Is It Worth Going to Vote? Another Look at Hindsight Statistics (Column 210)
With God's help
Today I received on the site a current question in the wake of the counting of the soldiers' votes:
I have read in several places in your name that, rationally speaking, there is no reason to go vote, since the chance that a single vote will make a difference is negligible. And yet in these elections Bennett is on the knife-edge of a single vote. What do you say about that?
Since this question is so common, and it touches on the essence of several probabilistic fallacies that have already come up here more than once, I decided, instead of answering the questioner directly, to devote a very short column to it and post it on the site in honor of the elections that have just passed, and to the glory of the State of Israel.
My argument
The questioner means to challenge my claim (see column 122 and in the forthcoming column) that, from the standpoint of outcomes, there is no point in going to vote, because one vote does not make a difference. The chance that it will have an effect is roughly 1 divided by the number of votes equivalent to one seat (more than 30,000 votes). The explanation is simple: my individual vote matters only in the case where the number of votes received by the party I favor, without me, is exactly one vote short of a whole-number multiple of the votes needed for a seat. In that case, and only in that case, my vote can change the situation. Clearly, the probability of obtaining such a result (assuming that the distribution of possible results is uniform—not accurate, but a reasonable assumption for our purposes) is, as stated, about 1/30,000. For a probability on that order of magnitude, people are willing to risk their lives every day (by going out onto the road). I would not invest a second of my life in it.
And we have not yet taken into account that in the case before us this involves crossing the electoral threshold and not merely changing the result, so the relevant resolution is four seats rather than just one (has stood against us to destroy us). That of course lowers the chance of making a difference still further.
Let us now return to the question.
A preliminary argument
First of all, I should note that we are not speaking here about dependence on literally a single vote. 'A few votes,' for this purpose, means hundreds or thousands. So long as the situation is not one of dependence on literally a single vote (that is, when the number of votes is not exactly a whole-seat number), there is not even a counterexample here. What is the difference between dependence on 300 votes and dependence on 12,547 votes? Each of these is one outcome out of 30,000 possible ones.
But for the sake of the discussion here, let us assume that the astonishing result really did occur, in which my poor, solitary vote carries Bennett and Shaked past the electoral threshold. Would that actually prove anything against my claim?
The law of small numbers
In column 38 I pointed out that counterexamples are a dangerous argument, one that is exposed to quite a few biases. In general, for a counterexample to function as an argument, it has to be representative. Sometimes it has unique characteristics, and therefore it teaches us nothing about the general principle. I gave examples there from alternative medicine and from the phenomenon of 'Breaking the Silence.'
But here the example appears, ostensibly, to be representative. It has no unique characteristic, and therefore it would seem to show that my argument—that such an outcome is not worth the effort—does not stand the test of the facts.
Rabbi Shach on Entebbe
I am fond of opening with Rabbi Shach's argument regarding the Entebbe operation. In his Iggerot U'Mikhtavim, sec. 2, it is reported that he was asked a similar question regarding Operation Entebbe. In the discussions beforehand he apparently opposed the operation, arguing that it involved risking soldiers' lives for the sake of a slim chance of success. After the operation succeeded, people came to him and proved to him that he had been wrong, since the operation had succeeded. Rabbi Shach gave an immortal answer: I too argued that there was a 5% chance of success, but a 95% chance of failure. When one says there is a 5% chance of success, that means that sometimes it does succeed; but given the conditions that prevailed before the operation, it was correct not to embark on it. That is hindsight wisdom.
Note carefully: he did not mean to justify his mistake by saying that only after the operation did we become wiser. He argues that it was not a mistake at all. Given the data that existed before the operation, it was not correct to embark on it. What happened afterward proved nothing at all about that consideration. The 5% possibility simply materialized here.
The die example
This is exactly like a dispute between Reuven and Shimon about the results of rolling a fair die. Reuven claims that 3 will come up five times in a row, and Shimon claims that this is improbable. They rolled the die, and astonishingly that exact result occurred. Does this prove that Shimon was wrong? Certainly not. The calculation remains correct even after the roll, except that the small probability was indeed realized (which by definition can sometimes happen). One must remember that any result we obtain from five rolls will have a very small a priori probability of occurring.
