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Electoral Math

# Dr. Strangevote or: How I Learned to Stop Worrying and Love the Bader-Ofer Law

The Bader-Ofer law is perhaps the most daunting and least understood aspect of Israel’s electoral system. The messy results of an election are thrown into a bubbling witches’ brew and somehow whole numbers of Knesset seats come out.

There will be math — but very little.

Consider: There are, of course, 120 Knesset seats. But the odds that we can actually divide the results of the election into 120 equal pieces are very, very slim. Voters aren’t cooperative enough to give the various party lists nice, neat percentages of the vote like 50%, 30%, or even 8.33333333333%. No, the electoral authorities have to deal with mutant fractions like 23.401452046% or 18.673347449% (both real examples).

Since the number of Knesset seats each list receives will never be exactly proportional, we have no choice but to approximate. The approximation that we use in Israel is fairly common in parliamentary systems, though each country’s version has its own quirks. Ours is called the Bader-Ofer law after the two Knesset members who proposed it, Yohanan Bader and Avraham Ofer. (Officially, the Bader-Ofer law actually governs how the Knesset distributes what are called “leftover seats”. This is actually the same thing, for reasons that will become clear.)

### Listen, bud, there are two ways we can do this…

There are two prominent ways to calculate the number of seats each party list gets under the Bader-Ofer law: the easy way and the difficult way.

The easy way is pure mathematical artistry. It gives out all 120 Knesset seats at once, through a very simple table that takes only three steps to fill out. It incorporates vote-sharing agreements with tranquility. It handles the threshold with grace. Even if a list wins more seats than it has candidates, it sidesteps the problem with the elegance of a professional ballet dancer. It’s even easy to play with: you can change the threshold or even the size of the Knesset at a whim and see what would have happened. The results are always easy to read and understand, and the way they are presented makes you feel good about how and why the system works the way it does. Truly, the easy way must be how the angels calculate Knesset seats.

Then there’s the difficult way.

The difficult way throws in a completely superfluous step at the very beginning that ruins everything that comes afterwards. Thanks to that extra step, you suddenly have to worry about something called “initial seats” and something else called “leftover seats”. You have to wait until halfway through the process before you can take vote-sharing agreements into account, and doing so is messy and artificial. You can’t adjust it to run different what-if scenarios, but rather need to recalculate every scenario from the beginning. It takes five times as long to set up everything and the results are more difficult to read. It causes the milk to go sour and the bread to go moldy. It makes the crops fail and makes the well dry up.

Worst of all, this extra aggravation is completely pointless – because the two ways always give the exact same results! The difficult way just introduces needless complication that is of no good to anybody. The difficult way makes no difference to the results of the election. Nothing is gained by doing things the difficult way. No sane country would ever use the difficult way.

So anyway, Israel uses the difficult way.

### Let’s not, and say we did

Forget the difficult way for now. I like the easy way much better.

The easy way is based on a concept that I will call Votes Per Seat, or VPS. Votes per seat is, very simply, how many votes a list got, divided by how many Knesset seats it wants to have.

When a list is “bidding” for its first Knesset seat, its VPS will be equal to how many votes it got – because you’re putting them all into one Knesset seat. But if the list wants 2 Knesset seats, its VPS will be equal to how many votes it got divided by 2. If it wants 3 Knesset seats, its VPS will be equal to how many votes it got divided by 3. And so on.

So, for example, a party list that receives exactly 120,000 votes has the following VPS values:

• 120,000/1=120,000 for its first seat
• 120,000/2=60,000 for its second seat
• 120,000/3=40,000 for its third seat
• 120,000/4=30,000 for its fourth seat
• 120,000/5=24,000 for its fifth seat

And so on.

Of course, most VPS values aren’t so clean. For a more realistic example, click on the chart to the left to see the VPS values that were offered by The Joint List in the last election, from 1 to 60 seats. (Why 60 and not 120? I’ll explain at the end of this article.)

You can see in the chart that VPS values usually involve fractions, often messy fractions. But we won’t need to actually do anything with those fractions – we just need to be able to tell whether one VPS is higher than another.

Now, if you look at the numbers closely, you may notice something interesting: VPS drops a lot from the first seat to your second, but as your party gets bigger you have to “pay” less and less for each additional seat. This fact will be important in a future post.

So what does VPS have to do with Knesset seats? Put simply, the 120 Knesset seats are given to the 120 best VPS scores across all party lists – not based on the percentage of the vote each list got.

Let’s try a simple example. The Freedonian Parliament has 10 available seats and no electoral threshold. Seven lists competed in yesterday’s election, and the results were as follows:

To find out how many seats everyone gets, we need to make a table showing the different VPS offers. Each list puts up ten VPS scores, offering their highest VPS (votes/1) when bidding for their first seat, and their lowest VPS (votes/10) when trying to pick up all ten seats:

Notice how the numbers drop as we move from left to right, because we’re taking each list’s number of votes received and dividing it first by 1, and then by 2, and then by 3, and then by 4, and so on.

Now we simply label the ten highest numbers in the table:

And that’s it! These are the winners of the election:

Three of the top ten VPS scores on the table belong to the Friends of Beer Party, so that list wins three seats in parliament. Two of the top ten VPS scores on the table belong to the Imperial British Conservative Party, so that list wins two seats in parliament. And so on until the last-place party, the Democratic-Republicans, which didn’t receive enough votes to win any seats at all. (I’ll spare you a walkthrough of the Freedonian coalition negotiations.)

