was a census-- the second census was 1999. Actually, it might have been the third already, and nothing has been changed.
So you have these kinds of discrepancies. The second district of Lozere, which is a tiny little department in the southwest, has 34,000 inhabitants. The second, the Val d'Oise, which is in the ring around Paris, 188. They both have one deputy. Now of course that's partly because of the apportionment of numbers of deputies to each department, as we have here to the states. And it's partially due to the cutting up of those districts within the department. But on the right hand here, you see, in the Department of Var, you see a discrepancy of almost 2 and 1/2, which is also pretty bad.
So this is a common problem, and I think it's a growing problem. And the analogy I would put is that the problem is we have these same systems. And you know, it's like with your locks at home. If you want to keep the thieves out, you have to keep up with the technology because they learn how to break the locks. But we're not changing the political technology. And the trouble is that the players in the political game are also the referees.
Now I'll hop back into history in order to first treat the problem of apportionment. And I would like to describe the story of the first presidential veto. Well, first of all, the Constitution. What does it say? It says, Article 1, Section 2, representatives and direct taxes shall be apportioned among the several states according to their respective numbers. The actual enumeration shall be made every subsequent term of 10 years. The number of representatives shall not exceed one for every 30,000. But each state shall have at least one.
Well, the first bill that was proposed by Congress was this. This is the result of the first apportionment. Populations are given. And what was in the air as a method, which was what is Jefferson's method, the idea was to do this. We chose a divisor, and because of the 30,000 given in Article 1, what they did was divide the population by 30,000 and round down. So the 21.019 becomes 21, 15.844 becomes 15, and so forth. And that gives us an apportionment of which turns out to be 112 seats. And that's a method.
Now one can ask what is the quota of 112 of each state? That means what is the proportional share of each state exact to three decimal points? And here you see strange things. Virginia's share is 19.431. On the other hand, the bill gave it 21. So why wasn't it 19 or 20? Well, that's a property of the method.
Now look at the red figures. You see that New York, with the right proportional share of 10.2, got 11. At the other end, Delaware, where a proportional; share of 1.7, got 1. That's a property of the method. It is a method which tends very strongly to favor big at the expense of small. That's true in general. You see it in this example.
Well, Hamilton immediately retorted-- of course, this is a time already where there's tensions between Jefferson and Hamilton-- and he proposed the following scheme. He divided the total population by 30,000 and found that and took that advisor, in fact-- well, let's leave that aside. What he did was he took the 30,000 and decided that that meant there should be 120 seats accorded to the states, and that what you should then do is compute the proportional share of everybody, of every state.
And then do what? Well, first of all you give them the integer part of their share. So for example, North Carolina has a share of 11.732. Therefore, to begin with, they get 11. Now you do this process and what happens? 111 seats get apportioned, so there's still nine to go around. And Hamilton explained-- and I'll give you his language in a moment-- that what you should do is give the leftover seats to those whose remainders are greatest. They are starred with the red. And the one that's underlined indicates the smallest remainder that got rounded up, just to help you look at it.
And again, don't look at the numbers. I mean, I think it's important to have real numbers so it's not some mathematical invention. It's the way it is. But it's the qualitative properties that should interest us. So the numbers are for illustration. But you can't get away from taking the numbers.
So that's the process. And what was the reaction of Virginians? Well, very harsh. James Madison wrote to his father, "The secret of the business is that by these different rules, the relative numbers of eastern and southern members is varied. The number 120 is made out by applying 1 for 30,000 and allowing to fractions of certain amount an additional member."
Richard Henry Lee wrote to his father, "Six eastern states have one apiece more than they ought. Jersey and Delaware the same. If the plain constitutional mode had been pursued by dividing the number of people in each state, respectively, by the agreed ratio of 30,000. But by a certain arithmetical-political sophistry, an arrangement of six to two against the south has been made."
Washington was upset. He asked for opinions. Hamilton wrote immediately, saying, "The following process has been pursued by the Act. One, the aggregate numbers of the United States are divided by 30,000, which gives the total number of representatives, or 120. Two, this number is apportioned among the several states by the following rule. As the aggregate numbers of the United States are to the total number of representatives found as above, so are the particular numbers of each state to the number of representatives of each state.
But as this second process leaves a residue of eight out of 120-- actually, it was nine-- members on apportion, these are distributed among those states which, upon that second process, have the largest fractions of remainders."
The same day, Jefferson replied to Washington, saying, "Though for taxes, there may always be found a device which will apportion among the states according to numbers exactly without leaving any remainder, yet for representatives, there can be no such common ratio or divisor which, applied to the several numbers, will divide them exactly without a remainder.
