Apportionment formula, Acts 1 & 2

A couple weeks ago, the Central Election Commission invited me to speak at a public hearing on changing the formula for apportioning legislative seats to the 22 cities and counties. I assume roughly half the people who read that first sentence are already nodding off to sleep. Let’s say that my reaction was exactly the opposite: my heart started pumping faster and my brain started racing. Change the formula! What in the world is going on!?!

I’m not quite sure how to best tell this story, so I guess I’ll tell it in three acts.

Act one: The technical details

Act two: The CEC public hearing and my testimony

Act three: The political machinations


Act one: the dry, boring, technical stuff that you need to understand everything else


Before the hearing, the CEC sent me an official notification of the meeting in which they laid out the reason for the meeting. The CEC is holding three public hearings in reaction to the demand from legislators, since legislators have raised concern that the current formula violates the constitution. The constitution sets out two criteria for allocating the 73 district seats: 1) every city and county should get at least one seat, and 2) the seats should be allocated according to the population of each city and county (not including the indigenous population, since they are represented by other legislators). The current formula uses a two stage process, and the legislators questioned whether the two stage process violates the second criterion. They suggested returning to an earlier formula which only uses one stage.

The current formula has been used for the three elections since 2008 (7th, 8th, and 9th Terms), so it has been dubbed the “7th Term Formula.” For simplicity, I’m going to label it T7. In T7, you take the national population and divide it by 73 to get a quota (Q1). Using June 2017 population, Q1=315019. There are six cities and counties whose population is less than one full quota, so we first give them each a seat (S1). Then the population of the other 16 cities and counties is summed and divided by 67 to obtain Q2=329837. This Q2 is used to apportion the remaining 67 seats. For every full quota, a city or county gets a seat (S2). If there are remaining seats after all the full quotas are allotted, they go to the cities or counties with the largest remainder (S3).


T7 Formula (using June 2017 population)

  Pop S1 Pop S2 R S3 S
total 22996448 6 22099102 60   7 73
Taipei 2673539   2673539 8 34843   8
New Taipei 3927273   3927273 11 299066 1 12
Taoyuan 2096623   2096623 6 117601   6
Taichung 2743103   2743103 8 104407   8
Tainan 1878892   1878892 5 229707 1 6
Kaohsiung 2744102   2744102 8 105406   8
Yilan 440228   440228 1 110391   1
Hsinchu Cn 528325   528325 1 198488 1 2
Miaoli 545096   545096 1 215259 1 2
Changhua 1278821   1278821 3 289310 1 4
Nantou 474319   474319 1 144482 1 2
Yunlin 690196   690196 2 30522   2
Chiayi Cn 507270   507270 1 177433 1 2
Pingtung 773446   773446 2 113772   2
Taitung 141245 1         1
Hualien 237493 1         1
Penghu 102908 1         1
Keelung 362548   362548 1 32711   1
Hsinchu Ci 435321   435321 1 105484   1
Chiayi Ci 268652 1         1
Kinmen 134527 1         1
Lienchiang 12521 1         1


The proponents of change want to go back to the formula used in the 1998, 2001, and 2004 elections, which I will label the “T4” formula. In T4, there is only one quota (Q=pop/seats=315019), which is identical to Q1 in the T7 formula. First apportion one seat to every city and county (S1), then if they have enough population for two or more full quotas, give them any additional full quotas (S2). Finally, allot the remaining seats according to the largest remainders (S3).


T4 Formula (using June 2017 population)

  Pop S1 S2 R S3 S
Total 22996448 22 46   5 73
Taipei 2673539 1 7 153387   8
New Taipei 3927273 1 11 147045   12
Taoyuan 2096623 1 5 206509   6
Taichung 2743103 1 7 222951 1 9
Tainan 1878892 1 4 303797 1 6
Kaohsiung 2744102 1 7 223950 1 9
Yilan 440228 1 0 125209   1
Hsinchu Cn 528325 1 0 213306 1 2
Miaoli 545096 1 0 230077 1 2
Changhua 1278821 1 3 18745   4
Nantou 474319 1 0 159300   1
Yunlin 690196 1 1 60158   2
Chiayi Cn 507270 1 0 192251   1
Pingtung 773446 1 1 143408   2
Taitung 141245 1       1
Hualien 237493 1       1
Penghu 102908 1       1
Keelung 362548 1 0 47529   1
Hsinchu Ci 435321 1 0 120302   1
Chiayi Ci 268652 1       1
Kinmen 134527 1       1
Lienchiang 12521 1       1


As you can see, the difference between the two formulae is not insignificant. In T7, Taichung and Kaohsiung both get 8 seats, while in T4 they get 9. In T7, Nantou and Chiayi County get 2 seats, but T4 only gives them 1 seat each.

