On Mathematics, Democracy, and Trojan Horses

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This time I will deal neither with philosophy nor with ideology or worldviews, but precisely with mathematics (very simple mathematics, with two exercises for the interested reader) and its applications to political ethics. In retrospect it became clear to me that this article is a direct continuation (though not planned in advance) of my two previous articles. Let us begin with a bit of mythology. The Odyssey tells of a brilliant stratagem by the Greek king Odysseus, through which the fortified city of Troy was conquered. The Greeks, who had besieged the city and failed to capture it, hid an entire unit of soldiers inside a large wooden horse that they placed at the foot of the besieged walls of Troy, and then ostensibly lifted the siege and sailed away. The Trojans concluded that they had won the campaign, came out beyond the walls with cries of victory, and brought the captured horse into the city. The rest is history—or rather mythology. From then until our own day, the expression ‘Trojan horse’ has been used to describe a foreign element, actually belonging to the rival, that is found within our camp and liable to bring about our defeat. What does this have to do with? A few months ago I received by email a circular letter from MK Motti Yogev in which he warned of undemocratic flaws in the proposed constitution of the Jewish Home (which has meanwhile been adopted by a large majority). In addition, he spoke about the importance of integrating Tkuma’s MKs into the party. I could not restrain myself (for fear of violating the injunction ‘do not make yourselves loathsome’), and in a reply email I remarked to him that since Tkuma’s representatives are determined by a committee of rabbis, Tkuma is not exactly a democratic model, and there is a certain contradiction in his words. In the course of the correspondence I argued that he and a few of his colleagues (such as MK Yoni Chetboun) are a Trojan horse within the Jewish Home party, since they are far closer in their views to Tkuma and its rabbis than to the Jewish Home under whose banner they were elected. On almost every issue over which the parties disagree—and in my opinion these are a very large majority of the issues on the table; see ‘The Anti-Religious Legislation’ and my article in that issue—he and Chetboun are on Tkuma’s side. So what are you doing in the Jewish Home? I asked. Why are you not placed among Tkuma’s MKs? To my amusement, the letter was published online and merited at least two articles on the Kipa website as though it were part of the argument over the constitution (which I had not known at all and whose very existence as a controversy I did not know about, so I had no position regarding it either). As a result of the publication, I received quite a few responses (including from Motti Yogev himself), and among other things many argued against me that these people were elected democratically by central committee members whose views these are (the Jewish Home is diverse and also has voters like that), and that in fact I am trying to impose my positions and views on the Jewish Home, which is composed of different shades (I will not weary you here with the slogans of unity that were thrown around there incessantly, whose relevance is itself open to question). The claim was that since they too are part of the Jewish Home and represent a proportional portion of its voters, there is no basis here for talking about a Trojan horse. Let me say in advance that I am certain that Yogev and Chetboun are honest people, and therefore I am left with no choice but to conclude that their words, like those of the other respondents mentioned here, are based on a mathematical misunderstanding. Since this is a fairly simple mathematical effect with broad ethical and political implications, I thought it worthwhile to clarify the error here and to point out some of its implications. Tkuma’s multiplied power. I have always wondered what people want from Moshe Feiglin and his friends who integrated themselves into the Likud and even seek to reach positions of leadership there. This is one current among Likud voters, and if its representatives are chosen there by due process then they are part and parcel of Likud’s democratic fabric. Ostensibly, those who criticize him want to impose their own views on the entire party. Although I disagree with Moshe Feiglin on several planes, I never understood what is ethically illegitimate about what he is doing. Democratic elections were held—primaries, a central committee, and other democratic institutions—and whoever is chosen is the one who best represents the voters. Simply perfect, no? But after my correspondence with Motti Yogev, the penny dropped. There is indeed a problem, a mathematical one—and in fact it is the same ethical error that I will now explain, though in the Likud it appears in a less problematic form. I will leave the explanation for that as an exercise to the reader at the end of my remarks. To understand the problem, let us return to the Jewish Home. The root of the matter is that Tkuma is a party that is run undemocratically. There is indeed some kind of central committee there, but there are no primaries; rather, a committee of rabbis determines the party’s representatives to the Knesset and the course of the party and its representatives. Let me clarify that even if I do not think this is the right way to act, that is not what I am discussing here. They have the right to be undemocratic, and I have no desire to interfere in Tkuma’s rabbinic leadership. But here is the problem: Tkuma’s potential voters seek a way to exert influence and do not find it within Tkuma, because the rabbinic committee determines everything there. What do they do? Obviously, they enter en masse into the central committee of the sister party, the Jewish Home, and vote in its primaries. The result is that the Jewish Home’s registered members are composed both of people whose outlook is that of the Jewish Home and of people whose outlook corresponds to Tkuma’s, and therefore the Jewish Home’s MKs are divided accordingly as well. In addition, on the list of the combined party—On Mathematics, Democracy, and Trojan Horses / Michael Abraham. Rabbi Michael Abraham holds a PhD in physics and works in general philosophy, Jewish philosophy, and logic. Psifas, Heshvan 5775, November 2014, 42.

