 Hello again. When we last left off, we were talking about Alexander Hamilton's method for apportioning representatives to our U.S. House of Representatives when the country was first founded. And in this video, we're going to look at three other guys you probably have heard of and methods that they suggested we use instead. All three of these methods are what we call adjusted divisor methods. It means we do not use the standard divisor that Alexander Hamilton proposed, but we start from that point and adjust as needed. So who were these men? Thomas Jefferson, John Quincy Adams, and Daniel Webster were the three statesmen who proposed alternative methods to Hamilton's. And it turns out that every single one of these methods depends upon the use of a modified divisor. So how do we go about using them? They're very similar, so I'm going to tell you about all three at once, and then we'll do all three with our data from the 1820 census. So we first determine the standard quota, just like you would have if you were using Hamilton's method. Then how we choose to round is what makes these three methods differently. Remember, with Hamilton's method, we always rounded down no matter what to the lower quota. With Jefferson's method, we do round down, but not less than one. We can't ever go less than one. With Adams method, we're going to round up. And with Webster's method, we're going to round according to the normal rounding rules. If you recall that, if you have a 0.5 or greater decimal, you go up to the next integer. If it's 0.4999, however many nines you want to put on there, or less, it goes down. That's the normal rounding rules. And so at this point, you may be wondering, well, how is Jefferson's method any different than Alexander Hamilton's method? Well, the key with Jefferson's method is that when you round down, it has to add up to the perfect number of items that you want to apportion. In our case of the 1820 census, that was 213. So we're going to go back to that data and see how this would work. We're going to use the resulting sum of our modified quotas. That's what we're going to call after we divide. Instead of dividing by the standard divisor, we're going to modify it, either up or down, and divide each of the populations by this new number so that the modified quotas add up exactly to the number that we desire. So this means we're going to be doing some guesswork. It may take a while. In practice, you can pick a smaller modified divisor if the sum is too small, and you can pick a larger modified divisor if the sum is too large when you choose your next one. So let's go back to our Excel spreadsheet. That's what all these other columns over here to the right dealt with. So first of all, we're going to deal with Jefferson's apportionment method and talk about it. Our standard divisor was 42062.5. If I were to plug that number back in, 42062.5, and then if I were to drag that number all the way down to fill in here, and this is what we get when we divide it, the same thing as Hamilton's apportionment method, and round it down, we remember it adds up to 200. That was, with Hamilton's method, his solution was, let's get rid of the extras by deciding who has the larger decimal pieces. Jefferson said, no, let's just divide by something else. What is that something else? Well, I don't know. We'll just keep dividing until we find what that something else is so that it adds up perfectly to 213 at this point. Because this column right here is each of these decimal numbers rounded down. Currently it adds up to 200. So what we're going to do is go change the number that we're going to divide by. So, well, which direction do I need to go? Well, this sum is too small, which means I need to divide by a smaller number so that these numbers would get larger. So we're going to go here and how small? Well, it's through trial and error that you kind of figure out what you might need to do. I'm going to skip about 2,000 and go down to a nice number of 40,000 at this point. And then we're going to allow Excel to propagate that number all the way down. There you go. And as it does this, again, the beauty of Excel, here's what we get as the modified quotas. I don't call them standard quotas anymore. They've been modified by the use of a modified divisor. And here's what happens when they round all down and we see that it adds up to 212, which is really close but still isn't what we wanted. If you recall, there were 213 members to be a portion at this point. So since 40,000 didn't work and 212 is still too small, we need to go smaller with the modified divisor. So perhaps we choose 39,000 because we're really close. 39,000, we propagate that all the way down and we find out 219, which is too big. And so now you get the idea here. We just have to narrow down until we find a number that works. Now the good news is that there's not just one number and you're trying to find the magic number that will work. There's a range of numbers that will work. It depends on the particular problem what that range is. Since we know that 39,000 is too big and 40,000 gave us a value that was too small, we need something between them, preferably closer to 40,000 than to 39,000. So maybe we try 39,700. Let's see what happens there. Drag that down and we get 213. So under Jefferson's method, in this particular case, we could have used a modified divisor of 39,700 and it would have worked. You could verify this yourself. You could also use something like 39,680. So there's a range of values that will actually work. What I want to point out at this time is that there are a few differences between Hamilton's apportionment method and Jefferson's apportionment method in this case. You'll notice up at Massachusetts, Jefferson's method awards Massachusetts 13 representatives, whereas Hamilton's method only gives them 12. Jefferson's method only gives Delaware one representative, but Hamilton's method gives Delaware two. There's several other places in there that differs. So you may not think this really matters, but when you think about bills that are passed in Congress, states want their power in Congress and the issue of one representative can really swing a lot of votes. This is actually a big deal. Let's talk about Adam's apportionment method. It works the same as Jefferson's except we're going to round up no matter the divisor. Remember when we used 42,062? It ended up being these numbers. Let's plug in 42,062 there again and drag that down. These are the modified quotas we get in this case, but now I've told Excel to round them up. Notice we get 224. 224 is way too big. We're supposed to get 213. Well, since this is too big, I need these numbers to be smaller, and the way that I get these numbers to be smaller is to divide by something that's bigger. So instead of 42,000, we go the opposite direction than we did with Jefferson's method. We go larger. Maybe we try 44,000. We'll drag that down and have Excel do all the work for us. I'm sure John Quincy Adams would have led to it, had Excel at this point, but he didn't. We get 215, which is still a little bit too big. So we're going to try a little bit larger. 44,000, maybe 300. We'll see what that does. Getting closer to 14. 