I answer, then, that taxes must be divided exactly, and representatives as nearly as the nearest ratio will admit, and that fractions must be neglected because the Constitution has left them unprovided for. The Bill does not say that it has given the representatives to the greatest fractions, though in fact it has done so. It seems to avoid establishing that into a rule lest it might not suit at another occasion.
Perhaps it may be found the next time more convenient to distribute them among the smaller states, or another time among the larger states, at other times according to any crotchet which ingenuity may invent and the combinations of the day give strength to carry. Or they may do it arbitrarily by bargains and cabals." Incidentally, when we come to gerrymandering, this is quite prophetic. "120 being once found, we must take the nearest common divisor, that divisor, which applied to every state, gives to them such numbers as added together, come nearest to 120."
Now of course, remember what Jefferson's method is. You take the number, you divide it into the population, you round down, and you add up the numbers, and you see. Now if you take a bigger number, what's going to happen? The sum of the rounded numbers will start dropping. If you take smallers, they start rising. So you can choose the divisor in such a way that you get 120.
Well, at that point, Jefferson gives an account of what happened the next day. "Washington observed that the vote for and against the bill was perfectly geographical, a northern against a southern vote. And he feared he should be thought to be taking side with the southern party. I admitted this mode of delicacy, but that it would not induce him to do wrong. He here expressed his fear that there would ere long be a separation of the union. He went home, sent for Randolph, desired him to get Mr. Madison. They came. Our minds had been made up before we drew the instrument."
Of course, Madison and Randolph were fellow Virginians. And Washington then delivered his first vetoment. "Gentlemen of the House of Representatives, I have maturely considered the act passed by the two houses, and I return it to your house wherein it originated with the following objections.
First, there is no one proportion or division which will yield the number and allotment of representatives proposed by the bill. Second, the bill has allotted to eight of the states more than one for 30,000." Jefferson recorded in his memoirs, "A few of the hottest friends of the bill expressed passion, but the majority was satisfied, and both in and out of doors, it gave pleasure to have at length an instance of the negative being exercised."
And so what was the upshot? Well, very quickly, a few days later, the both houses passed the bill, which was using Jefferson's method to allot a House of 105. The one difference between Jefferson's method and Hamilton was what you see signaled in red. Virginia got one more seat and Delaware one less. And of course, that raises this question, does one make a difference? Of course, it does.
Take the presidential election of 1876. Hayes had a minority of the vote, 300,000 less votes than did Samuel Tilden of New York. And yet he lost by one in the electoral college. Had the correct apportionment been made, that result would have changed by one and it would have been the other way around.
The method today being used was decided upon in 1940 when there was a safe Democratic House, Senate, and president. And the one change between the two methods which were then in the law, saying that it had to be one of the two, in 1930 it had given the same result, so there was no problem. In 1940 there was one difference. One more seat would be given by the method that was adopted, which is Hill's, to Arkansas. Arkansas is a safe Democratic vote. And so that was what was decided.
So what is the problem? Who is right? Both of those methods sound reasonable. Or maybe was is there another method? And obviously, the question is why.
Well, I'll come back now and go back into antiquity. Aristotle-- it's a very tautological discussion that he gives. But he says, "This, then, is what is just. It is the proportion. The unjust is what violates proportion. The justice which distributes common possessions is always in accordance with the kind of proportion mentioned above. And the injustice opposed to this kind of justice is that which violates proportion."
And he goes on in this light. "The just is a species of the proportionate, for proportion is equity of ratios and involves four terms at least." And I won't go through the rest. It's just sort of a repetition of this. And yet I put it to you that when we, in our daily lives, want to divide something up above people, we always are starting to look for proportion.
And I ask, why is it that is a well-nigh universal belief that when something is to be shared, what is fair is what is proportioned. Why? Now is it merely what Blaise Pascal asserts? "Custom makes equity for the sole reason that it is received. It is the mysterious foundation of its authority."
And my answer is no. There is a much more fundamental principle that is lurking in the background, and that is the main concept I'd like to get across today. It is coherence. It's a very simple notion. It says every part of a fair division must be fair. So the idea is if, say, some five of us here are brothers and sisters, and there is an inheritance, and we use a rule.
And by that rule, each of us gets a certain share. Then three of us could get together, repool that, and say, how should we be fair among the three? And I say, well, it should be the same solution unless there happens to be two solutions, let's say, more than one. Every part of a fair division should be fair.
Now it's clear that this is true for proportionality. That is, I can take any number-- 100 people, 100 objects-- and the vector, the list of numbers A1 to AN is proportional to B1 to BN means simply that one of those lists is a multiple of the other, a constant multiple of the other. That is, you preserve the proportion among every pair.
Now I think that is the reason. The question, then, is are there other ways that would satisfy this same principle? Well, we'll now look at the Talmud. Now I'll present you to two teachings of the Talmud concerning property rights.
