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Electoral Math

# The ‘Why’ of the Bader-Ofer Law

After two posts in which I explained in great detail what the Bader-Ofer law does, I didn’t address one basic question.

### Can’t we just give out seats proportionately like normal people?

The short answer is there’s no such thing.

As I said in the first Bader-Ofer post, the chances are infinitesimal that every party list will happen to get a percentage of the vote that corresponds exactly to a whole number of Knesset seats. So what do you do when a list wins 28.081742456 seats?

Well, you should obviously give them 28 seats to start with (this is how the “difficult way” begins). Do that with each of the lists and you’ve given out almost all of the Knesset seats.

But those leftover fractions – the “.081742456” parts – can add up to four, five, six, maybe even ten seats. And regardless of how we fill those seats, some parties will be awarded more than the actual percent of the vote they got, and other parties will be awarded less. How do we decide who gets them?

### The problem with rounding

The simplest way to give out those extra seats, one could argue, is to simply round to the nearest whole number. If a party gets 38.6224801 seats, round up to 39; if a party gets 21.28774541, round down to 21.

But that won’t always work. In many cases, rounding won’t give you the correct number of seats. Sometimes you’ll fall short (in the 2015 election, rounding each party to the nearest whole number would have given you a total of 117 seats), and other times you’ll end up with too many.

For example, imagine a three-party election where the parties get vote totals equivalent to 40.7, 30.6, and 48.7 seats. Those three numbers add up to 120 – but if you round each of them to the nearest whole number, you get 41+31+49=121 seats. There aren’t enough Knesset seats for that.

### Which one’s closest?

Aha, you say. But two of those parties are closer to their next seat than the third one is! If we only have two leftover seats to give out before reaching 120, we should just give them to whoever’s closest to their next seat – regardless of rounding.

In this case, This Party and The Other Party are each .7 of the way to their next seat, while That Party is only .6 of the way there. So why can’t we give the two leftover seats to the two who are closest, and That Party will simply be out of luck?

You make a persuasive argument. (Thanks!) So persuasive, in fact, that this was the system that Israel used until it was changed in 1973 by Yochanan Bader and Avraham Ofer.

The law proposed by MK Bader and MK Ofer implemented what is called the “Hagenbach-Bischoff system” – which is not the name of the greatest ice cream store I’ve never been allowed to eat at, but rather a common method of distributing parliamentary seats that is used in over 50 countries. Hagenbach-Bischoff (also known as the D’Hondt or the Jefferson method) is responsible for all the “VPS bid” and “initial seats” and “leftover seats” shenanigans that I spent my last two posts explaining.

But why is such a complicated system so much more popular than the simple-and-obvious idea of simply choosing whoever’s closest to the next seat?

Because the whoever’s-closest method, technically known as the Largest Remainder Method, has several problems:

### Problem #1: Splitters

Imagine a party whose share of the vote was worth 13.3 seats. Normally, it would receive 13 in the initial allocation, and its fortunes would probably end there.

But if it were to split into two parties that receive 6.6 and 6.7 seats each, both of these might round up to 7 – which means they would get 14 seats in total.

Of course, it’s not very easy for a party to take advantage of this. It’s hard enough to accurately predict how many seats a party will get – much less predict it down to the first decimal place! But merely the fact that it can happen implies that there’s something unfair going on here. It shouldn’t be possible to take the same number of votes and split them among more parties so that they’ll be worth more.

The Largest Remainder Method allows for this sort of problem – but Hagenbach-Bischoff ensures that two small parties will never get more seats than they would have gotten if they had run together.

### Problem #2: Votes per seat

There is a perennial complaint in the UK about how unfair the electoral system is, often based on comparing how many votes a party got to how many seats it won in parliament. In 2017, for instance, the Conservative Party “got one seat for every 43,018 votes, while for Labour the figure was one seat for every 49,152 votes“. The Liberal Democrats had it even worse, with a staggering 197,665 votes per seat.

You can see, based on these complaints, that a party’s votes per seat (VPS) is considered a “fair” way to distribute seats in a parliament. But then again, it’s also fair if you try to hew as close as possible to the percentage of the vote each party received. There are even other types of measuring fairness that are outside the scope of this article. So which one is preferable? Each type of fairness contradicts the others; it is impossible to fulfill them all. That’s part of the debate between the Largest Remainder Method, Hagenbach-Bischoff, and other seat distribution systems.

