In August 23, 2022, Meretz political party held primary elections for the list of candidates for Israel’s parliament, the Knesset. The internal elections were divided to two parts—one part to determine the identity of the party’s leader at the head of the list, and a second part to determine the order of candidates in the list following the party’s leader. Out of the twenty two candidates in the second part, six were former members of parliament.

It turned out¹ that the top places in the list were occupied solely by the former members of parliament. As one journalist summarized (Zaken, 2022) : “In practice, there is no real change in Meretz’s list, except for the return of Galon. All the realistic positions chosen are members of Knesset except Galon that was in the past, and there are no new faces”.

From the above summary and indeed from the resulting list, one may conclude that the voters (more than 15,000 members of Meterz party), preferred to have a list with no new faces in realistic places in the list. However, looking at the detailed election results, this conclusion appears to be incorrect. The primaries voting mechanism was quite simple: each voter chose exactly 4 candidates, without order between them and with equal weight. Of the total number of votes, 45% were given to “new faces”. So how come there were no new faces in the top of the list? Is such a result expected or should it be considered a rare case? This is what we’ll try to study here.

The representation of “new faces” in the list is a special case of a representation consideration having two options (say option A and option B). So we can frame a theoretical question as follows:

● Assume the existence of primaries in a political party to rank candidates in a list.

● Assume the candidates can be grouped according to the representation consideration options A and B.

● Assume each voter votes for a number of candidates (with equal weight between them), where the number of votes per voter is smaller than the number of candidates in each of the groups A and B.

● Assume that the top N places in the list are “realistic” places (i.e., good enough chances to have a place in the next parliament following general elections), desirably reflecting voters’ preferences.

● Define the ratio between total votes to candidates from groups A and B to the total number of votes as the preference of the voters for the relative numbers of groups’ A and B candidates in N first places in the list.

Under the above definition and assumptions, we can ask the following research question: “how well are voters’ preferences expected to be manifested in the election results?”.

Figure 2 shows the simulation results for identical parameters except for changing the number of places at the top of the list. While at 50% preference the simulation results do predict that half of the places at the top of list will be from group B, the results for different ratios differ from the expectations. For example, when the number of places at top of the list is defined as 6 (red line) and the preference for group B is 1/3, the voters’ expectation is that 2 places will be occupied by candidates from group B, but the simulation predicts an average of less than 0.3. The deviation is opposite for preference of 2/3 for group B, where instead of 4 group B candidates out of 6, the simulation predicts an average of

about 5.7. The reason for the discrepancies is that when the relative representation preferences differ from the ratio between the sizes of groups, the votes are factorized accordingly. Then, the non-linear sort mechanism for ranking the candidates amplifies this effect (as in “vote splitting”).

If the number of candidates in groups A and B is smaller than the number of places in the top of the list, then the latter will always be occupied by members of both groups, so the proportion cannot be zero or one irrespective of the voters’ preferences. This creates another type of discrepancy between the preferences and the expected average election results, as shown in Figure 3. For example, for number of places of 18 in the top of the list (yellow line), say the voters prefer to have 1/3 of the list composed of group B. In such case the list should have been composed of 6 candidates from group B and 12 candidates from group A. However, the simulation results show that the average number of group B candidates in such case would be close to 8.

In the former scenarios, the support for candidates in both groups was fully random with uniform distribution. How will a fixed bias affect the election results? By fixed bias, we mean that the support distribution is composed of a fixed part and a random part. Say the proportion of the random part in group A is α. Then, each of the candidates in group A will have an evenly distributed random

support on a scale between 1 − α and 1 calculated using the following formula: α·R + 1 − α, where R is a random number uniformly distributed between 0 and 1.

Figure 4 shows the simulation results when the proportion of randomness of support inside one of the groups is varied while the distribution of support inside the other group is random in the range [0, 1] with uniform distribution. It turns out that the less random the support inside the group is, the more its representation is augmented with respect to the expectation when it is favored (i.e. when voter would like to have more in the top of the list from this group than from the other group).

The deviation from voters’ preference is augmented further if the amount of randomness of support is reduced in both of the groups, as shown in Figure 5. For example, for 50% randomness in both groups, for preference to have 1/3 of the placed in the top of the list occupied by members of group B, the simulation predicts a negligible average of less than 0.1 instead of the desired 4 candidates.

In the former scenarios, the support for candidates in both groups was fully random with uniform distribution. How will a fixed bias affect the election results? By fixed bias, we mean that the support distribution is composed of a fixed part and a random part. Say the proportion of the random part in group A is α. Then, each of the candidates in group A will have an evenly distributed random support on a scale between 1 − α and 1 calculated using the following formula: α·R + 1 − α, where R is a random number uniformly distributed between 0 and 1.

In typical elections, the number of candidates in each group differ. Figure 6 presents the predictions of the simulation for changing divisions of a total fixed number of 24 candidates to group A and group B, from a minimum of 6 members to maximum of 18 members. The number of places at the top of the list in this simulation is selected to be 6 and both support distributions within the groups are fully uniformly random. As can be seen, all the expected election results differ from the voters’ preferences, with “benefit” to the smaller group in most of the preference ratios (while a small preferred ratio for a group always results in under-representation).

For example, for 9 members in group A (red line), if the voters’ preference is to have 40% percent of the places at the top of the list occupied by members of group B (i.e., 2.4 on the average), the simulation predicts an average of less than 0.5.