But there is an important point here. With regard to the die example, I assumed that it is known that the die is fair, and the dispute concerns only the question of the result of the probabilistic calculation (what is the probability that 3 will come up five times in a row). In such a case, the results teach us nothing about the dispute, because the a priori calculation remains exactly as it was. But if the dispute between Reuven and Shimon had been about the fairness of the die itself (Reuven claimed that it was not fair, and Shimon claimed that it was), then in such a case the results of the roll certainly do lend support to Reuven's claim (though of course not absolutely).[1]
Back to Entebbe
If we return to Entebbe, ostensibly there too the matter depends on the question of what exactly the dispute was about. But now it seems that this is not quite so simple. If the dispute was about the justification for undertaking the operation given a certain agreed probabilistic calculation, then the result of course proves nothing. And if the dispute was about the probabilistic calculation itself (what the chances of success were), then there is room for discussion. Ostensibly this still proves nothing, because the result could be accidental. But that is not precise. To understand this, we must note that the probabilistic calculation in this case is far more complicated than the one for the die. The calculation for the die is certainly correct (assuming it is fair), and therefore if Reuven disputes it, he is simply mistaken. The result cannot prove that a valid mathematical calculation is incorrect. But in relation to a military operation like Entebbe, the calculation is far more complicated. Therefore there is room here for a dispute about the assessment, and if the operation succeeded, that lends some support (though of course rather slight support) to the claim that there was a good chance it would succeed.
In columns 144–145 I discussed the issue of conditional probability (see also column 176). Here too, the question on the table is not what the chance of success was for Operation Entebbe, or what the chance is that the die will land on a sequence of five 3s. The question is the reverse: given that this happened, what is the probability that the die is not fair (or that the calculation of the chances of success in an operation like Entebbe is such-and-such). The non-trivial and confusing relation between these two questions was discussed in those columns.
And back again to Bennett
My argument about the lack of influence of a single vote was described above. Now the true results arrive, and we find that the New Right party is in a situation where it depends on a few votes (I do not know what 'a few' means. A few hundred? A few thousand? See above). Does this prove that I was wrong?
Again, ostensibly this depends on what the dispute was from the outset. If the dispute was whether it is worth investing in such a small chance, the result proves nothing. It happened because even a small chance is sometimes realized, but that does not prove that a priori it was correct for me to decide to invest my time for such a small chance. And if the dispute was about the chance itself, then we are dealing with a mathematical calculation, as in the case of the die. That calculation is completely technical, and the practical result proves nothing about it. So in our case, whatever the dispute may have been, the result is irrelevant to it.
And in shorter form: any result that occurs in the end will have an extremely tiny probability. Suppose the New Right had received 95,437 votes. The probability of that is negligible. Does that prove that the probability calculation is mistaken? No. It only means that a small probability was realized (in this case every result has a small probability, and therefore one can say in advance that here a small probability will be realized. Exactly like the results of a hundred rolls of a die).
[1] If we were to roll the die a thousand times, and within that obtain the sequence under discussion once (five 3s), that of course would not prove anything even with respect to the second dispute.
Discussion
Interesting as usual!
I agree in principle, but with one small reservation. You wrote: “And we have not even taken into account the fact that in the case at hand we are talking about crossing the electoral threshold and not just an ordinary change in the result, so we are dealing with a resolution of four seats and not just one (someone rose up against us to destroy us). This of course lowers the chance of influencing things even further.”
In my opinion, the calculation is different:
Before the election, polls are published that give us a certain estimate regarding the number of votes each party may receive. Therefore, if according to the polls Bennett and Shaked are supposed to be near the electoral threshold, then the estimation error is not 30,000 votes times 4 (seats). I take as given the expected behavior of the environment around me (which does not read your articles) and is expected to vote. Therefore I am quite certain, with a high confidence level, that the New Right will get three and a half seats (say). Now, the chance that my vote will matter is one in half a seat, that is, 1:15,000 (and not one over 4*30,000).
The consequence that depends on this probability is significant: if by voting for Likud or for Gantz I can, with a tiny probability, add one more seat, then here, with a greater probability, I can add 4 seats.
What do you think?
Personally, I did not vote, and as R. Gedaliah Nadel replied to someone: “If it doesn’t disgust you, vote.”
Read the comments and corrections in his latest column and you’ll see that no. He is calculating something different from what I’m talking about.
First, my articles do not say not to vote. They say that the obligation to vote cannot be based on consequentialist considerations.
Beyond that, indeed, when there is additional information (such as the polls), the probability changes. But that is only with respect to the comment about the electoral threshold. Beyond the question of the threshold, the fuzziness of a seat is always there. The polls cannot provide information at a resolution finer than one seat.
