Switzerland is known as one of the world’s foremost democracies. In fact, when the United States sought a framework of government for its 13 states in the late 18th century, it was the Swiss model of cantons that was adopted. Switzerland consists of 25 cantons, each one of which wants, and gets, a say in the affairs of state. Every 10 years the citizens of all the cantons elect their delegates to the Federal Council. Article 149 of the Swiss Federal Constitution stipulates among other things that the Federal Council consists of 200 representatives and that the seats are to be allocated in proportion to the population of each canton.

One may think that nothing could be easier than to fulfill these clear stipulations, but this would be far from the truth. The constitution’s apodictic instructions cannot, in general, be obeyed. The reason is that each canton can only send an integer number of representatives to the Federal Council. Let us take the canton of Zurich, for example. At last count it had a population of 1,247,906, which corresponded to 17.12 percent of the Swiss population. How many representatives can Zurich delegate to the Federal Council? For a house size of 200, can the canton send 34 or 35 representatives to the capital, Berne? And once the issue of regional representation has been resolved, how should the 34 or 35 seats be apportioned to the parties that participated in the elections in Zurich?

One simple way to allocate seats in parliament is by rounding the results. This method is unsatisfactory, however, since rounding up or down too often can lead to a change in the total number of seats, thus violating article 149 of the constitution. So some other method must be found.

According to theoreticians who deal with apportionment problems, a fair method of allocating seats in a council must fulfill two requirements. First, the method must yield numbers of seats that equal the computed numbers, rounded either up or down. Hence, Zurich would be apportioned not less than 34 and not more than 35 representatives. This is the so-called quota rule. Second, the allocation method must not produce paradoxical results. For example, the number of seats allocated to a growing constituency must not decrease in favor of a constituency whose population decreases. This is called the “monotony requirement.”

At first glance, the requirements appear to be reasonable. But when they are investigated mathematically or tested in practice, they are nothing of the sort. In 1980 the mathematician Michel Balinsky and the political scientist Peyton Young proved a very sobering, if disappointing, result: No ideal method of apportionment exists. A method that satisfies the quota rule cannot satisfy the monotony requirement, and any method that satisfies the monotony requirement fails to satisfy the quota rule.

So what is to be done? Article 17 of Swiss federal law spells out how the seats of the Federal Council are to be apportioned to the cantons. First, the cantons whose populations are too small to warrant a delegate of their own are apportioned one seat each. Then the number of seats that the other cantons are due is computed, fractions and all. Next, a preliminary distribution takes place: Each canton receives the rounded-down number of seats. Finally, in the so-called remainder distribution, the left-over seats are allocated to those cantons whose dropped fractions are the largest. The method seems acceptable even though it favors larger parties ever so slightly: Rounding 3.3 seats down to 3 is more painful than rounding 28.3 down to 28.

But even though the method seems so sensible, it can cause enormous problems. This first came to light in the United States, which used the same allocation method in the 1880s. A conscientious clerk who also happened to be good with numbers realized, to his utter amazement, that

the state of Alabama would lose a representative if the size of Congress were increased from 299 seats to 300. This nonsensical situation, which contradicted the monotonicity condition, was henceforth called the Alabama paradox. In the example illustrated in Table 1 an increase in the size of the House of Representatives from 24 to 25 has the ridiculous consequence that State A loses a representative.

As far as Switzerland is concerned, this problem ceased to exist in 1963 when the number of representatives, which heretofore had varied, was fixed at 200. In the United States, the size of Congress has been fixed at 435 representatives since 1913.

The Alabama paradox is not the only potential problem, though. Another paradox, referred to as the population paradox, may appear under certain circumstances. Constituencies whose populations have increased could

lose representatives in favor of other constituencies whose populations have decreased. In the example in Table 2, State C, whose population decreased, is awarded an additional representative at the expense of State A, whose population increased. In this paradox and in the Alabama paradox, the culprits are the fractional parts of seats. The population paradox still looms in Switzerland, albeit without having caused any problems so far. And in the spirit of “if it ain’t broke, don’t fix it,” the issue has been put aside. For the time being, the method of rounding down and then distributing the remaining seats according to the largest fractions is still in force.

But the problems are not over, even after the 200 seats of the Federal Council have been apportioned to the cantons. Each canton’s seats must now be assigned to the individual parties. Articles 40 and 41 of Swiss federal law say how this is to be done. The key for the distribution is based on the proposal of Victor d’Hondt (1841–1901), a

Belgian lawyer, tax expert, and professor for civil rights and tax law at the University of Gent.

D’Hondt suggested a rule which assures that the greatest number of voters stand behind each seat. It works as follows: For each seat the number of votes cast for a party is divided by the number of seats already allotted to the party, plus one. The seat then goes to the highest bidder. The process continues until all seats have been filled. (Table 3 should make this complicated-sounding procedure clearer.)

It did not take long for the Swiss to realize they had reinvented the wheel. As it turned out, d’Hondt’s method is computationally equivalent to the method that had been put forth a hundred years earlier by President Thomas Jefferson. He had used the method to apportion delegates to the House of Representatives in the United States. As far as the Swiss were concerned, they refused to relin-

quish ownership of the title to either the Belgians or the Americans and decided to name the method after Eduard Hagenbach-Bischoff, a professor of mathematics and physics at the University of Basle. Hagenbach-Bischoff had came across the method while serving as councillor of the city of Basle.

The fact that the Jefferson–d’Hondt–Hagenbach-Bischoff method slightly favors larger parties was not considered a major flaw. In fact, only when it is applied in a cumulative fashion will a disadvantageous impact be felt by smaller parties—for example, when a house uses the d’Hondt method a second time to fill various commissions. Paraphrasing Winston Churchill, one could say that the Jefferson–d’Hondt–Hagenbach-Bischoff method is the worst possible form of allocating seats … except for all the others that have been tried.