44,600. Oh, not good enough either, right? So you can see this time takes. How long does it take? It depends on how good you are at guessing. You can actually create computer programs to figure this out for you much quicker than we're doing right now, but let's go with 44,900. And we're off the other side again. Okay, so we're going to try 44,760 and see what happens here. Perfect, 213. So these are the representatives that Adams' method would propose that each of the states have. Again, it's different than Jefferson's and it's different from Hamilton's. They all have their differences. The last one, Webster's method, if we go up here to the modified divisor, let's just try our original standard divisor. If we click over here and put in 42062, propagate that down, I've instructed Excel to round according to normal rounding methods. It's perfect the first time, 213. So with Webster's apportionment method, sometimes the modified divisor is larger than the standard divisor, sometimes it's smaller and sometimes it can work as the exact same thing. With Jefferson's apportionment method, the modified divisor is always smaller, and with Adams' apportionment method, the modified divisor is always larger. But we can find a number that will make all these methods work and they all give out some slightly different variety of who gets what. Maybe as you're looking at this data, you're wondering which method favors which types of states. We could compare, say, if we scroll back over so that we can actually see the states here for a second. This really large state here in row 11 is Pennsylvania, I believe. Let's look. Jefferson's method gives Pennsylvania 26. Adams' method gives it 24. Let's find another large state, Virginia. Jefferson's method gives it 22. Adams gives it 21. Jefferson's method tends to favor the larger states. If you know any of your history, Jefferson was from the state of Virginia, which was one of the original largest states. Adams' method tends to favor the smaller states. So let's talk about when these methods were used and see if we can figure out which method is best if we can answer that question. Actually, before we do that, I was going to show you one more method of Jefferson's. Just to show you one more example of how to do that. If we go back to our state university problem from the first video, where we had a state university wanting to disperse their scholarships among the different schools, according to their population, we had a standard divisor of 456. I've just gone ahead before this video and played around with a few numbers and found 445 as a modified divisor that would work with Jefferson's method. We would divide each of the enrollments by 445, get these numbers, and round every single one of them down, 28, 13, 16, 36, and 7, no matter what the decimal is. Even this one, 36.99, it goes down to 36 if we're using Jefferson's method. And those are the seats that Jefferson are apportioned. You can check this out again. It adds up exactly to 100. So now let's look at the history. We've already looked at the fact that Jefferson's method seems to favor the largest states, and it was used in our country's history up and through the census after 1830. Then Adam's method was offered as a counter to Jefferson's method at some point in the early 19th century, but it was never actually formally adopted for use by Congress. Webster's method was used at two interesting times. It was used in 1840 census. It was left alone for a while. It was brought back at the beginning of the 20th century. A lot of political scientists and mathematicians feel that Webster's method is the most unbiased. The quota rule. Most political scientists and mathematicians believe that when you're doing apportionment, each group should receive either its lower quota or its upper quota. It doesn't make sense for a group to get more than one of those two things. The beauty of Hamilton's method is that it always satisfies the quota rule. There will never be a state that would get more than its upper quota or less than its lower quota. Unfortunately, that's not the same with the other methods. In fact, Jefferson's method actually violated the quota rule following the 1820 census, where one state got two above its lower quota and the rest of the states cried foul. That's one of the reasons Jefferson's method was discontinued for use. It can be shown mathematically, not in this video, it needs to be shown, that both Adams and Webster's method can violate the quota rule under certain circumstances. That brings us to this question. Is Hamilton's method the best method of apportionment? Because it always satisfies the quota rule. And the answer is maybe. It depends on what you think about these. The Alabama Paradox, the Population Paradox, and the New States Paradox were actually discovered when we were using Hamilton's method. As we were apportioning representatives to the U.S. House of Representatives, throughout history, we ran into these three paradoxes. And I'm going to tell you what these three are, and then you can decide for yourself whether you think they're important. First is the Alabama Paradox. It says that when you add a new item to be apportioned, that causes a group to lose an originally apportioned item if that happens and the Alabama Paradox has occurred. This actually happened following the 1870 census. The house size, if it were kept at 299 representatives, Alabama would have gotten eight representatives. But if the house size was increased to a nice round number of 300, the state of Alabama only got seven. And that just seems paradoxical. Why should that happen? If we increase the number of representatives, doesn't that mean that somebody else just gets an extra seat? Why should Alabama have to lose a seat of representation? So that was one of the first paradoxes that was discovered. The second one is population paradox. This states that when group A gains an item at group B's expense, even though group B's population is growing at a faster rate than that of group A, then the population paradox has occurred. This was discovered after the 1900 census. Virginia's population was growing at a rate of 1.07 percent per year, while Maine's population was only growing at a rate of 0.67 percent per year. The country's population was growing faster than both of them, but after one year, after that census, the standard quota for Maine would end up giving a larger decimal piece than that of Virginia and Maine would deserve the extra representative if we were using Hamilton's method. Virginia didn't like that. And lastly, we have the new state's paradox. When the addition of a new group causes the loss of an originally apportioned item for one of the original groups, that's when the new state's paradox has occurred. It's called the new state's paradox because it happened when we brought a new state into the Union. The state of Oklahoma in 1907 was welcomed into the United States, and because of her population, she deserved five representatives in the House based on her population. But adding those extra five to the total actually would have caused a change in representation level for both the state of Maine and the state of New York. And so when this one was discovered, this was kind of the last straw for Hamilton's method, and we changed our mind as a government, I guess, and Congress passed that we would be using Webster's method for a while. We've got one more method discussed, and if you come back for the next video, we'll talk about whether there is a perfect voting method that we haven't talked about yet.