Seats are also reserved for Tkuma’s own MKs (who, as noted, are determined by a committee of rabbis). What have we got? Tkuma’s path enjoys upgraded influence: it has substantial representation within the Jewish Home, and in addition it has Tkuma’s own Knesset members. In other words, the Jewish Home’s outlook has built-in underrepresentation in the Knesset. One must understand that when people discuss integration between the parties and the construction of a joint list for the Knesset, they examine in the polls the electoral strength of Tkuma and of the Jewish Home, and discover that Tkuma is worth a certain number of seats (say four, just for the sake of discussion). In such a poll, as also in Knesset elections (if the parties were to run separately), all the Jewish Home members who hold these views are counted as Tkuma voters. Therefore the electoral power of Tkuma’s outlook is built not only on its own voters but also on Jewish Home voters and central committee members who hold those views. Tkuma then demands proper integration, that is, a place proportionate to its electorate, which, as noted, stands at four seats. A most justified demand, no? And thus it receives representation in the Knesset that is not proportionate to the degree of public support it actually has. Even without entering into general formulas, a simple calculation can show the problem. Let us assume that the religious-Zionist and secular public that votes for the Jewish Home constitutes a certain segment of the Israeli population, say one million voters in all. Let this million be divided into two components, which we shall call X (those who hold the Jewish Home outlook) and Y (those who hold the Tkuma outlook). For the sake of discussion, let us further assume that Y has an electorate of about 30% of the million, and the rest support X. X is a democratic party whose representatives are chosen in primaries, whereas Y’s representatives and course are determined by a central committee (without primaries). How many representatives in the Knesset, dear student—argue, explain, detail, and prove—will the Y outlook have? Answer: if we assume that the combined party has 20 MKs, then 6 of them will be reserved for Y MKs (in accordance with its level of support in the population—the proper integration). Of the remaining 14 that represent X (again, the proper integration), there will of course be another 30% from among the central committee members who hold the Y outlook, since 30% of those who choose the X MKs espouse the Y outlook. That means another 4.2 MKs are added who represent Y. All in all, Y will have more than 10 MKs who represent holders of that outlook—more than half of the MKs on the joint list—even though its support among all voters is, as noted, only 30%. That is the entire theory of the Trojan horse. We have received an explanation of why, structurally, a non-democratic party always upgrades its power in the Knesset at the expense of its democratic sister party. For the reader (this time seriously, Exercise No. 1): prove that if the rate of public support for party Y is p (a fraction between 0 and 1), then its representation in the Knesset will be p(2-p), that is, always greater than its share of public support. If you are still not convinced, look at how many of the Jewish Home’s voters and registered members hold Tkuma views. In addition, see how many of the Jewish Home’s MKs hold Tkuma views (cf. Yogev and Chetboun, and from the above calculation I would wager there are roughly one or two more). All of these are in fact Trojan horses (and again, I believe unintentionally, because of a mathematical misunderstanding). Tkuma is represented in the Knesset by Uri Ariel and his colleagues, in addition to Motti Yogev and his colleagues, far beyond its true power in the public. This is the ethical implication of the mathematical misunderstanding. Now any child can understand that the claim that they were elected democratically and therefore represent one shade among the Jewish Home’s voters is irrelevant; it stems from a purely mathematical error. Without question, this is blatant unfairness (though in my assessment an innocent one). The Trojan horses among party X’s representatives were indeed elected democratically by the Trojan horses among its voters in the primaries. In war such a maneuver is certainly legitimate (so far as I recall, a Trojan horse does not violate the Geneva Convention; Odysseus will not be brought before the court in The Hague). After all, if it is permitted to ambush or kill in order to win a war, then it is certainly also permitted to deceive and carry out brilliant tricks of this sort in order to win. But in fair politics this is utterly illegitimate. In fair politics the aim should be faithful representation of the true distribution of opinions in the public, so that the decisions made will reflect what the public wants. There is no doubt that that is not what is happening here. Let us now descend from the heights of theoretical Olympus. Psifas, Heshvan 5775, November 2014, 43.

