Daniel Sterman
Electoral Math

Counting those ‘lost seats’

Netanyahu blamed right-wing voters for squandering 7 mandates on parties that didn't pass the electoral threshold. That's not even close to being true
Illustrative. Prime Minister Benjamin Netanyahu waves to his supporters after polls for Israel's general elections closed in Tel Aviv, on April 10, 2019. (AP Photo/Ariel Schalit via Jewish News)
Illustrative. Prime Minister Benjamin Netanyahu waves to his supporters after polls for Israel's general elections closed in Tel Aviv, on April 10, 2019. (AP Photo/Ariel Schalit via Jewish News)

Last week, Prime Minister Benjamin Netanyahu wrote the following in a front-page opinion piece for Israel Hayom:

We cannot repeat the mistake of the last election when right-wing voters squandered seven mandates on parties that failed to pass the electoral threshold.

This is not even close to being true.

Netanyahu isn’t alone in this claim, of course. He is only the latest in a long line of people who have misrepresented the impact of the two parties who failed to cross the threshold in April’s elections, the New Right and Zehut. Other culprits: Susan Rolef, Caroline Glick, and David Bitan. Dishonorable mention goes to David Israel, whose recent article used polling estimates for the next election to lament seats that haven’t even been lost yet – most of which, as you will see below, won’t be lost at all.

Trurl’s Machine

The problem is that everybody in Israeli politics has a vested interest in playing up the importance of the threshold. Large parties go on and on about it to scare people away from voting for small parties, while small parties go on and on about it to convince voters of how important it is to help them cross it.

But the truth is that the threshold really does not matter as much as everyone says.

Now, it is indeed true that the New Right received 4 seats’ worth of votes – votes that ended up being ignored when the seats were allocated. It is also true that Zehut received 3 seats’ worth of votes, totaling 7.

But that doesn’t mean there were 7 seats missing from Netanyahu’s coalition.

As I explained in my article about the threshold, Knesset seats have to go somewhere. If a party list fails to cross the threshold, those seats are instead picked up by other parties. This means that before you get too mad (or gleeful) at Bennett’s and Feiglin’s failure to enter the Knesset, before you blame (or credit) them with Netanyahu’s failure to put together a coalition, you should first check who ended up sitting in their seats instead.

After all, three of the four largest parties in the short-lived 21st Knesset were right-wing, and the fourth didn’t have a vote-sharing agreement. Since larger parties have a better chance of getting leftover seats than smaller parties do, one can easily picture a scenario in which all seven of those lost Knesset seats went instead to other right-wing parties, making the entire thing a wash.

Furthermore, one mustn’t ignore the other party that fell below the threshold, Gesher – which picked up two seats’ worth of left-wing votes. If you want to assess the damage done by Zehut and the New Right to their bloc, you have to take Gesher’s lost seats into account as well.

So how do we find out what really happened? Simple: We give out all 120 seats as if there were no threshold, and see which parties get fewer seats. I won’t go through the full calculations so as not to bore you, but here are the results:

The parties that picked up the lost seats are Blue & White (3), the Likud (2), and Shas, UTJ, Chadash-Ta’al, and the URP (1 each). That’s 4 seats going to the Left and 5 seats going to the Right.

And where did those seats come from? 7 from right-wing parties (the New Right and Zehut) and 2 from a left-wing party (Gesher).

In other words: thanks to the threshold, the Right gained 5 seats but lost 7, while the Left gained 4 seats but lost 2. In other other words: the right-wing bloc lost a grand total of two seats to inefficient voting.*

Now, two extra seats would have been enough for Netanyahu to form a coalition without Lieberman. But to get those two seats, Netanyahu would have had to negotiate coalition deals with two more parties! And given the very public enmity between the prime minister and both Bennett and Feiglin, it’s not clear that he would have succeeded.

In which case we’d have ended up exactly where we are now: awaiting new elections.

That’s Exhibit A of my thesis, entitled “the threshold doesn’t really matter”. Now for Exhibit B.

Gaining seats

There’s another angle by which the threshold is frequently misunderstood, one that the larger parties have successfully embedded in the Israeli psyche over the course of generations: the idea that votes below the threshold are somehow “wasted”.

Consider Jonathan, who crawled over broken glass to make it to the polling place and cast his vote for Blue & White. Had he stayed home and not voted, Blue & White would have had one less vote, taking it from 1,125,881 votes to 1,125,880. This would have decreased Blue & White’s Knesset seats from 35 to… still 35.

Was Jonathan’s vote “wasted”? What if he had voted for Gesher? Or the Pirate Party? In all three cases, his vote wouldn’t change the outcome of the election.

So why do people feel as though voting for either of the latter two is a “waste”, merely because they didn’t cross the threshold – but voting for Blue & White (or Labor, or Shas, or any other party where that extra vote wouldn’t get them an extra seat) is perfectly valid?

Well, part of the reason is that Jonathan doesn’t have perfect information. He didn’t know exactly how many votes Blue & White would get. In making the effort to vote, he was playing the odds, hoping he’d happen to be the one extra vote that made a difference in the number of seats.