Now the first lesson of the Babylonian Talmud is the principle of the contested garment. Two hold the garment. One of them says, I found it. And the other, Lou Billera, says, I found it. One of them says, it is all mine. That was me. And the other says, it is all mine. That was Lou. Then the one shall swear that his share in it is not less than half, and the other shall swear that his share is not less than half, and it shall then be divided between them.
Second aspect. But if one says it is all mine-- that was Lou-- and the other says, half of it is mine-- that was me because I'm more reasonable-- he who says it is all mine shall swear that his share in it is not less than 3/4. And he who says half of it is mine shall swear that his share in it is not less than 1.4. The former then receives 3/4 and the latter receives 1/4.
So what does this say? Well, we have two claims, A and B. One claims one, the other claims 1/2. So the total claim is 1 and 1/2. But what's available, there's an estate of one, the coat, the garment. Now the proportional rule would do what? 2/3 for one, 1/3 for the other, in proportion to the claims. But the contested garment rule says 3/4, 1/4.
Now what's the general idea? Well, there are two ways of explaining this. And please keep in mind that always, no one is allowed to claim more than there is. In all that we're going to be discussing here, you can claim a million, but if there's one available, it's as though you've claimed one.
So the first explanation is this. I conceded 1/2 the coat to Lou. And now he concedes nothing to me, but 1/2 of it goes to him. There's 1/2 left. What will we do? We split it. I prefer to look at it in the following, though, which is equivalent, that it is a solution in which we each suffer the same loss because he claimed all and he got 3/4, so he lost 1/4. I claimed 1/2, I got 1/4. So I lost 1/4. That's the notion of equity here.
So that's the principle of the contested garment. Now let's look at this in general. I'm going to take two claims. One is for 16 and one is for 12. Maybe I'll use this. And this line shows how the total estate is to be divided. When we start here, there is a state of 0, so each person gets 0.
Now as we start moving up, and let's say there's an estate of 2. Then what is that? The estate is 2. One claims 16, one claims 12, so they're both claiming how much? 2, right? So what do they do? Nobody's giving up anything to anybody. They each get 1/2.
So we do that. We start climbing up until we get to this point. Now at this, 6 goes to one, 6 goes to the other. We're at 12. As soon as we go above 12, what happens? Well, one of them, the second, is claiming less. So when we get up to 13, the one who is claiming 12 concedes one to the other, right? So what's left? 12. They're split.
So what happens? As we keep increasing, we'll start going on like this. The extra amount is always going to the first player, the one who claims the 16, until we get to where the total amount that has been allocated is 16.
Now we're up again against the 16 and 12. Soon as we go beyond it, what happens? Nobody's claiming the entire amount. Both are claiming half of the incremental amount. And so the solution looks like this.
Now notice if I take this curve, and I spin it around this point-- that is, I'm going to spin the whole diagram around-- it's perfectly symmetric. So what is that telling us? That's telling us that the rule for allocating the gains that you get is the same as the rule for allocating the losses, because if I was thinking of coming up here and coming down, it is how do I allocate losses to each of them? And then I see what the gain is because I know what they were claiming. And if I know what their loss is, I take the difference, and that is what their gain is.
So there's a perfect symmetry around this point right here. Therefore, we can write what the formula is in a very simple way. The formula that describes this curve up to here is simply the following, that each share, the share of A and the share of B, is going to be the minimum of a number lambda and half of the claim, the minimum of lambda and half of the claim, where the lambda is chosen so that we ascribe the amount we want to allocate.
So for example, if I want to allocate 10, if the total amount I'm going to allocate is 10, what do I do, I will choose the lambda to be 5 because then the minimum of 5 and 8 is 5, of 5 and 6 is 5, the sum is 5. Soon as I raise that lambda a little bit, I get off 10-- or lower it, I get off 10. Or if we take the point where the H is 13, which would be right here, then I would take a lambda equal to 7 because then it would be the minimum of 7 and 8, which is 7, and the minimum of 6 and 7, which is 6. And that gives me the 13.
So I can give a formula for that. And as soon as I have that part, the other part we have by symmetry. We just take those as the losses. So since, as I say, the contested garment rule allocates losses exactly as it does awards, then what do we do for H's that are above 1/2 the total, 1/2 the sum of the claims? Then we calculate the losses for 28 minus what we want to ascribe and subtract them from the claims.
So OK. I've already said if it's 10. But if H is 18, which is 28 minus 10, then here's what I was just saying. We would take 5 away from 16 and 5 from 12. That gives us 11 and 7, or 18.
