Line by Line

The results themselves: Polling Place by Polling Place and Station by Station.

Israel’s Knesset has 120 seats, and a government needs the confidence of 61. No list has won 61 alone, so every government is a coalition. Since 2019 Israel has held six elections, five of them in under four years. Between April 2019 and the summer of 2020 there were three elections and, for a year, no government. The question is why, when the seats themselves allow hundreds of majorities.

Lists as points, red lines as edges

Draw each list in the Knesset as a point. Whenever two lists will not sit in the same government, join them with a line. A government is then a set of points with no line between any two of them, worth at least 61 seats. Mathematicians call a set with no lines inside it an independent set, and the question “is there a government?” becomes “is there an independent set weighing 61 or more?”

That question is hard in general, but not with eleven lists. There are $2^{11} = 2{,}048$ sets of eleven lists, and a computer checks them all in a fraction of a second. So does a short argument by hand, as below.

Two kinds of constraint are needed, not one. A veto says two lists will not sit together: that is an edge. A loyalty says a list will sit only in a government that includes another one: Shas will not join a government without Likud. Loyalty is not an edge, but it is just as easy to check.

Four red lines

These four lines are applied the same way to all six elections, changing only which lists they cover each time:

Each is a simplification of something parties said and did. Benny Gantz “had promised during the election not to join a Netanyahu-led coalition”; the April 2019 negotiations broke on “fractures over Haredi conscription within the Netanyahu coalition”; in 2021 the Religious Zionist Party “ruled out being a part of any right-wing majority which would rely on” Ra’am’s outside support. The lines are not laws. The interesting thing is what happens when one of them gives way.

Six elections

“Ways to 61 by arithmetic” counts the minimal winning coalitions: sets of lists with 61 seats or more in which every list is needed, so that dropping any one of them falls below 61. The rest of the table is what is left once the four lines are drawn. In every election the best the lines allow is Netanyahu’s bloc, and the question each time is whether it reached 61, and if not, who would cross a line to make up the difference.

April 2019. Netanyahu “received recommendations from leaders representing 65 seats”, including Avigdor Lieberman’s five. But the bloc with Lieberman was a government only if the draft line gave way, and it did not. Without Yisrael Beiteinu the bloc had 60, one short. Breaking the draft line would have given the whole bloc, 65. The Knesset voted 74 to 45 to dissolve itself in the early hours of 30 May.

September 2019. The bloc fell to 55. Lieberman, now holding 8 seats, called for a “broad liberal government” of Likud and Blue and White, and did not want any majority with the Arab parties. Gantz and Yair Lapid would not join Netanyahu with the religious parties, and Lapid said that “if Netanyahu steps aside, we’ll have a unity government”. Every route needed a line to break, and none did. Netanyahu and then Gantz returned the mandate, and when the deadline ran out at midnight on 11 December, the Knesset dissolved again.

March 2020. Gantz was asked to form a government after he “received support from 61 of the 120 MKs”, with the Joint List among them. But a recommendation is not a coalition. Two of his own members, Zvi Hauser and Yoaz Hendel, said they “would not support a government that relied on the Joint List’s support”, and Orly Levy would not back a minority government. In the graph, a list is not one point when its members disagree. Take those three members out of their lists and the other side, even counting the Joint List, falls below 61. So Gantz broke the Netanyahu line instead. He signed with Netanyahu on 20 April, with Netanyahu first in a rotation, and the government was sworn in on 17 May. It lasted until the Knesset dissolved on 23 December, before Gantz’s turn came.

March 2021. This was the most fragmented Knesset, with thirteen lists and 742 minimal majorities. With all four lines holding, both sides fell short: the bloc with Naftali Bennett’s Yamina had 59, and the other side 58. Each side had the same way out, Ra’am’s four seats, and on each side it meant breaking the Arab line:

Religious Zionism refused. Bennett, Gideon Sa’ar and Lieberman did not. The second line of that table is the government that was sworn in on 13 June, with Bennett as prime minister on Yamina’s seven seats, and it is the only coalition the model finds on that side when the Arab line gives way for Ra’am but not for the Joint List. It was confirmed by 60 votes to 59, with one Ra’am member abstaining and one Yamina member having already ruled out supporting it. The Knesset voted to dissolve itself a year later.

November 2022. The bloc won 64 seats, and the model finds exactly one government: Likud, Religious Zionism, Shas and UTJ. No line had to break. Part of the reason was the threshold of 3.25% of the vote that a list needs to win any seats. Meretz won 3.16% and Balad 2.91%, so both elected no one. With no threshold, the four lists of the bloc would have had 60 seats.