This hints at a deeper truth: whether or not a vote is wasted depends on the probability that your vote will change how many seats they got – not whether this ended up actually happening. If you knew with 100% certainty, for example, that a party will get between 12.1 and 12.3 seats, then voting for them is definitely a waste, because it won’t affect the outcome. But we don’t have that sort of certainty. Generally we only know that a party will get somewhere between 10 and 14 seats. Which means that voting for them is worth a try, because you just might be the one to tip them over from one number into another. It’s not a waste if there was a chance.

Most parties have ranges of similar width, which encompass three or four different numbers of seats. (Sectorial parties such as UTJ have a much narrower ranges.) So what are the chances that your vote will be worth something?

In the case of Blue & White, or any other party that is comfortably ahead of the threshold, the chances are about one in thirty-five thousand – because that’s approximately the number of votes you need to go up a seat. Those are the odds that your vote will hit an inflection point, that magic number of votes that makes a difference. A party that polls between 6 and 8 seats will have inflection points from 5 to 6, from 6 to 7, from 7 to 8, and from 8 to 9; any individual’s vote has the same small chance of hitting each of those four points.

But a party whose range of possible results touches the threshold has a different kind of inflection point: one that jumps it straight from 0 seats to (usually) 4. If you hit that magic number, yours is a far more powerful vote – worth four times as many seats as any vote belonging to a larger party!

At the same time, however, such a party is missing other inflection points: between 0 and 1 seats, between 1 and 2 seats, between 2 and 3 seats, and between 3 and 4 seats. If a party is below the threshold, it won’t make it into the Knesset at all regardless of which side of those inflection points it’s on.

So is it worth it? Is it worth trading four inflection points for a single inflection point of quadrupled strength?

We can find out using a probability distribution: a graph that shows the likelihood that a party will reach any given status over a range of possibilities. (For probability experts in the audience: I will be deliberately simplifying things here. Don’t send me angry letters.)

For instance, a run-of-the-mill party’s probability distribution might look something like the graph to the right. It shows a range of potential vote counts for the party, from a vote count worth less than 10 seats (all the way to the left) to a vote count worth more than 13 (all the way to the right). The blue curve tells you how likely each of those possibilities is: the higher the curve above a given possibility, the greater the chances that that possibility will come about.

Most possibilities aren’t particularly interesting. But we care about the four inflection points, which are the vertical lines at the bottom of the graph. That’s where the number of votes makes a difference. A party moves from left to right on the graph as it accumulates more votes, and if your vote pushed it just past one of those vertical lines, congratulations – you gave your party an extra seat.

In this case, the curve isn’t very high at points 10 and 13, but point 12 is very close to the peak, and point 11 also has a respectable value. This means that your vote has a pretty good chance of being the one to help the party reach 12 seats, a smaller chance of helping it reach 11, and very little chance of helping it reach 10 or 13 seats.

How can we measure this more precisely? Take the height of the curve at each inflection point. Stack these four heights on top of each other, and you get a relative assessment of how valuable your vote for this party will be. The bulk of your vote’s value comes from the red line (above 12); notice the very tiny contribution that the green line (above 10) and the purple line (above 13) provide.

But that’s for a party comfortably ahead of the threshold, with regular inflection points. What about a party that’s just close to the threshold?

The graph to the right shows a similar probability distribution, but for a much smaller party: one that is struggling to make it to the Knesset. Any result lower than 4 means it doesn’t cross the threshold, and gets no seats. That means that it has far fewer inflection points – none at 1, 2, or 3 seats. But the first inflection point that it does have, at 4 seats, is far more valuable: it’s worth four times as many seats!

In this case, therefore, we can’t simply measure the height at each inflection point; since one inflection point is worth four times as many seats, we need to multiply its height by four to see how valuable it really is.

And the answer: ridiculously valuable. More than twice as valuable as a vote for a standard party above the threshold.

So it would seem that the conventional wisdom is wrong! On the contrary, you should davka – I wish there were a good English word for davka – you should davka vote for parties that are under the threshold!

But not so fast. What if a party is firmly below the threshold and only has a faint hope of reaching it? That party’s probability distribution might look like this:

You can see that, when the chances of reaching 4 seats are fairly remote, even multiplying that inflection point’s value by four doesn’t help much. A vote for this party is far, far less valuable than a vote for a party above the threshold.

Now, I can’t give you exact numbers for any of the above. I can’t say that a vote for UTJ is worth X seats while a vote for Meretz is worth Y seats and a vote for Otzma Yehudit is worth Z seats. The actual probabilities are far, far more complicated, and usually aren’t nice, symmetric curves like the ones depicted above.

But the important takeaway isn’t the numbers, it’s the general concept. You can’t take a simplistic view. “A vote under the threshold is wasted” (the position taken by the largest parties, for obvious self-serving reasons) is obviously wrong. But then so is the opposite position, that “a vote under the threshold is more valuable” (espoused by Moshe Feiglin, for equally obvious self-serving reasons).

You have to look at the full range of possibilities. And vote for who you want.


* One of Gesher’s lost seats went to the Likud, so in actuality the right-wing bloc lost three seats to its own inefficient voting but got one back because of the left-wing bloc’s inefficient voting.

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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