OK. That's the contested garment. But now there's another principle in the Talmud, and this has to do with family rights. And here's that rule. If a man who was married to three wives died and the ketubah-- that is, the estate-- was 100-- excuse me. And the ketubah was [INAUDIBLE]. That is, the respective claims of the three women was 100, the other 200, and at the third 300, and the estate was worth 100, then the three wives are to divide equally.
If the estate was worth 200, then the claimant of 100 would get 50 and the two others would get 75. Finally, this [INAUDIBLE], if the estate was worth 300, then the claimant of the 100 would get 50, the claimant of 200 would get 100, and the claimant of 300 would get 150. So it's a proportional solution. And I put in red to emphasize it, similarly, if three persons contributed to a joint fund and they had made a loss or profit, they share in the same manner.
So here's the problem, put into our notation, so to speak. We have three claimants. The claims are 100, 200, 300. So the total claim is 600. In one case, the man dies, leaves 100. That's case one, equal division. Case three, when the estate is worth 300, we have proportional diversion. But now there is this puzzling case two, when there's 200. It's 50, 75, 75.
Now this has led to a debate which has lasted 20 centuries and was only resolved recently via a very sophisticated of game theory by Bob Aumann and Michael Maschler. But it has a very, very simple answer as to why and where this ruling comes from, and that is coherence. Specifically, coherence with the rule of the contested garment. That is, this solution, as I will show you in a second, every two women share what they have, are given by these solutions, according to the principle of the contested garment.
So let's check that. Case 1. Each pair-- well, look at the first case. Look at A and B, for example. How much do they get together? 66 and 2/3. That is, both of them, therefore, claiming all of it. So what does the contested garment say? Split. That's true for every pair here, obviously.
Let's look at case two. Women A and N are accorded 125 together. But A claims of 125, 100. So she concedes 25 to the other. B, however, claims all. So 25 goes to the other. What's left? 100. The rest is split. So we get the solution 50-75.
Now the same analysis is exactly true for A, C, since 300 is more than 125. Now what about B,C? Well, the total claim is 150. Therefore, what? They both claim all. Therefore contested garment says, what do we do? Split.
Finally, the third case. A and B receive 150 together. A concedes 50. B claims all because she's claiming 200 and there's 150. Therefore, 50 goes to B, and then the rest is split. I can go on to each of the cases. Each one of those cases is going to give you exactly the same thing.
So we have a general rule. What is it? Well, here's the problem. 10, 100, 200, 300. And let's look at all possible estates. Well, what would the proportional rule do? The proportional rule would say I'm going to assign what is left proportionately. That is, I'm going to multiply the sums 100, 200, and 300 by the same factor and choose that factor so that the total sum that is to be allocated is allocated.
But the ketubah rule does the following thing. Well, we know what it does. It's the contested garment. So we can use the formula we had before. And what it does is simply you take the minimum of lambda and 1/2 the claim of the first, lambda and 1/2 the claim of the second, lambda and 1/2 claim of the third. And you choose the lambda so that it adds up right.
And of course, that's for less than 1/2 the total claim, less than 600 in this case. Excuse me, so the half is less than or equal to 300. And the other we get by the symmetry we already observed. So if H is equal to 100, you take lambda equal to 33 and 1/3, and that does the trick. If H is 200, we take lambda equal to 75. And if H is 300, we take lambda equal to 150.
And for the reverse, as we said, we subtract them, and so we get this solution. So for example, if H is 500, then what do we do? Well, we know the solution for 100. So we're going to subtract 33 and 1/3 from each of their claims to get that solution, and so on for the other possibilities.
So here we have a complete rule. And what do we have here? Again, for a rule to be fair, every part of the division must also be fair. It must be coherent with itself. Here we have a rule that is coherent with the contested garment. Now notice that the important part about this notion of being coherent with itself is that it must therefore be coherent with every pair.
Well, that means that all we need to do is have a rule for deciding how do we split between two? And as soon as we have it between two, we have it for everybody because then we have it by this extension.
Now notice also that how-- and this is important for the kidney problem that I hope I will get to-- that how this establishes, because you're making comparisons between two, what the priorities are in a pair whenever there's an extra amount. You remember we had the curve up there. And as we increased the amount to be shared, no one ever lost.
So there was a claim for who would get the extra amount that is available. And there is a priority that was set up. It could be equal priority, or we saw that at one point, all the priority went in one direction. And obviously it is much simpler to think about and devise a rule of division between two claimants than among many.
Now I'll tell you right away, there are infinite numbers of coherent rules. On the other hand, not all rules that are used, including the kidney rule, if we get to it, are coherent. And now we'll come back to our problem. Now, in some sense, we know what we should do. The problem is how should two states share any numbers of seats? If we know that, then we can easily generalize the thing. And I say that is obvious.
What do you do? You use, essentially, Hamilton's idea. You compute the quota. That is, you compute the proportional share. And you round to the nearest integer. So if you have-- here I've taken populations of 1,900 because it's a particularly nice example.