Assuming that you care about the VPS version of fairness, it’s easy to see that the implementation of Hagenbach-Bischoff in Israel was a success (I have left out one-seat parties in the following analysis because they can have wonky effects):

• In 1969, the last Knesset election before the Bader-Ofer law was passed, the parties ranged from 14023 votes-per-seat for the Progress and Development Party to 8196.5 votes-per-seat for the Free Center. The crazy part is that both these parties got the same number of seats!
• In 1973, under the first Bader-Ofer election, the VPS range was much tighter: from 14140 for the Independent Liberals to 11302 for Progress and Development.
• In 2015, the higher threshold helped stabilize the VPS range even further. It went from 35818 for Yisrael Beiteinu to 31536 for Kulanu – a very tiny spread, proportionately.

So it’s clear that Bader-Ofer succeeds in ensuring that party lists “pay” a relatively equal number of votes for each seat they get in the Knesset.

I should reiterate, though, that it’s not technically a “problem” that the Largest Remainder Method’s emphasizes the percent of the vote rather than the VPS; it’s simply a matter of where you want to put your priorities.

### Problem #3: Too many leftovers

Under normal circumstances, the number of leftover seats is about half of the number of party lists who crossed the threshold. It’s not hard to give those out, one seat at a time, to whoever is closest to the next seat under the Largest Remainder Method.

In my previous posts I described how a party can win more seats than it has candidates, and how those extra seats are then redistributed according to VPS bids. But it’s easy to picture a scenario in which a party with, say, 3 candidates wins 15 seats – leaving 12 seats left over. What if there simply aren’t enough parties to take those 12 seats? Will we round everybody up to the next seat… and then round some of them up again to the seat after that?

### Problem #4: The Alabama Paradox

This is a weird one.

The Largest Remainder Method can cause a strange paradox in which increasing the number of seats in a parliament can cause some parties to lose seats. It was discovered after the 1880 census in the United States, when Census Bureau worked on determining how many seats each state would get in the US House of Representatives. At the time, the number of seats in the House was not fixed in place, and it was discovered that Alabama would get 8 seats if the House of Representatives had 299 seats in it… but only 7 if the size of the House were increased to 300 seats.

How is this possible?

To determine a party’s share of seats in parliament before any rounding, you have to multiply their percentage of the vote by the number of available seats. That number will always be higher if you’re multiplying by a higher number. For example, Party A might have gotten 18.678097% of the vote, which is worth 70.603 seats in a 378-seat parliament but 70.789 in a 379-seat parliament.

But different parties rise at different rates. If Party B got 25.802492% of the vote, that’s worth 97.533 when there are 378 seats and 97.791 when there are 379.

Notice how Party B overtakes Party A:

If we are allowed to choose only one of these parties to receive a leftover seat, Party A would have 71 seats if the parliament were smaller, and only 70 if the parliament were larger!

Again, this is not a practical issue – we’re not exactly changing the number of seats in the Knesset on a regular basis. But it signals a deeper unfairness in the Largest Remainder Method.

### Problem #5: Who gets to choose the system?

I pointed out in my first Bader-Ofer article that a party’s VPS bids drop dramatically at first, but the drops become less and less drastic as the number of seats get higher and higher. This makes sense intuitively: half a pizza pie is much bigger than a third of a pizza pie, but if you’re cutting the pie into 100 pieces there isn’t much a difference between that and cutting it into 101.

Let’s apply that logic directly to a Knesset election. Imagine that the first two leftover seats have just been given out, one to a party with 48 seats and one to a party with 6 seats. Which party is more likely to get another leftover seat, assuming there are enough?

We can figure it out with a little algebra. If both parties have just won leftover seats, they must have submitted similar VPS bids. Call the value of that bid x.

• If the larger party has 48 seats, that means it must have divided its vote total by 48 in order to bid x. So its vote total must have been roughly equal to x * 48. Since its next VPS bid is for a 49th seat, that VPS bid is equal to the vote total divided by 49, or x * 48/49, which is approximately x * .98. In other words, its next VPS bid is 98% of its most recent one – the bid barely dropped at all!
• If the smaller party has 6 seats, that means it must have divided its vote total by 6 in order to bid x. So its vote total must have been roughly equal to x * 6. Since its next VPS bid is for a 7th seat, that VPS bid is equal to its vote total divided by 7, or x * 6/7, which is roughly equal to x * .86. In other words, its next VPS bid is 86% of its most recent one – a much bigger drop. The larger party is almost certain to get another leftover seat before the smaller one is.

What’s clear from this is that the Hagenbach-Bischoff system tends to favor larger parties. But that’s not just true when there are a lot of leftover seats. Recall that the easy way proves that, from a different point of view, all of the Knesset seats are given out via VPS bids. Since even the initial set of seats can be said to have been given out that way, that means that from the very first leftover seat all of the parties’ VPS bids are based on drops from a relatively equal starting point – and larger parties drop a lot less than smaller ones.