To reiterate, the normative argument that proponents of T4 are pushing is that the two quotas (Q1 and Q2) in T7 are somehow violating the requirement that seats be apportioned according to population. Since the constitution and election law say nothing about discarding the populations of the small cities and counties and calculating a new quota, they argue that it is forbidden to do so.

Before we move on, there is one more (quite important detail). We are all using June 2017 data. However, according to the election law, seats are apportioned according to the population 26 months before the end of the term. That means that critical data is the November 2017 population. We still have five months to go. As my friend Donovan Smith keeps pointing out, Taichung is growing faster than Kaohsiung and is about to overtake it as Taiwan’s second largest city. For this exercise, that doesn’t matter. What does matter is that Taoyuan is growing fastest of all. Over the past year, Taoyuan has increased an average of a little more than 3000 people per month. By November, it should have at least 15000 more people. Right now, Taoyuan’s remainder is 206509. With another 15000, it will have a remainder of about 221000, which might even pass Kaohsiung for the fourth largest remainder. The loser will probably be Hsinchu County, which is growing by about 300 people per month. If you add 1500 people to its current remainder of 213306, it will be at about 215000, about 6000 behind by Taoyuan and Kaohsiung. Poor Hsinchu County! For a couple of years, it has been excited that it will get a second seat and now that second seat is about to be ripped away. Worse, nobody seems aware that this is happening. I also have no indication that the Taoyuan politicians are aware a seventh seat might fall into their lap.

From here there are two things that might happen. After the three public hearings, the CEC will assess the public feedback, hold a meeting, and vote on which formula to adopt. For reasons that I will delve into in Act Two, I’d be shocked if they didn’t vote to maintain the current formula (T7). Following that decision, the legislators have a choice. If they are dissatisfied with the CEC’s decision, they can choose to override it by writing the formula into the election law. I suspect most legislators will favor T4, and so this will be the eventual outcome. Those calculations are the subject of Act Three.


Act Two: The CEC public hearing

I’ve never been asked to speak at a public hearing before, so I really didn’t know quite what to expect. I was told that everyone would get only five minutes, with the possibility of another round if time permitted. Five minutes is not long, so I had to be concise. I think this is the shortest powerpoint presentation I’ve ever made. Here it is in its entirety.




The first point is simple. Neither of these formulae violate the constitution. The constitution mandates the CEC to allot seats according to population, but there are many ways to do that. The constitution does not specify which formula to use, so the CEC has leeway to use its judgment as to which is the most appropriate. Every person who spoke today agreed on this point. In other words, we all explicitly rejected the current normative argument for changing from T7 to T4.

The second point is that there is, in fact, a solid normative argument to be made in favor of T4. T4 reduces the overall level of malapportionment relative to T7. My hope is that, since the politicians will probably continue to press for T4, I hope they will use this argument instead of the constitutional one. I’d like for all discussions of electoral system fairness to be conducted in terms of disproportionality.

I will say that one of the other scholars, Huang Kai-ping 黃凱苹 (NTU), and the KMT representative, Huang Teh-fu 黃德福 (who is also a political scientist and was one of my teachers two decades ago when I was a MA student at NCCU), disagreed with me about which system was fairer. They focused on comparing the four cities and counties that would change seats, looking at the ratios of voters in each one. For example, Huang Teh-fu, looked at the ratios of the each of the three larger districts to the smallest district among the four, adding up the differences to get a deviation index as follows:

T4 formula

  Pop Seats Ave Ratio Ratio-1
Taichung 2743103 9 304789 1.0000 .0000
Nantou 474319 1 474319 1.5562 .5562
Chiayi Cn 507270 1 507270 1.6643 .6643
Kaohsiung 2744102 9 304900 1.0004 .0004
Sum         1.2209

T7 formula

  Pop Seats Ave Ratio Ratio-1
Taichung 2743103 8 342887 1.4458 .4458
Nantou 474319 2 237159 1.0000 .0000
Chiayi Cn 507270 2 253635 1.0695 .0695
Kaohsiung 2744102 8 343012 1.4463 .4463
Sum         0.9616

As you can see, T4 yields a deviation index of 1.22, much higher than the T7 deviation index of 0.96. Huang’s conclusion was that T7 is fairer.