But together, if New York and Iowa were to receive 49 seats together, then you would compute their share of 49 exact-- that is the quota-- and you run to the nearest integer. And you will always get a solution to this unless, obviously, each remainder is exactly 1/2. And then you spin a coin. That's a tie. There's nothing you can do about this.
There's one problem, one little exception, and that is the Constitution says each state must have at least one. Therefore-- here's an example-- you compute the codas, and you see that rounding Wyoming to the nearest integer would give Wyoming 0. That's not allowed. So you have to give it 1. So what's a rule that will give us this? Well, notice it's exactly the same spirit as before except that instead of a min, we have a max.
What we'll do is, the rule is we want to choose a lambda so that the share of New York and the share of Iowa comes to be 49, where the square brackets mean that we multiply the population, or divide the population-- comes to the same thing, but here I've written it multiply by lambda. And we round to the nearest integer. Now you can do this simply by computing what their exact quotas are, and so we get this solution.
So in general, what do we do? Well, same thing. That is, choose a lambda so that when you multiply-- now we could be talking about dividing, obviously, right-- by the population of each one, by this number, then round it. And then if that number is less than 1/2, then we say it's got to be 1 because we've got to assure everybody 1. Then we want to choose the lambda so that everything adds up to exactly the number of seats we want to allocate. Today that would be 435.
And this is Webster's method, and here's the way he described it. "To apportion is to distribute by right measure, to set off in just parts, to assign in due and proper proportion. The apportionment of representative power can never be precise and perfect. That which cannot be done perfectly must be done in a manner as near perfection as can be. Let the rule be that the population of each state be divided by a common divisor." I wrote multiply. Excuse me. I should put divided.
"And in addition to the number of members resulting from such a division, a member shall be allowed to each state whose fraction exceeds a moiety of the divisor. So that's exactly round to the nearest integer.
Now Hamilton's method, which was used very much in US history, is incoherent, and it caused its downfall. Its incoherence caused its downfall. And I'll take the example again of 1900 because that's when it happened. Hamilton, according to the census of 1900, gave the following solution.
I've underlined Virginia here. That was the lowest remainder which got rounded up. Remember, that's the way Hamilton's method works. You compute the quotas, you give the integer part, and then you give extra seats according to the size of the remainder. So this was the apportionment. And you'll notice, of course, Maine's remainder and Virginia's remainder are very, very close. It's a very close thing here.
But now if I look at how New York and Maine should split 41 seats because they got 41, I compute their quotas of 41, and what happens? Well, Maine should get 4 and New York should get 37, according to Hamilton's method. So Hamilton is not coherent. It's an incoherent method because of that, so that this notion that every part of a fair division should be fair is violated.
Where? Fairness here would be saying my method is fair for everybody. Therefore it's got to be fair for any subset. Same method. And here I see no, it doesn't agree with itself. Now this has lots of effects. And the famous Alabama Paradox-- this is called the Alabama Paradox because it was first noticed with the method of Hamilton several years earlier, in 1860, I believe.
But according to the census of 1900, it's particularly interesting because what happens is that as the House size grows, starting at 350 and going up to 400, Maine first gets three seats. One, the House size is between 50 and 82. Then it gets 4, then it goes down to 3 again. Then it goes up to 4, then to 3, and finally gets up to 4 again.
So it's hopping up and down. And this caused a representative from Maine, Representative Middlefield to say, "Not only is Maine subjected to the assaults of the chairman of this committee, who had chosen H equals 386-- that was the number the chairman chose-- but it does seem as though mathematics and science had combined to make a shuttlecock and battledore of the state of Maine in connection with the scientific basis upon which this bill was presented. In Maine comes and out Maine goes. God help the state of Maine when mathematics reach for her and undertake to strike her down."
Coherent rules guarantee that when the size of the House increases, no state can lose seats. We saw that in the curve with the principle of the contested garment, and this happens here as well. In fact, all coherent methods must look like this. That is, they look exactly like Webster's method, except that where you round in an interval can be anything.
That is, we could have Lou's method, and Lou would simply take a random number generator and say, well, between 0 and 1, you round at .7. Between 1 and 2, you round at if the remainder is above 0.3. And any sequence, and that gives you a divisor. Obviously, that's not a very reasonable thing to do. But that's the characterization in terms of the mathematics of the affair.
The method that is presently used is not the method of Webster. It is the method of Hill, which has been very cleverly rebaptized by mathematicians as the method of equal proportions. It is the law of the land the United States since 1940, as I mentioned before, and its thresholds, where the rounding takes place, is at the square root of the product of the two ends of the interval. Now as you get the numbers become bigger and bigger, the square root of that product gets closer and closer to 1/2, but it's always below. And it's a lot below for the small states.
