How the Bader-Ofer Law Really Works

In my last post, I explained how to use the raw vote counts to figure out how many seats each party gets — but I used a simpler method than the one called for in Israeli law (“the easy way”). In this post, I’ll walk you through how the Bader-Ofer law actually distributes Knesset seats (“the difficult way”), and you’ll see that the two methods give the same results.

There will be more math than there was in the last post, but I’ll try to keep it under control.

Overview

Under the Bader-Ofer law, Knesset seats are distributed in two phases: first an initial set of seats, followed by what is known as “leftover seats.”

Initial Seats

In the first phase, a large number of Knesset seats are given out all at once, based on the each party’s share of the total relevant vote. The total relevant vote is the number of votes that were received by only the successful party lists, i.e., lists that crossed the threshold. (In contrast, remember that when we calculated the threshold it was equal to 3.25% of all valid votes for all lists.)

The total relevant vote is then divided by the number of Knesset seats (120) to figure out how much each seat is worth, and then each party list’s vote count is divided by that number.

Confused yet? An example will hopefully make it clearer. We’ll use the same fictional Freedonian election we used in the Threshold section of the last post:

Freedonia has a 10% threshold, which means that three party lists in the election didn’t receive enough votes to cross it: the Rhinoceros, Bull Moose, and Democratic-Republican parties. Their votes are thrown away.

Next, we calculate the total relevant vote, which is the number of votes that went to all the other lists, the ones that passed the threshold: Friends of Beer (690 votes), Imperial British Conservatives (656), Rent Is Too Damn High (497), and Best Party (364). The total relevant vote is therefore 690+656+497+364=2207.

If the total relevant vote is 2207, how much is each Freedonian Parliament seat worth? The Freedonian Parliament has exactly 10 seats in it, so each seat is worth 2207/10=220.7. It’s tempting to call this “votes per seat,” but since we’re already using that term to mean something else, we’ll say that 220.7 is the number of Votes a Seat is Worth, or VSW.

The next step is to figure out how many whole VSWs each party list received. You do this by dividing each party list’s vote by the VSW, and chopping off the remainder:

The Friends of Beer Party gets 3 seats, the Imperial British Conservatives and the Rent Is Too Damn High Party get 2 seats each, and the Best Party gets only 1 seat.

Notice that it doesn’t matter how close you are to your next seat. In this first stage, you only get seats by whole numbers. This is particularly annoying for the Imperial British Conservatives, who are excruciatingly close to their next seat; they needed only seven more votes to make it to 3 VSWs.

But don’t cry for them just yet. You may have noticed that we only gave out 8 of the 10 seats in the Freedonian parliament. There are two seats remaining: the famous “leftover seats”.

Leftover Seats

For the second stage of seat allocation, we will be bringing back our friend from the last post, the concept of Votes Per Seat (VPS).

(If you didn’t read the last post–and I suggest you do–your VPS is the number of votes you got, divided by the number of seats you’re trying to obtain. For example, if you have four seats and are trying to get a fifth, your VPS is the number of votes you got divided by 5.)

In this phase, the leftover seats are put up for “auction” one at a time. In the auction, each party list submits a VPS “bid” to try and get that seat. Whoever bids highest gets the seat, and submits a new (lower) bid for the next seat. Repeat until there are no more seats left to give out.

To continue with our Freedonian example:

The Friends of Beer Party got 3 of the initial seats; to win a leftover seat, it must divide the number of votes it received by 4, and hope the result has the highest VPS. Notice that we are dividing the number of votes by one more than the number of seats the party list currently has. That’s because Votes Per Seat depends on the number of seats you’re trying to get, not the number of seats you already have.

This idea was easy to understand in the last post, because we had an entire table in which we kept dividing the number of votes each party got by 1, by 2, by 3, by 4, etc., and the number of seats you got depended on how high you climbed the ladder. But in this case, we’re jumping into seat allocation in the middle, which can cause confusion: you can forget to add 1, whether because you’ve misunderstood the law (which is a mistake I see often) or because you copied the number of seats unchanged out of pure instinct (which is so common that while putting together that tiny four-line table I did it myself. Twice.). So this is important not to forget: Each list’s VPS offer depends on how many seats it is trying to get, which is the number of seats it already has plus one.

To return to the election:

The highest VPS offer is clear: it belongs to the Imperial British Conservatives. This shouldn’t come as much of a surprise, because we already established that they were excruciatingly close to their next seat. So the first leftover seat goes to them.

Who gets the second one?

You’ll notice that all of the VPS offers are unchanged, with one exception. The Imperial British Conservatives divided their votes by 3 when they were trying to get their third seat – but now that they have it and are trying to get a fourth, they need to instead divide their votes by 4. But all of the other party lists have the same number of votes and the same number of seats that they did before, so their VPS offers don’t change.

Anyway, the winner of the second leftover seat is clear: the highest VPS belongs to the Best Party, which wins its second seat. This is the tenth seat in the Freedonian Parliament, which means seat allocation is now ended. The final results:

You’ll notice that these are exactly the same results we got in my previous post.

