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Electoral Math

# What vote-sharing agreements actually do

The list system, the threshold, the Bader-Ofer law. The only thing left to explain about how Israeli elections work is the vote-sharing agreement.

Most articles about vote-sharing agreements will explain what they are using some variation of this sentence: “Vote-sharing agreements allow two parties to pool their leftover votes to see if they are enough to win an extra seat.”

Now, this is true as far as it goes. But there’s actually more to them than that. When two party lists share a vote-sharing agreement, what they’re actually doing is submitting all of their VPS bids jointly.

### A quick refresher

VPS bids are how an Israeli election is decided. A single Knesset seat is put up for “auction”, and is awarded to the party that deserves it the most, based on how many votes-per-seat (VPS) it would have if it won that seat. For example, imagine we’re partway through seat distribution and a party has received 9 seats so far on the strength of 440,000 votes. Its bid for a tenth seat is therefore 440,000/10=44,000 votes per seat. If no other party is bidding higher and it therefore picks up that tenth seat, then its bid for an eleventh seat will be 440,000/11=40,000 votes per seat.

If you use the easy way to calculate the results of the election, all 120 Knesset seats are given out through this auction process; if you use the hard way, this auction is only used for the last few “leftover” seats. But whichever way you calculate it, you’ll come to the same electoral results.

So how do vote-sharing agreements affect this process?

When two party lists sign a vote-sharing agreement, they don’t bid for seats on their own. Instead, they submit VPS bids based on the total number of votes they received between them. This is the case every time they submit a VPS bid, whether we’re giving out Knesset seats the easy way or the difficult way.

### The easy way

In the easy way of calculating the election results, all of the parties start out at zero seats, and we give out each of the 120 Knesset seats, one by one, to the highest VPS bidder. If two parties have a vote-sharing agreement, however, they submit all of their VPS bids jointly.

The best way to understand this is to see an actual example.

Imagine two ideologically similar parties: Party A received 65,000 votes, and Party B received 38,000. Their opposition, Party C, received 100,000 votes – far more than either of them. Nobody has a vote-sharing agreement. How would the first five seats in the Knesset be given out?

To answer this question, we set up a standard table of VPS bids: each party submits the votes-per-seat it would have if it won that many seats, and we mark which parties have the top five values in the table:

It’s obvious that Party C has the advantage here. It wins the first Knesset seat with its high bid of 100,000, then follows it up with three of the first five seats (see the red entries in the table). In fact, it wins three of the next five seats as well:

So Party C’s advantage is pretty clear-cut. Even though Party A and Party B together would have slightly more votes than Party C does, Party C will always be in the lead.

So how can A and B actually make use of their electoral advantage? By signing a vote-sharing agreement. If they do, that means the two parties submit VPS bids jointly:

The situation has now turned on its head. The first seat goes to the A+B alliance, thanks to their high bid of 103,000. Then Party C’s bid of 100,000 gives it a matching seat to tie the score. The third seat goes to A+B, and the fourth to Party C. The fifth seat goes to A+B,  the sixth to Party C. And so on – the A+B alliance always gets a tiebreaking seat before Party C matches them.

In the previous scenario, Party C had an obvious lead over its opponents. But in this one, thanks to the A+B vote-sharing agreement, Party C is now struggling to keep up. (And it will finally fail to keep up when they each get to seat #34, when A+B will pull out ahead by two seats and stay ahead for good.)

So we can see that signing a vote-sharing agreement can provide a structural advantage in seat allocation.

### The difficult way

In the difficult way of giving out Knesset seats, the parties don’t start out at zero. Rather, each one gets a number of Knesset seats according to their percentage of the vote, and only a few seats (generally about half of the number of parties) are left over. Those seats are given out one by one – again to the party with the highest VPS bid.

During initial allocation, the parties receive seats as in the chart on the right:

You can see that one seat remains to be given out, so each party competes for it by submitting a VPS bid. This bid is equal to the number of votes they got, divided by the number of seats they have plus one – in other words, it’s the number of votes per seat they would have if they get that extra seat. The bids are as follows:

The highest VPS bid (barely) is submitted by Party C; 1830 votes per seat is higher than the 1829 offered by Party A or the 1772 offered by Party B.