And in fact-- and now I'll get-- because Cornell has a story here-- there were five divisor methods that were considered in the US debate over which method should be used, which was mainly waged in the 20s and 30s. And I'm listing them in order going from the one which most favors the small to the one which most favors the big.
The first is John Quincy Adams' method. And I point out, he was terribly concerned-- this was in the 1830s-- that New England, Massachusetts was losing ground. And he was, as you see, "I was was all night"-- he writes this in his diary-- "meditating in search of some device, if it were possible, to avert the heavy blow of the loss of representatives from the state of Massachusetts and from New England."
And what he says is, just always round up. So any number you get, say 10.0001, it gets up to 11.
Another proposal was James Dean. He was a professor at Dartmouth and sent his proposal to Daniel Webster. And he said, you should round at the harmonic mean. The harmonic mean is the-- I hope I got this right-- the product of the endpoints divided by the average.
Then there is Hills' method, which is round at the geometric mean, Daniel Webster's, which we say, which is round at the arithmetic mean, and of course, Jefferson's, which we saw, which was round down. It was very much used, but at a certain point it was observed that it was really favoring the large so much that a representative exclaimed that it was committing a classic rape on a cloud of statistics right in the face of the House.
And in this regard, Lou, you mentioned there was a Japanese translation. When I learned about this Japanese translation, I received a letter from a lawyer who was doing this. And was a sequence of questions that was addressed to me. And he asked me to explain exactly, exactly, what is a classic rape? I did not know the gentleman. Therefore I had to answer this in as serious a way as I possibly could.
At any rate, here is, again, the census of 1900 and what these various methods give as solutions. And you see that they give very different solutions. Now for example, there New York's share was 37.6. Jefferson says 39, which seems to be too much. Adams is 36, which is too little.
And as you march from left to right, basically-- and that's what these arrows are meant to indicate-- you're giving seats that are going where? From states which are lower in population to states that are higher in population. So you get a very definite progression from left to right here.
The numbers themselves are not the important thing. The important thing here is the qualitative sense of what's going on. Hamilton I just put for interest. But it no longer enters the picture because of the famous Alabama Paradox and Maine coming in and out.
Now the history comes to this big fight between Wilcox and Huntington. In direct violation of the Constitution, there was no reapportionment in 1920. This was the effect of the war. And there was a lot of confusion. There's a very great upset of the smaller states, especially the smaller western states, to lose seats. And as Emanuel Selover of New York explained, "The issue and struggle underlying reapportionment is between large states with large cities on one side and the rural and agricultural states on the other side. That thread of controversy runs through all the political struggles of evidence in this House. The issue grows more and more menacing."
Walter Wilcox was president, at different times, of the American Economic Association, the American Statistical Association, and the American Sociological Association. I don't know what memories you have here in the lure of Cornell. But we found that-- this was with Peyton-- that he made this statement at a certain point because apparently he was a great walker. "Unfortunately," he said, "there is some danger that I will be remembered more for my feet than from my head."
But he championed the method of Webster from the beginning in 1900 and right to his final article on this in 1952. And his main reason was summarized in 1915 in his presidential address to the American Economic Association. He said, "The use of Hill's method has recently been advocated. To use it would result in defeating the main object of the Constitution, which is to hold the scales even between the small and large states, for the use of it inevitably favors the small state."
Edward V. Huntington of Harvard-- so this was Cornell versus Harvard-- was at different times president of the American Association of America, Vice President the American Mathematical Society and of the American Association of the Advancement of Science. And he was reputed to be a very charming and witty person, and a very adept expert witness.
He led the mathematicians in support of Hill's method. He said the following. "Statistical experts who have examined Hill's method have pronounced it the only scientific method. The method of Webster has a distinct bias in favor of the larger states, while the method of Dean has a similar bias in favor of the smaller states. Between these two methods stands the method of Hill. Which has been shown to have no bias in favor of either the larger or the smaller states."
There was a sequence of, I don't know, eight, nine, 10 articles on both sides appearing in Science. There were hearings in Congress and so on between these two. And Wilcox studied the numbers, and he prepared tables painstakingly. He was not a good mathematician. And he made a lot of errors in mathematical statements, which the mathematicians just killed him on. But he showed-- he showed totally convincingly that, if you look at the numbers, it was clear that his statement was correct.
On the other hand, Huntington studied the mathematics. [INAUDIBLE] there are five methods. They can be ordered from most favoring small to most favoring large. There are five. One's in the middle. It's Hill. Therefore it was the only unbiased method. Now the two gave identical results, as I said, in '30, so there was no choice. And then, of course, as I've already mentioned, the choice was made in 1940 on political grounds.