Now here’s an interesting fact about the two MKs responsible for the Bader-Ofer law: Mr. Bader was a representative of Gachal, a precursor to the Likud, and Mr. Ofer was a representative of the Alignment, a precursor to today’s Labor Party. These were, then as now, the two largest parties in the Knesset.

I’m not saying these two MKs changed the Knesset over to the Hagenbach-Bischoff system to benefit their own parties at the expense of smaller ones… actually, I am saying that.

The Hagenbach-Bischoff system isn’t perfect; it prioritizes one type of fairness over another that is arguably more intuitive, and it is biased against small parties (or, depending on how you look at it, biased in favor of more political stability).

But it’s also not insane, the way it might have appeared to be at first glance. There are some good reasons to prefer it over the Largest Remainder Method, including some additional ones I didn’t go into in this article. It’s not an accident that many, many countries use it.

There is an entire field of political science that discusses how to distribute seats proportionately. New methods are being proposed all the time, and debates over how fair they are have raged for decades.

Before I close out this series on the Bader-Ofer law, I want to address one particularly strange misconception that I noticed.

Recall that the Bader-Ofer law gives out Knesset seats in two phases:

1) An initial batch of seats are given out based on percentage of the vote

2) Leftover seats are given out based on VPS bids.

Apparently some people confuse this with the fact that Israel’s vote totals are reported in two phases:

1) Regular votes are counted the night of the election.

2) The so-called “soldiers’ votes” are counted over the course of the following week. These are actually absentee ballots cast by soldiers on military bases, sailors on IDF naval vessels, prison inmates, hospital patients, and Israeli diplomatic envoys abroad.

These are not the same thing! The adjustments to the seat totals that trickle in days after the election have nothing to do with the Bader-Ofer law. They simply represent the fact that, as more votes are counted, parties might find themselves with a slightly lower or slightly higher percentage of the vote than they initially thought, depending on how well they do among the new ballots coming in. In this case, those new ballots are the absentees.

The mistake is understandable. In 2015, for example, on the day after the election Meretz appeared to get only four seats. However, a few days later Meretz was reported to have risen to five seats. It’s easy to conclude, as these articles did, that they somehow got this extra seat due to the distribution of leftover seats when Yachad didn’t cross the threshold.

But the articles’ mocking of Yachad for failing to cross the threshold and thus giving “greater proportional weight” to left-wing votes and winning Meretz an extra seat is complete nonsense. I’ve already demonstrated that when Yachad failed to cross the threshold its three seats went to the Zionist Union, Kulanu, and the Likud – not to Meretz. And there’s no such thing as “greater proportional weight”. The way the Israeli system works, a party that doesn’t cross the threshold is simply deleted and removed from the election along with all of its votes – and it won’t affect the results one bit.

The articles’ claim is also illogical based on the sequence of events. Yachad didn’t fall below the threshold some time after the election; it had failed to cross the threshold on the very first night of vote-counting, when Meretz still had only 4 seats. Meretz picked up its fifth seat only later – but seats can’t appear out of nowhere. If Yachad didn’t have any seats to lose, then Meretz could not have gotten its fifth seat from Yachad.

In fact, the party that lost a seat to Meretz when the absentee ballots were counted, days after the election, was UTJ.

The night of the election, when all the regular ballots had been counted, UTJ had had enough votes for 7 seats. But it would end up receiving relatively few votes among the absentee ballots, and couldn’t keep up with the increased vote totals that other parties received there. UTJ therefore lost a seat, while Meretz – whose vote count among absentee ballots was disproportionately high – gained a seat.

I say “lost” and “gained” in a figurative sense, of course. The seat counts that are published after the regular ballots are counted aren’t actually given out; they are merely the seats that each party would have received, had the regular ballots been all of the ballots in the election. It’s no different from saying, in an American election, that a particular candidate has “48% of the vote so far” and then discovering later in the night that they had in fact risen to 53% and won the election.

### Next up

My wife is expecting, which is why I haven’t had the opportunity to write as many articles as I intended in the last two weeks. As a result, I’ve had to forgo the series of articles I had hoped to write on the accuracy and transparency of Israeli polling. Perhaps I will go into that after the election.

In the meantime, however, you can expect an explanation of vote-sharing agreements, another set of inaccuracies and curious questions arising from Knesset reporting, and an explanation of the other way that a party can get into the Knesset with fewer than four seats — the first way being, of course, if its list had fewer than four candidates.