So Party C wins the auction and the extra seat, rising from 58 to 59 seats. Remember that in my explanation of why the Bader-Ofer law exists in the first place I pointed out that the VPS bid system gives an advantage to larger parties. So it’s no surprise that in a competition between a party of size 58, a party of size 40, and a party of size 21, the largest party would win the most often. (When there are ten or twelve parties, and the largest ones have 30 seats and the smallest ones have only four, the differences are even more pronounced.)

Now: what if Parties A and B signed a vote-sharing agreement?

The initial seat allocation is exactly the same as above; vote-sharing agreements only affect VPS bids. But now, Parties A and B have more votes – and more seats. The VPS bids are as follows:

With 61 seats, the A+B alliance now has a slightly (very slightly) better chance of winning the next seat than Party B. And win that seat it does – the final seat goes to either Party A or Party B.

### But who gets it?

Whenever a seat is awarded to a pair of parties who have a vote-sharing agreement – whether we’re using the easy way or the difficult way – the party that gets it is determined via… wait for it… VPS bids.

In this case, though, it’s a private auction between those two parties.

For example, in the last election Yisrael Beiteinu and Kulanu had a vote-sharing agreement. If you used the easy way to allocate Knesset seats, their alliance submitted 16 out of the 120 VPS bids, and therefore received 16 out of the 120 Knesset seats. How many go to Yisrael Beiteinu, and how many go to Kulanu?

Here’s the two-party VPS chart that decides the matter:

With 16 seats available to the alliance, the seats are awarded according to the top 16 VPS bids. Ten of those sixteen were submitted by Kulanu, so Kulanu won ten seats; the other six were won by Yisrael Beiteinu.

If you use the difficult way to calculate Knesset seats, you get the same results, but it’s more annoying to work out. I’ll skip several steps in order to present it concisely. We’ll take it as given that Yisrael Beiteinu received 6 seats and Kulanu received 9 seats in the initial seat allocation, and that their alliance helped them receive one leftover seat.

That seat will go to whichever one of the two parties submits the higher VPS bid:

The winner is quite clearly Kulanu, so Kulanu gets the tenth seat. Same result as the easy way, just more annoying to calculate.

To prove that the difficult way is not without merit, this method actually provides an interesting nugget of information. You may have noticed, two charts up, that Yisrael Beiteinu’s vote count was worth 6.4129 seats, while Kulanu’s was worth 9.4105. That means that Yisrael Beiteinu was ever-so-slightly closer to its next seat (.4129 vs. .4105). If the extra seat had been given out on that basis, it would have gone to Yisrael Beiteinu.

But, as I said above, the Bader-Ofer law give an advantage to large parties. This advantage outweighed that extra .0024 of a seat, and awarded the extra seat to Kulanu.

### To share or to merge?

In the 2015 election, all four major Arab parties ran on a single joint list that was called The Joint List. This election, they have split into two separate lists: Ra’am-Balad and Hadash-Ta’al. This led to a disagreement between two analysts:

If we’re to analyze this, we’ll have to acknowledge a number of complicating factors:

• There may be voters who were attracted to the image of Arab unity that the four parties showed last election, but will be less inclined to vote this time around.
• There may be voters who refused to vote for the Joint List last time, because one of the parties in the alliance represented an absolute deal-breaker for them; whereas this time, with two separate lists, they can feel free to vote for the one they like better.
• There is a risk that the Ra’am-Balad party may not cross the threshold.

But let’s put these aside, and assume the two lists receive exactly the same total number of votes that they would if they ran together. Who’s right, Harkov or Sales? After the split (bad) and the vote-sharing agreement (good), are the Arab parties worse off or better off?

As you may have already guessed: it’s a wash. Two parties that sign a vote-sharing agreement are in effect running together on a joint list. This is made clear if you look at the election through the lens of the “easy way” of allocating Knesset seats: a pair of lists who have a vote-sharing agreement compete in the election exactly as if they were one list.