But Congress was confused, so they asked the National Academy to give its opinion. Four prominent mathematicians, George Bliss, Ernest Brown, Luther Eisenhart, and Raymond Pearl--I think still names that are well-known-- reported in favor of Hill's method because it occupies mathematically a neutral position with respect to emphasis on larger and smaller states.
The same advice was given by von Neumann, Morse, and Eisenhart in 1948 because it stands in the middle position as compared with the other methods. What would they have said had there been an even number? Of course the answer depends on what bias means. Wilcox had the right idea. The mathematicians neglected to define the concept.
There are various definitions. All reasonable definitions lead to the same conclusion. But the basic sense is that you want to look at this, well, what happens over lots and lots of problems? Now of course, here you're stuck with weighted distributions, probability distributions, and all the rest of it. But basically the notion is a method is unbiased if, over many problems, the average of their exact fair share equals the average of what they got, that somehow these things average out and they come down to the same theorem. Webster's is the unique one. Now this can be proved in various ways.
Now let's look at what does this say in practice about what actually has happened? Now to measure the bias of a particular apportionment, you can do this in lots of different ways. Here's one. Divide the states into thirds, the large, the middle, and the small. Well, you can't do it exactly into thirds, so 16, 18, 16. The measure of bias will be the percentage by which the small states' representation per capita differs from that of the large states.
There have been 22 apportionments in US history. Here you see the number of times. The small was favored by Adams every time, by Jefferson never, Dean 17 times, Hill 15 times, Webster 10. What was the average bias in favor of the small over all of history? Now taking each of the years and then averaging, you get 18.7% in favor of the small for Adams. Then goes to Dean, 5.3, Hill 3.4, Webster 0.6%. And of course, Jefferson is in the other way, 16.2 in favor of the large.
And then you can also do-- this is a theoretical computation based on the apportionment that we actually have today. That is, take Hill's method, which is the law of the land, take that apportionment, and now ask the question. The essential idea is to say I'm going to look at all the populations that would give this solution by Webster's method, by Hills method, etc. And I'm going to compute, over all populations, how often the small would be favored.
And to give that solution, the computation gives you, of course, theoretically, Webster is perfectly fair, Jefferson in favor of the large 19.5, Adams in favor of the small, 24%.
And so I conclude Walter Wilcox's view was true. Webster's is the method to you. Of course, it isn't used.
How am I on time? Five minutes?
OK. Well, then I'll conclude on this and quickly give you a couple of things about the kidneys because it's of interest. And it's of importance.
Senator Arthur Vandenberg was a great proponent of permanent legislation. That is, going back to what Jefferson was saying, and people can invent whatever method they want every 10 years. And no, we need permanent legislation, which is of course what we have since 1940. But he said, "To identify any one method in this permanent act would be to assume that science itself has traversed the subject with finality. Science is not thus static. The last word by no means has been spoken. A permanent ministerial apportionment act should be susceptible of accommodation to the progressive state of knowledge." I would say it has not been.
OK. A quick five minutes worth, which is about all it takes, on allocating kidneys. I don't know if you know there is an organization called the United Network of Organ Sharing, which was established by US Congress in 1984 to manage the distribution of organs as they become available. And I looked, as I started to prepare these lectures on August 24, at its website. And I found that there are 103,000 patients all waiting for organs, of which almost 70,000 are waiting for a kidney.
Now I'm going to say all of this in a very succinct fashion because I'm obviously not an expert in these things. But I'm going to give you the essence of what is going on. The success of a kidney graft, or any of these grafts, depends on factors which are related to the donor and the recipient, the blood types, the type of kidney, the size, and all those other parameters that have to do with the human body.
The UNOS established a rule for each type of available kidney that assigns points to waiting patients. Its goal is to ensure the success of the graft, on the one hand. Yet it wants to avoid to penalize patients with rare characteristics who therefore would be always left behind. So somehow they want them to be rising faster in the list so as not to have to wait too long. And they also want to respond to patients who are in the greatest need.
So they have a system--and again, I'm giving you a very short notion, but qualitatively it's exactly this-- in which two points are given for each of six possible matches of antigens, one point for each 10% of the population whose characteristics are incompatible with his, so the measure that's going to be difficult to fit. And then there are a bunch of other conditions. And finally, there's a bonus of 10 points minus 10 times the fraction of patients who have waited longer.
So notice that all of the first list-- and the dot dot dot is supposed to somehow indicate this-- always has something to do with the individual patients. The last has something to do with them all. And then the priorities are determined by the total points of a patient, his score, all points that are independent of time of waiting, plus his bonus.