And take a short contemporary time-out. Naftali Bennett inserted into the constitution a clause that allows him to place a representative of his choosing in each group of five party candidates. As is well known, this is one of the clauses that aroused the anger of the opponents of the proposed constitution (Yogev and Chetboun and their friends). Very undemocratic, right? Let us now calculate: out of 20 MKs, 10 represent Tkuma instead of 6. One out of every five on Bennett’s account would remove 4 representatives here and insert Jewish Home representatives in their place. And presto—we are back at exactly fair representation. It is surprising to me that it is precisely the Tkuma supporters within the Jewish Home (Chetboun and Yogev), who created this representational distortion, who are protesting the attempt to correct it. As decent people (and in my opinion they really are such), they of all people should be pushing for adoption of the proposed constitution. My claim above about the contradiction in Motti Yogev’s words receives mathematical confirmation here. How does one correct the distortion? It is worth noting that this phenomenon occurs because of a merger between a democratic party and one that is not. The assumption is that in combinations of two non-democratic parties, or of two democratic parties, this effect does not exist. Proofs: a. The symmetry between the parties prevents the effect, because either both parties can plant Trojan horses in each other or neither can do so. But when there is one democratic party and one non-democratic party, the democratic party cannot influence the other’s ‘no primaries.’ b. If supporters of party X decide to vote in party Y, they lose the option of voting in party X (assuming the system carefully prevents double registrations, as in Israel today), and therefore if party Y’s representatives are also chosen by a democratic body—primaries—this of course will not happen. A person will not register with a party that does not suit him if by doing so he loses the possibility of influencing his own party. QED. But that is not the whole story. Ostensibly there is a distortion here created by a coalition between a democratic party and one that is not, and therefore the obvious solution is not to make such combinations. In a case of such asymmetry, let each party run separately with its own voters, and then there will be no underrepresentation of the democratic party and all will be well. But as the perceptive reader has surely noticed, this is a mistake. The Trojan horse effect is created by the very existence of a non-democratic party in our political arena, with no connection at all to the question whether it is combined with another party. We can see this again through the example accompanying us: even if the Tkuma party were to run separately, the problem would still remain in force. Its voters would still register with the Jewish Home and participate in choosing its representatives to the Knesset (because Tkuma is a non-democratic party and they have no ability to exert influence within it), while at the same time voting for Tkuma at the ballot box. In this way they achieve the same upgraded influence in the Knesset according to exactly the same formula. The surprising conclusion is that a merger between the parties is not a condition for the existence of the Trojan horse effect. The very existence of a non-democratic party on the map is what creates the effect (the electoral threshold introduces the merger between parties into the effect in an interesting way, but I will not go into that here). It is worth noticing the following non-trivial implication. The United States is a two-party state (Democrats and Republicans), and ostensibly if one of these two parties reaches a support level of 30%, it ought to despair. But do not worry: such a party has a sure way to take over the leadership of the state. It only has to abolish the primaries, become non-democratic, and send its supporters to vote in the primaries of the rival party (in fact they will go there even without being sent), and that is all. Alternatively, here in Israel as well, if there is one non-democratic party in the Knesset whose support in the general population—It is precisely the Tkuma supporters within the Jewish Home (Chetboun and Yogev), who created this representational distortion, who are protesting the attempt to correct it. As decent people (and in my opinion they really are such), they of all people should be pushing for adoption of the proposed constitution. Psifas, Heshvan 5775, November 2014, 44.