How hard can it be?

Freedonia had it easy. There are only 10 seats and 7 lists, only 4 of whom crossed the threshold. Under those circumstances, you can expect to only end up with two leftover seats. Which means that to calculate the results of a Freedonian election you usually need to make three tables:

  • The table we used to give out the large block of initial seats
  • A table of VPS offers for the first leftover seat
  • A table of VPS offers for the second leftover seat

Israel is… not so lucky. We have 120 Knesset seats, and each election sees 10-15 parties cross the threshold. This means that to calculate the results of the election we need on average anywhere from five to 10 tables, just so we can give out each of the leftover seats one by one!

Now, I like tables (big surprise). But I like tables that are useful for something. And I hate the idea of making 10 tables when only one will do. I definitely prefer the easy way, which requires only one table — no matter how big your parliament is or how many parties are in it!

Drawback #2: Size of the Parliament

Notice how early in the process we had to bring in the number of seats in the parliament. In the easy way, we only brought in the number of seats at the very end, after the table had already been created – so if we wanted to play around with the results by increasing or decreasing the size of the Knesset, it was just a matter of coloring a bit more or a bit fewer boxes.

With the difficult way, it’s a lot more problematic: if you want to change the size of the Knesset and see how it affects things, you have to go all the way back to that early step and do the entire calculation over again.

Drawback #3: Short Lists

In the last post, I discussed what happens if a party wins more seats than it has candidates on its list. If you’ll recall, the easy way handled unexpectedly short candidate lists by simply sidestepping the problem: from the very beginning, each party list only filled out entries in the VPS table according to the number of candidates it actually had. There was no way to accidentally give a list more seats than it had candidates, because it simply couldn’t submit a bid for those seats.

You can do the same thing with the difficult way — but only if the list runs out of candidates during the Leftover Seats phase. In such a case, the list will simply stop submitting VPS bids, and the other party lists will continue to compete for seats without it.

But what happens if the list already had too few candidates in the Initial Seats phase? The difficult way won’t warn you if this happens; before proceeding to the Leftover Seats phase, you need to check each party list manually to make sure none of them are short on candidates. If they are, then you have to backtrack, count how many extra seats the list received that it shouldn’t have, and set those extra seats aside to be given out again as additional Leftover Seats.

For example, imagine a particular list wins 12 Knesset seats in the initial phase, but after investigation we discover that it only had eight candidates on it. Those four extra seats are now additional Leftover Seats, to be given out one by one to whoever submitted the highest VPS bids. Hooray, more tables!

And finally, the majority quirk

As stated in my last post, Israeli law does not allow a list to obtain a majority of the Knesset seats unless it also had a majority of the vote. This is implemented here in an interesting way: a party list stops submitting VPS bids if it reaches 60 seats during the Leftover Seats phase. If it already reached 60 seats or more during the Initial Seats phase, it bids for seats like everybody else.

(Incidentally, a small correction to my previous post: the law appears to refer to the majority of votes going to parties over the threshold. In other words, if you got 49.9% of the vote, but find yourself at over 50% once all the below-threshold parties are eliminated, you can submit VPS bids as much as you like. Otherwise, you are capped at 60 seats.)

Summary

Hopefully you now have an understanding of how the Bader-Ofer law distributes Knesset seats:

  1. Eliminate all the party lists below the threshold
  2. Figure out how many votes each seat is worth by dividing the total relevant vote by 120
  3. Figure out how many whole seats each party list obtained
  4. Double-check to make sure no list got more seats than it had candidates
  5. Ask each list for VPS bids, and give the first leftover seat to whoever has the highest bid
  6. Recalculate the winner’s VPS bid
  7. Repeat steps 5-6 until all leftover seats have been given out.

It is, in my opinion, an unnecessarily overcomplicated method, but that’s Israel for you. Personally, I prefer the easy way:

  1. Fill out a table showing each list’s VPS bids, cutting them off at whichever of the following comes first:
    • 0 seats, if they didn’t cross the threshold
    • 60 seats, if they didn’t get a majority of the vote
    • X seats, where X is the number of candidates on the list
  2. Label the top 120 VPS bids on the table.

Much simpler, right? Again, it is important to emphasize that both methods will always give the same results.

Next steps

In my next post, I will explain why we use this bizarre VPS-based method of allocating seats in the first place, discuss the pros and cons of some of the alternative systems, and review some common misconceptions. In future posts I also have a few other fascinating questions and scenarios to explore, questions to answer — and, most importantly, an explanation of vote-sharing agreements.

If you have any questions about Bader-Ofer or the threshold or about any of the other topics I’ve covered, or even a general question about the Israeli electoral system, please feel free to submit them here.

About the Author
Daniel Sterman lives in Jerusalem with his wife and five children. By day he is a not-so-mild-mannered technical writer; by night he dons a cape and mask and sends strongly-worded emails to news organizations to complain about minor mathematical errors.
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