2026: a proof that there is no government

On the final count: Likud 27, Yashar! 21, Together 12, the Joint List 11, the Democrats 8, Ra’am 8, Otzma Yehudit 7, United Torah Judaism 7, Yisrael Beiteinu 7, Religious Zionism 6, Shas 6.

The four lines here are:

• N: Yashar!, Together, the Democrats and Yisrael Beiteinu will not serve under Netanyahu.
• D: Yisrael Beiteinu will not sit with Shas or UTJ.
• A: Likud, Otzma, Religious Zionism, Shas, UTJ and Yisrael Beiteinu will not sit with Ra’am, and the Joint List joins no government.
• L: Shas, UTJ, Otzma and Religious Zionism sit only with Likud.

With those, the proof needs two cases.

1. Likud is in. By N, Yashar!, Together, the Democrats and Yisrael Beiteinu are out. By A, so are Ra’am and the Joint List. What is left is Likud, Otzma, Religious Zionism, Shas and UTJ: 53.
2. Likud is out. By L, Shas, UTJ, Otzma and Religious Zionism are out too. By A, so is the Joint List. What is left is Yashar!, Together, the Democrats, Yisrael Beiteinu and Ra’am, 56, but by A Yisrael Beiteinu and Ra’am cannot both be in. The best is 49.

Neither reaches 61, so with all four lines holding there is no government. ∎

The proof is short because the lines cut the Knesset into two pieces that cannot be joined: Netanyahu’s bloc on one side, everyone else on the other, and neither is big enough. That is also the shape of 2019 and 2020. What has changed is the size of the pieces.

Which line gives way

Three routes open when lines break, and each is a scenario already being discussed in the Israeli press:

The first route now has a public advocate inside Likud. On 29 October Haim Bibas, the Likud mayor of Modi’in and chair of the Federation of Local Authorities, and Shai Hajaj, head of the Regional Councils Centre, called on the party leaders to form “a broad national unity government”, because “the public has had enough of repeated rounds of elections” and another round “will not change the picture”. In the language of this post their call asks for a line to break: “a dispute cannot become a permanent boycott”. It does not say who would lead, and that is the one question on which the Netanyahu line turns.

The Haredi route is the only one that needs two lines to break, not one. With Ra’am in place of Yisrael Beiteinu it reaches 62 instead, but then the Haredi lists have to sit with an Arab list. Breaking the loyalty line alone also gives 61 on paper, with Yashar!, Together, the Democrats, Otzma Yehudit, UTJ and Religious Zionism, but only because the four lines say nothing about Otzma sitting with the Democrats. Nobody is proposing it.

One route does not appear in the graph: a minority government, with the Arab lists supporting it from outside rather than joining. Investiture needs more votes for than against, not 61 members in the coalition. Yashar!, Together and the Democrats, with the Joint List and Ra’am voting for them, have 60; Netanyahu’s bloc and Lieberman have 60 against. A tie fails. With Lieberman voting for, it passes 67 to 53. So the minority route also comes down to the Arab line, and to Lieberman.

What the mathematics says

1. Arithmetic is never the problem. There are always dozens or hundreds of majorities. Fragmentation does not stop coalitions forming; red lines do.
2. Four lines are enough to explain six elections. Applied the same way each time, they leave no government in 2019 or March 2020, two in 2021 once one line gave way, one of them the government that formed, and exactly the one that formed in 2022.
3. Since 2019 the graph has had two pieces. Netanyahu’s bloc, held together by loyalty, and everyone else, held apart by the Netanyahu and Arab lines. A government forms when one piece reaches 61, as in 2022, or when someone builds a bridge by breaking a line, as Gantz did in 2020 and Bennett, Sa’ar and Lieberman in 2021.
4. Bridges are expensive. Gantz’s ended before his turn as prime minister came. Bennett’s made him prime minister on seven seats, and its majority of one lasted a year.
5. In 2026 both pieces are small. The bloc has 53, its fewest since 2019, and the other side’s best without an Arab list is 49. So the question is again who builds the bridge, and at what price.

How this was done

The seats for 2019 to 2022 are the official results, as computed by bechirot.fyi from the committee’s files. The 2026 seats are the committee’s final count; the official result goes to the President on 4 November. The four lines are a model: each election’s version names the lists on each side of each line, and every coalition in this post was found by checking all the sets of lists against them. Changing one line’s lists changes the answer, and the scripts make it quick to try.

Quotations are from the Wikipedia articles on the April 2019, September 2019, 2020, 2021 and 2022 elections and the thirty-fifth and thirty-sixth governments, and on 2026 from Mako, i24NEWS, Ynet, Srugim and Al Jazeera.