I have two responses to this. First, a deviation from the mean in Taichung is not equivalent to a deviation of equal magnitude in Nantou. In T7, Taichung has four times as many seats as Nantou, so a deviation of equal magnitude affects four times as many people in Taichung as in Nantou. If you think about the total number of people affected, it is more important that Taichung not be too far over or under whatever the fair number is than Nantou. Second, you should not consider the four cities and counties that will be affected in isolation. Malapportionment is a national level problem, and you have to look at it from a national perspective.

The standard measure used in the academic literature to measure is the Loosemore-Hansby Index, which considers the difference between how many seats each district should have according to its population and how many it actually has.

Taiwan has 79 nominal seats (73 district and 6 indigenous). I made a spreadsheet with 79 rows to calculate this index, but since that is a tad long I’ll just do one line for each county. Taipei has eight seats, so you should imagine eight identical rows for Taipei 1 through Taipei 8. (After the actual districts are drawn, Taipei 1-8 will each have a slightly different population, so each one will deviate slightly differently. This will increase the overall level of malapportionment slightly. For now, we will ignore that part.)

T7 Formula: modified Loosemore-Hansby Index

  S Ave Pop Exp obs Abs(diff)
Taipei 8 334192 0.0142 0.0131 0.0011
New Taipei 12 327273 0.0139 0.0130 0.0009
Taoyuan 6 349437 0.0148 0.0133 0.0015
Taichung 8 342888 0.0146 0.0132 0.0013
Tainan 6 313149 0.0133 0.0129 0.0004
Kaohsiung 8 343013 0.0146 0.0132 0.0013
Yilan 1 440228 0.0187 0.0145 0.0042
Hsinchu Cn 2 264163 0.0112 0.0122 0.0010
Miaoli 2 272548 0.0116 0.0123 0.0008
Changhua 4 319705 0.0136 0.0129 0.0006
Nantou 1 237160 0.0101 0.0119 0.0018
Yunlin 2 345098 0.0147 0.0133 0.0014
Chiayi Cn 1 253635 0.0108 0.0121 0.0013
Pingtung 2 386723 0.0164 0.0138 0.0026
Taitung 1 141245 0.0060 0.0107 0.0047
Hualien 1 237493 0.0101 0.0119 0.0018
Penghu 1 102908 0.0044 0.0102 0.0058
Keelung 1 362548 0.0154 0.0135 0.0019
Hsinchu Ci 1 435321 0.0185 0.0144 0.0041
Chiayi Ci 1 268652 0.0114 0.0123 0.0009
Kinmen 1 134527 0.0057 0.0106 0.0049
Lienchiang 1 12521 0.0005 0.0090 0.0085
Plains Indig 3 86959 0.0037 0.0100 0.0063
Mountain Indig. 3 98382 0.0042 0.0101 0.0059
Sum/2         0.0729

In Taipei, each of the eight districts has 334192 people, which is 1.42% of the total population of Taiwan. Since each Taipei district has 1.42% of the population, each should also have 1.42% of the representatives. What do they actually have? There are 113 legislators and each district gets one, so Taipei 1 has 1/113 of the representation. But wait, what should we do about the 34 party list legislators? We assume that they are perfectly apportioned to each district according to its population and thus do not create any further malapportionment. (This is the “modified” part of the modified Loosemore-Hansby Index.) Thus, Taipei 1’s total representation is (1/113) + (.0142*34/113). Taipei 1 thus has 1.31% of the total legislators, or somewhat power less than its 1.42% of the population would receive under perfect apportionment. The eight districts in Taipei are thus collectively underrepresented by 0.88% of the total legislature.

To get the total malapportionment, you add up the absolute value of all these deviances and divide by two. T7 will thus produce 7.3% malapportionment. What this means is that 7.3% of the seats in the legislature are apportioned to places that would not receive them under perfect apportionment. From a cross-national perspective, 7.3% is fairly high. When you consider that Taiwan is not federal system with an upper house (two factors that tend to be associated with higher malapportionment), Taiwan’s malapportionment problem is actually quite serious. I’ll skip the table for T4, but the answer is that T4 yields 6.7% malapportionment, or a slightly fairer outcome than T7. The difference is not enormous, but neither is it insignificant. If you care about malapportionment, T4 is better.

What’s going on with this? The essence of the problem is that the constitution requires a significant level of malapportionment. Politicians have made value judgments that it is important to overrepresent indigenous voters and voters from sparsely populated cities and counties. The six indigenous seats and six seats from cities and counties that are below the quota account for 6.3% malapportionment. That is, these voters are getting an extra 6.3% of the representation in the legislature beyond what their raw numbers would confer. Lianchiang County is the most extreme; it gets 0.90% of the power with only 0.05% of the population. Since the constitution mandates that these voters be overrepresented by 6.3%, voters in the other 67 districts must be underrepresented by 6.3%. Remember how Taipei is collectively underrepresented by 0.88%? In essence, Taipei is making up for Lianchiang’s 0.85% overrepresentation.