In the examples thus far, most of the factors were maintained as the original symmetric and fully random case, (where the elections expect to reflect the voters’ preferences well as shown in Figure 1), and one or two factors were changed to demonstrate their effect relative to this “base case”. Now we can combine a few factors to generate predictions for a few more elaborated cases of elections (and presumably more realistic or typical scenario).

Figure 7 shows a case of where 1/3 of the 24 candidates are in group A and the rest 2/3 of the candidates are in group B. The distribution of support in each of the groups is mostly random (0.7) but also has a substantial non-random bias (0.3). As can be seen, the average number of members of group B at the top of the list of 6 candidates is negligible below support level of about 50%. And this, if not understood correctly, may cause frustration and disappointment to the voters as they may be puzzled by the large discrepancy between their evaluation of the public preference and the results of the elections.

The distribution of votes for former members of Knesset (“MK”) and for “new faces” in the primaries in Meretz party on August 2022 (described in the introduction), are presented in Figure 8. Looking at the graph, we can evaluate that the distribution for MK’s has a bias of 0.5 (10% minimum relative to maximum of 20%), and the distribution for new candidates is without bias (while not perfectly evenly distributed).

Fixing the distribution inside each group and changing only the relative number of votes between the groups, we can simulate the election results assuming that

only the voters’ preferences related to the proportion of “new faces” change from the actual proportion of 45% to other ratios. The results are presented in Figure 9. It so happens, that with small additional increase of support for new faces, at least one new candidate would have entered in the top 6 places. Nevertheless, “new faces” would still suffer significant under-representation even if 60% of the votes would have been allocated to members of this group. On the other hand, if the ratio of votes to new faces would have been 69% or more, all the top list would have been occupied by new faces.

Now we can compare the speculated dependence of the results shown in Figure 9 with predictions of the simulation with approximations for distributions of

internal support in the groups—for example, using a fully random distribution for group B (“new faces”) and random with bias of 0.5 for group A (former members of Knesset). The results of simulation with these inputs are presented in Figure 10, and show negligible average “new face” candidates when the votes ratio for “new faces” is up to 45% (as when the simulation is based on the actual election distributions), as well as a list fully occupied with “new faces” candidates when the ratio is 75% or more (compared to ~70% when based on the actual distributions in the elections).

The representation of the Arabic minority in the list of candidates for the general elections is traditionally considered to be of high importance for Meterz voters. The 22 candidates can be grouped according to their ethnic origin: five of the candidates from Arabic ethnic origin (a somewhat higher ratio than in the general population) and the rest are of Jewish ethnic origin. The internal distributions of support within the groups, according to the number of votes for each candidate, are presented in Figure 11.

As can be seen, amongst the Arab candidates there was one dominant candidate as well as one candidate with markedly smaller support than others. Such distribution deviates from most of the distributions that are generated by the simple model of fixed base support plus uniform random distribution. Therefore, a simulation was carried out with the support for Arab candidates set according to the real election results, and the support for Jewish candidates modeled

by fully random uniform distribution. The size of “top of the list” is chosen as 5 places, since there are only 5 Arab candidates. The results are presented in Figure 12. The actual ratio of votes for Arabic origin candidates was about 25%, and indeed one candidate entered the top of the list of 5 places (in reality won the third place). Interestingly, the results are predicated to be close to the expectations (shown in broken line) around this ratio, which is not so surprising as the proportion of Arab candidates is close to the preference (5/22 = ~23%), so the effect of vote splitting is small. The simulation predicts that if the voters’ preference was smaller than 15% for Arabs then the representation was zero (although the expectation is an average of 0.75), while for voters’ preference of ~35%

or more, the representation of Arabs would have much exceeded the expected value.

A common representation consideration, also considered highly important in Meretz party, is gender representation. Meretz’s constitution forces the representation of both genders to be at least 40% of any 5 consecutive places in the list³. This complicates the comparison between voters’ preferences and election results, as voters may take the gender balance limitations into account in their allocation of votes⁴. However, for the sake of completeness of the study of Meretz primaries results, we will still assume that such considerations are either not highly prevalent, or are balanced between the genders.

³If the results do not fulfill the required proportions, candidates would be pushed up and down the list to meet them.

⁴For example, if a voter considers it highly important to have at least 40% women in the list but have no particular preference amongst the women candidates, the voter may decide to vote only for men, knowing that this choice will not jeopardize women representation and cause it to fall below 40%.

The distributions of support inside the group of 13 men and group of 9 women candidates are shown in Figure 13. Here, we note that two candidates have significant dominance over other women. The distribution of support for men is divided mostly to two main levels. Therefore, we chose to base the predictions of the simulation fully on the actual support distributions within the groups, and only modify the ratio of total votes between the two groups.

About 32% of the votes in the elections were given to women, which constitute

about 40% of the candidates. This, together with the specific distributions of support (in particular for women), results in relatively good correlation between the simulated predicted results and the expectations over most of the simulated ratios of votes for women, and an excellent matching for the actual case of 32% votes for women, as shown in Figure 14.

These observations still hold even when the distribution of votes in the men group is replaced by uniform random distribution with 0.3 fixed bias (Figure 15), but does not hold if both distributions are replaced by the simple uniform random plus bias model (Figure 16).