Now, one of the greatest benefits of the easy way – aside from the fact that it’s easy – is that it gives you all sorts of interesting extra information. As you saw in the animation above, we not only know how many seats each list got but also in what order. For example, we can tell at a glance that the third candidate on the Friends of Beer Party’s list got into Parliament by the skin of their teeth, taking the last of the 10 available seats. We can also tell which party just missed out: the Imperial British Conservative Party’s third candidate has the highest VPS score among the losing candidates, meaning it would have gotten the next seat if the Freedonian Parliament were just a little bit bigger.

Below are the results of another election in Freedonia. If you want to test yourself, set up a table of VPS scores based on these vote counts, mark who won each of the ten available seats, and then click here to see the answers.

Now let’s introduce some complications. The most obvious are the vote-sharing agreements – but those deserve their own post, so we’ll put them aside for now.

But there are three other items to address:

### The threshold

The best way to account for a threshold is to replace a party list’s VPS offers with zeroes whenever it falls below the threshold. For example, if Freedonia has a 10% threshold, the Rhinoceros Party and the Bull Moose Party don’t make it into parliament anymore:

Why do I replace the VPS scores with zeroes instead of removing the party from the table altogether? A couple of reasons – one, it will make our lives easier when we introduce vote-sharing agreements, and two, it lets us play with various scenarios. For example, by leaving all of the parties in the table, it’s easy to compare the results of the election with and without the threshold:

We can see, for example, that when the two animal-themed lists fell below the threshold, their seats were picked up by the Best Party and the Imperial British Conservatives.

If I may return to Israel for a moment, this type of comparison is how you prove that the three seats lost to the threshold by Yachad in the 2015 election were picked up by the Likud, the Zionist Union, and Kulanu.

### Short lists

There are fourteen lists in Israel’s upcoming election with a very small number of candidates (ten or fewer) – including two lists with only a single candidate each. What happens if a party wins more seats than it has candidates?

The difficult way of calculating Knesset seats has to deal with this problem head-on. But the easy way sidesteps it altogether. Put simply: When filling out the table of VPS offers, a party is only allowed to submit a VPS offer if it actually has enough candidates.

For example, imagine if the Rent Is Too Damn High party list only had 8 candidates on it, instead of the full set of 10. Without enough candidates to fill a ninth or tenth seat, it is not allowed to submit VPS offers for those seats at all:

Notice the two empty spaces in the table, at the end of the Rent Is Too Damn High row.

Now, removing those two VPS offers didn’t affect the election, because the Rent Is Too Damn High Party wasn’t going to win nine or ten seats anyway. But what if the Friends of Beer Party was a passion project set up by a single individual?

With only one candidate on the Friends of Beer Party list, it isn’t allowed to submit VPS offers for seat #2 or higher. Which means it can’t win any more than one seat, even though (as you may recall from above) it got enough votes for three seats.

So who gets those seats instead? As usual, the ten available seats always go to the ten highest VPS scores in the table:

If you compare these results to the previous table, you’ll see that Friends of Beer dropped from three seats to one (because it only has one candidate), and as a result the Rent Is Too Damn High Party and the Imperial British Conservatives each picked up a seat.

This bizarre situation actually happened once in Israel’s history. In 1977, a man named Shmuel Flatto-Sharon started the Flatto-Sharon Party. France had him up on embezzlement charges, and he wanted to become an MK so that he could avoid being extradited thanks to parliamentary immunity. For some reason the Israeli public approved of this platform enough that his party received two seats’ worth of votes! But with only one person on the Flatto-Sharon Party list (namely, Flatto-Sharon), he only got to take one seat; when everything was calculated out, his second seat ended up going to the Left Camp of Israel instead.

### The majority

Israeli law has one more interesting quirk when it comes to allocating Knesset seats. A list is not allowed to have a majority of the 120 Knesset seats unless it received a majority of the votes. In other words, unless you actually got more than 50% of the vote, all of your VPS offers from seat #61 and onward are void.

Now, this might sound a little bit odd (why would you get more than 50% of the seats if you didn’t get more than 50% of the vote?), but remember that Knesset seats are given out based on VPS counts, not based on vote percentages. I’ll go into the details of why in a later post, but if you got 49.9% of the vote you very probably will end up with 61, 62, or even 63 seats – enough to form a government without needing to create a coalition with other parties. But the Bader-Ofer law says that you can’t make the jump from coalition-building leader to sole majority party if the voters didn’t actually give you a majority themselves.

Unlike the other situations we’ve described so far, this scenario has so far been entirely theoretical. No party has ever received 60 seats in an election; the closest was 56 in 1969, before the Bader-Ofer law was passed. And no poll shows any party coming anywhere close to this number in 2019.

### Questions?

I hope that in this article I successfully guided you through the ins and outs of Knesset seat allocation. The Bader-Ofer law is at the core of the Israeli system, and understanding it is central to understanding how our elections work under the hood. But if anything I said is unclear, or if you have questions about specific scenarios or are curious about “what happens if…”, feel free to leave a comment and I’ll address it in a future post.

### The difficult way

I should repeat that everything that I described in this post is not the actual procedure by which Knesset seats are divided. It’s the easy way. But remember that both ways always give the exact same results. If you understand everything in this article, you understand enough to calculate the results of any Israeli election on your own – and you’ll come to the same conclusions as the official method (not including, of course, the vote-sharing agreements, which I will explain later.)

In my next post I will explain the difficult way: what the Bader-Ofer law actually says we do. Now, the difficult way will involve more math. But it’s worth it, because it gives a different perspective on how the seats are allocated. There are certain things you can see from one angle that you’ll miss from another. And besides, it’s always good to know what the law says – even if you’re not planning on following it.