So why is T4 better than T7? In T4, other than the 12 constitutionally mandated overrepresented seats, there are also four other overrepresented districts (two in Miaoli and two in Hsinchu County). This means that in addition to making up the 6.3% deficit, the remaining 63 districts have to further give back an additional 0.4% of the total power. In T7, Nantou and Chiayi County both get two seats, making them overrepresented. This means that the remaining 59 districts now have to spit up 7.3% of the total power. Think about this from the perspective of Taichung. Under T4, Taichung gets 9 seats. Each of those nine seats accounts for 1.29% of the population but only gets 1.27% of the power. Under T7, Taichung gets only eight seats, each of which has 1.46% of the population but only 1.32% of the power. Taichung is already underrepresented under T4, but under T7 it becomes much more underrepresented.

(As noted above, if current population trends hold, T4 would take one seat away from Hsinchu County and give it to Taoyuan. This would reduce the malapportionment to 6.5%, an even fairer outcome. A reduction from 7.3% to 6.5% is not too shabby.)


Remember when I said my remarks at the CEC hearing were concise? Suffice it to say I didn’t have time to go into all that detail.


My third point was that T4 is vulnerable to a mathematical problem. If the populations are distributed just right, you can actually apportion more full quotas than there are seats. The slide gives an example of how you might apportion 74 seats with full quotas.

Now, this probably won’t happen. You need all the remainders to be pretty small. However, it isn’t impossible. I was able to manipulate the numbers to apportion 75 seats. And then with even further manipulation I got up to 76 seats. And in an extreme case, I got to 77. Theoretically, I think with six cities or counties below the quota, the absolute limit would be 78 seats. So 74 seats is quite possible, even if it isn’t likely. But as the world has seen repeatedly over the past couple years, improbable things happen all the time.

This isn’t necessarily a fatal flaw. As long as the bureaucrats are aware of the problem, they can write a rule for how to deal with it. The obvious fix is that the lowest remainder should lose a seat. However, you have to have this rule clearly spelled out in advance. Otherwise, if you try to tell a city that, even though it has a full quota for each seat, it isn’t getting all those seat, all hell will break loose. Remember, you can’t just compromise and allot 74 seats because the constitution clearly says that there will be exactly 73 district seats.

Fantastic! Now we are aware of the problem, and we can avoid the great constitutional crisis of 2037.

By the way, I suspect this mathematical vulnerability is probably the reason the CEC switched from T4 to T7 a decade ago.


My fourth point was a simple, normative point. In electoral systems, stability is a value. You shouldn’t change the rules of the game unless there is some compelling reason to think that the system is significantly unfair. The electoral rules are the playing field, and you shouldn’t modify the playing field for a particular party or area’s short-term interests. This time you lose; next time you might win. Populations grow or shrink in ways that are hard to predict. Further, it is entirely possible that we might have some administrative reform. Ten years ago, there were 25 cities and counties; now there are only 22. Ten years ago, Kaohsiung was a winner because it was still split into two pieces. Kaohsiung City had enough people for 4.7 seats and Kaoshiung County had enough for 3.8. Because they were independent, both numbers were rounded up. If they had been combined, as they were in 2010, Kaohsiung would have only had enough people for 8.5 seats, and this would not have been enough to get the 9th seat. Tainan had exactly the opposite situation. In 2006, Tainan County had the population for 3.4 seats, which Tainan City had enough for 2.4. Both remainders were too small to get an additional seat. After the two were combined in 2010, Tainan had enough people for 5.8 seats, so this time it will get a sixth seat. If you change the formula every time based on your immediate political interests, people will lose faith in the neutrality of the system.


You might have noticed that my four points don’t all point in the same direction. Points 1, 3, and 4 favor T7, while Point 2 favors T4. As a scholar, it isn’t my job to make the value judgement about which priority is most important. Rather it is my responsibility to point out whether the system has any serious flaws and what the impact of the various options might be. It’s up to the CEC and the legislators to make the value judgements.

(Don’t worry, I’ll take off my scholarly hat and tell you what I really think in Act Three. However, I’m too tired to write that story now.)

CEC hearing.php

One Response to “Apportionment formula, Acts 1 & 2”

  1. Irwin Says:

    This is great stuff for election buffs and political junkies. Look forward to part 3.

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