OK. Let's take four patients, A, B, C, and D. And I'm assuming they are listed in order from the one who has waited the most-- A had waited the most-- to the one who has waited the least, D. And suppose their scores are as I've given them, 16, 21, 20, 23. And therefore the bonuses, by the rule I just gave, is 10. No one's waited longer than a A, and now on down for the other bonuses.
So then I make the sum, I get these totals, and that says what's the priority order? Well, B has the highest total. Then comes A. Yes? Then comes D, then comes C. OK. A kidney comes along. B gets it. So we're left with these three. But now the bonuses have changed. First the bonus of A is 10. But C's bonus, somebody is off the end of the queue. So his bonus goes up. And D's bonus goes up.
And now what happens? Here are the scores. The priority order is exactly the opposite. So at the beginning, B was first, C was last. Once B drops out, C becomes first. Something screwy about this. Why? It's not coherent. And it's not coherent because of this dependence on the entire list.
So this is incoherent, therefore, I submit, inequitable. And yet, of course, it is quite a simple matter to define a coherent rule. For example, you could give points for every week or every month. I wouldn't know what the correct period is. But medically speaking, perhaps you could put in the correct period. You give points for months that are individually assigned to the person. Then you've got a coherent rule.
And so that's the lesson I would like to leave, if we can call it a lesson, that what is fair-- well, an essential property of what is fair is that every part of a fair division should be fair. And we'll see that this theme-- and in certain sense, if you look at it a little bit larger-- applies to the problems that I'll be talking about tomorrow and the next day. Thank you.
Have I talked with the UNOS people? No. No. I gather their rule was changed and improved and yet still had some of this element in it. It's a very funny thing in all of these. As we'll see tomorrow also, in many applications of ranking-- I'll talk about figure skating, I'll talk about politics. Maybe I won't talk about all these things, but I've looked into them with my colleague, Rida Laraki. We looked at piano competitions. We've looked at all of these things. Everybody invents his own rule, and so in some sense, there's no expert knowledge that has been built up. They're doing their own thing, thinking that this is an innocent problem. But it's not innocent at all, and they get into big problems.
For example, in figure skating, there was that big scandal of 2002, I think it was in the west or in Canada, I can't forget. But it was between a Russian pair and the Canadian pair. And a French woman was accused of cheating in her giving scores, which she first admitted, then retracted. Anyhow, it was all a very peculiar affair having to do with the method that was used.
And as a result, there's been a total change of method, not a very good one chosen this time round either. But that's another question.
AUDIENCE: I'm curious, when you described this Talmudic method [INAUDIBLE]
MICHEL BALINSKI: A little louder, please.
AUDIENCE: In this Talmudic method that you described [INAUDIBLE], it struck me as something of a leap of faith to impute this type of logic to the people who formulated it. And it seemed that you measured two data points. And as you said, I believe [INAUDIBLE], I understood correctly, that there's a whole class of methods consistent with this, depending on how you resolve-- where within the [INAUDIBLE].
MICHEL BALINSKI: Well, in one sense you're right. But look, the way this was discovered was-- there's a very nice paper that was written by Barry O'Neill-- oh, I don't know, about 10, 15 years ago-- where he discussed many of the division-- let's call it fair-- but anyhow, division rules that were to be found in the Talmud. There are lots of them.
And this particular one, no one had any much notion as to what to do. And what actually happened is that Aumann and Maschler started looking at the problem. And they started using solution concepts in game theory to see whether could they formulate it as a game? Could you-- is there one of the solutions, the Shapley value, the this, the that, the other. And they found that what is called the nucleolus which is a fairly sophisticated concept, gave exactly those solutions.
Now of course, I don't think they knew about the nucleolus 20 centuries ago. It's a concept that is very sophisticated. On the other hand, yes, there may be a bit of a leap of faith. But I don't know. I guess I'm willing to take it.
AUDIENCE: No, but my question was more or less answered. Are there other data points from decisions of that era?
MICHEL BALINSKI: No. There are no other data points for this particular ruling as far as I'm aware. I am not a Talmudic scholar. So I don't want to pretend to answer what I don't know. But not that I know of.
But it was-- these great rabbis, Maimonides, he discussed this rule in the whatever he lived, in 14th century or something, the 12th, I don't know exactly. But there has been a lot of discussion over the years about this, trying to understand where does this come from? And this gives a lovely explanation.
And of course the purposes of here, it makes a point. And it makes a point, is this notion of how you divide between two. And you sort of then lift this up to [INAUDIBLE]. Now you can't do this with any rule, obviously. There's certain continuity properties and things of this sort. But roughly speaking, this is a reasonable idea and a reasonable way of describing the thing.
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Why the current method of apportioning United States Representatives to the respective States is not equitable... and what to do about it.
The first in a series of three Messenger Lectures on representing, electing and ranking by Michel Balinski, Professor Emeritus, École Polytechnique, Paris.