Selected trips for the coming winter: Vietnam, Thailand, Portugal, Brazil, Argentina, Mexico

Is 30% (Arabs, ultra-Orthodox, kibbutz members, redheads, or any other sector), it can activate the Trojan horse maneuver and receive the leadership of the state with an absolute majority (without a coalition) on a silver platter. The conclusion from what we have said so far is that the only thing that can prevent this problematic effect is the democratization of all the parties in the country. There must be a law requiring all parties that run for the Knesset to conduct themselves democratically (through primaries); otherwise the very existence of parties that do not operate that way automatically brings about underrepresentation of the democratic parties. Another possibility, of course, is to determine that all of them will be run by a central committee or a nominating committee, but there must be symmetry among all the parties that run for the Knesset. What do we do until there is such a law? There is no choice but to rely on the fairness of the representatives and of their voters from all the non-democratic parties (that they will not employ the Trojan horse maneuver), that is, that each person vote for the party for which he is registered, and register only for the party for which he votes. That is a requirement that is hard to enforce, and it certainly does not hold in our case. The concluding sentence is that if a person supports a party that is conducted undemocratically, he must be very fair and very skilled in basic mathematics. Alternatively, in order to allow non-democratic parties to exist, we must have full trust in the fairness and mathematical competence of all their voters. I assume I will be forgiven if I say that I would still prefer a law. In closing, I will also connect these matters to my article (and thereby turn this series into a trilogy). In my article there I defined the philosophical root of fundamentalism as the suspension of critical thinking. A citizen who supports a party that is run not through his influence but by a committee of rabbis is essentially giving up his ability to influence and criticize, and in that sense he is a kind of fundamentalist. From this it follows that the Trojan horse effect expresses an inherent advantage of a fundamentalist group over a parallel group that advocates critical and open (and rational) thinking. A similar phenomenon can be seen in two additional contexts: the Muslims flooding Europe and taking part in the democratic process that is taking place there, thereby creating overrepresentation of Islam on the world map and in effect underrepresentation of democracy at the UN (the implications for our standing there, in the press, and in international discourse are clear). This is also the situation between us and the Palestinians. Palestinian society, even if we assume that there is some measure of democracy in it, certainly is not prepared to accept Jewish residents into its midst, even those who purchased their homes legally, and certainly not to give them political power (think of a Jewish party running in elections to the Palestinian Authority). In the discourse between us and them, the situation is that facing this group stands—meaning fights, or negotiates and bargains—an Israeli society that is democratic, part of which consists of Palestinian Arabs who are citizens of Israel and who receive influence, civil rights, and representation in the Knesset and in government. Therefore the Israeli position itself partially reflects the Palestinian position. In more pictorial language, there are in fact Palestinians on both sides of the negotiating table (interesting why it is nevertheless stuck?), and now the Palestinian citizens of Israel come and claim—not in a way disconnected from the facts—underrepresentation and inequality in Israeli governmental institutions. Does that remind you of anything? To conclude, let me clarify that I do not support such discrimination on the formal level, that is, to anchor in legislation that Arabs will not be able to participate in the Knesset or even in the government. I am in favor of democracy and equality for all citizens, and the character of our state should be determined by all the citizens and not only by some of them. In other words, it is hard to accept a demand for democracy in the Palestinian Authority as a condition for including Israel’s Arabs in our own democratic game. By the same token, it is not reasonable (nor practical) to demand that they move there and vote there. So what should we do in order to prevent the Trojan horse effect—give up our democracy? That too is unacceptable to me. Therefore the similarity to the case of Tkuma and the Jewish Home is not complete. But I would be untrue if I said that I am surprised that in practice Zionist parties and leaders are not inclined to include Arabs in the government. Who wants his rivals to conduct negotiations for him from both sides of the table and jointly reach decisions about his fate?! Troy is here. For the reader (again, this time seriously, Exercise No. 2): think now about Feiglin’s case in the Likud, and explain why the problem there is smaller than in the case of the Jewish Home and Tkuma. In order to allow non-democratic parties to exist, we must have full trust in the fairness and mathematical competence of all their voters. I assume I will be forgiven if I say that I would still prefer a law. Psifas, Heshvan 5775, November 2014, 46.