But there are crucial differences between a vote-sharing agreement and an actual joint list, even beyond the obvious public relations aspects.

### Difference #1: The threshold

As I hinted to earlier, and as I said explicitly in my previous article about the threshold, a vote-sharing agreement can’t save you if you don’t have enough votes. If you sign a vote-sharing agreement but fail to receive 3.25% of the vote on your own, all of your votes are voided, and your partner party competes for VPS bids alone. In their split, the Arab parties are risking losing a large number of seats if Ra’am-Balad doesn’t cross the threshold.

### Difference #2: Seat allocation

When parties run together on a joint list, they undergo extensive negotiations to figure out exactly how to share the spots on that list. Some examples:

• In the Knesset elections that were held in 2013, the Likud and Yisrael Beiteinu ran together on a list colloquially referred to as “Likud Beiteinu”. It was agreed that every three names on their list would be made up of two candidates from the Likud and one from Yisrael Beiteinu.
• Every election, the two parties that make up UTJ (Agudat Yisrael and Degel Hatorah) renegotiate how many candidates each one will be allowed to place and in what slots. These negotiations are often accompanied by a lot of animosity and argument, though in the current elections they decided on a simple, alternating system: one candidate from Agudat Yisrael, one candidate from Degel Hatorah.
• In the 2015 elections, the four Arab parties that formed the Joint List implemented a very complicated coalition agreement in which some of the candidates agreed to resign midway through the Knesset’s term in order to be replaced by the next candidate on the list, coming from another party.

A vote-sharing agreement cuts through all of that. Rather than politicians deciding in advance how many candidates each party will submit and in what order, a vote-sharing agreement in effect enables the voters to decide: how many candidates do you want from Party A vs. how many from Party B?

We saw a striking example of this above, when I analyzed how many seats went to Kulanu and how many went to Yisrael Beiteinu. There’s no guarantee that a joint list of those parties would have resulted in the same proportion of MKs from each of those two parties.

### Difference #3: MK replacements

When multiple parties run on a single list, the resignation or death of an MK may lead to their replacement by a member of a different party.

For example, the Zionist Union’s list in the last Knesset contained MKs from two parties: Labor and Hatnuah. As a result, when Labor-derived MK Danny Atar resigned from the Knesset, he was replaced by Hatnuah-derived MK Yael Cohen Paran, because she was next on the list.

In most cases, the bond between the parties in a joint list is strong enough that its voters don’t care about something like that. But in other cases, it can present a problem. Some voters for the Jewish Home party, for instance, would not be thrilled to see the resignation of one of that party’s MKs result in a candidate from Otzma Yehudit entering the Knesset. Or one can picture a scenario in which the Blue & White party splits sometime in the next Knesset’s tenure, with Gantz and Lapid going their separate ways – and if somebody in Gantz’s party resigns, somebody from Lapid’s party replaces them.

With a vote-sharing agreement, however, two parties can run together – yet keep their separate identities and separate lists, so that an MK leaving the Knesset is always replaced by an MK from the same party.

### Difference #4: No more than two

This is an easy one: a vote-sharing agreement can only be signed by a pair of parties, whereas a joint list can be made up of as many as it likes.

### Conclusion

A vote-sharing agreement is, essentially, a way for two parties to run together without actually running on a joint list. It has advantages and disadvantages over running jointly, but the seat allocation benefits are equal to the ones you have when running as a single, larger party.

This concludes my series of articles explaining the Israeli election; with the election tomorrow (or possibly today, by the time you read this), I’m at least glad to finish explaining the basics in time.

I had to give up on a lot of posts, but some of these will be able to go up after the election:

• The Three Seats Scenario
• Assorted nonsense I’ve seen in articles reporting on the election
• The fundamental flaws in Israeli polling practices
• Analysis of voter movements

I deeply apologize for not having had the time to write and submit these. I’ve been rather busy.

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