After reading a serious body of literature, the bottom line is that there is indeed still no convincing explanation for why millions of people bother to vote when the probability of being pivotal is negligible. In the literature this is called the “paradox of voting,” and it is compounded by the fact that in all Western countries (except one, I forget which one…) turnout rates in general elections are much higher (sometimes by dozens of percentage points) than turnout rates in municipal elections.
This is analogous to the behavior of a member of a cartel. Every member’s interest is that all the other members keep the agreement and do not exceed the quotas allotted to them, while “only he” sells “a little” more (since his “small” deviation will not significantly affect the price). The problem is that this reasoning applies to everyone, and therefore cartels tend to collapse (unless there is coercive power holding them together, such as the mafia and the like). In game theory this is called an “unstable equilibrium.” So too in our case: every eligible voter’s interest is that everyone else who thinks like him vote for the party he holds dear, while “only he” enjoys a day off without bothering to go to the polling station, “since in any case the probability of being pivotal is negligible.” But… that reasoning applies to everyone! Yet here, for some reason, millions of people around the world still stand in line at polling stations.
By the way, years ago I heard Prof. Yisrael Aumann speak about this issue, and he distinguished between “the rationality of individuals” and “the rationality of rules.” I do not remember everything clearly, but it seems to me that he mentioned evolutionary game-theoretic theories that explain that we are “programmed” to behave altruistically, according to the group’s interest and not necessarily according to the selfish interest of each individual. You see this kind of behavior among animals as well (for example, wolf packs), which behave as a cohesive coalition even though the interest of each individual is to “free ride” on the efforts of the others. Maybe after Passover I’ll have time to dig through the literature and bring some interesting references…
That is exactly what I wrote in the column here about the categorical imperative, and see also neglected column 209.
With God’s help, on the eve of the holy Sabbath, Tohorot, 5—
The chance may be small or not, but the very ability to act imposes on us the duty to exhaust that ability. What is this comparable to? In the overwhelming majority of cases, no disaster occurs if so-and-so loses concentration while driving or while standing guard. But if, God forbid, at that very second someone suddenly appears… what shall we answer our conscience? Had we fulfilled the duty of caution, which obligates us to take account even of remote harm,
we would have spared ourselves and others the anguish…
And in our case, anyone who saw the results of the previous elections should not have been surprised. The polls predicted success both for HaBayit HaYehudi and for Yachad—11 seats for the former and 6 for the latter—and what happened in the end? A large portion of the votes leaked to Likud, and among the clearly Religious Zionist parties there remained one party with 7 seats and the other with about 4 seats, but below the electoral threshold.
So what changed in these elections? Was Netanyahu’s “gevalt” campaign not foreseeable—that he would take a large portion of the Religious Zionist vote and the traditional and secular right-wing vote, and we would be left with two small parties hovering between 4 and 6 seats, with a not-insignificant chance that both of them, or one of them, would fall below the electoral threshold? It is a shame that the people of the Union of Right-Wing Parties and the New Right did not understand this and did not reunite before the lists closed, and it is a shame that we did not mobilize more to prevent this scenario, which had a very high probability.
All we can do is hope that the soldiers’ votes will ultimately save the New Right from going down the drain, and that in the future every voter will understand that his single vote has decisive force—“If I am not for myself, who will be for me?”—and that the two parties will understand that what they share is greater than what divides them, and “if I am only for myself, what am I, what am I”; and the more we increase unity, the greater our strength will be.
Let us keep before our eyes the guidance of our sages (cited by the Rambam) that a person should always see himself and the whole world as being exactly balanced, half and half, so that any one of his actions can tip his own fate and the fate of the entire world for the good. Each person should guard his own small corner, and together we will join into a mighty force.
In pain and hope, Sh. Tz. Levinger
It reminds me of a heavy smoker who argues: what are the chances that one additional cigarette will harm my health? One in a hundred thousand? And after the first cigarette he asks and answers the same thing about the second cigarette (quite rightly, of course), and the third… and the hundredth and the thousandth, etc. etc.
The analogy is not all that similar to the case at hand. With cigarettes there is a cumulative effect. The second cigarette is added to the first, the third to the second, and so on. By contrast, although voters’ votes are aggregated, their acts of voting are independent (at least according to the assumptions of the simplistic model).
Not only are their votes independent, but each one by itself has no effect whatsoever (and not just a little effect). But it seems to me that Roni only wanted to illustrate a principle, not make a full comparison.
Indeed so, there are similarities and differences.
What they have in common is that the isolated decision is not rational; there is no logic or benefit in it, but in the end the sum total of isolated choices changes reality. Whether we are talking about many negligible choices by one person, or one negligible vote by the aggregate of individuals in society. (And whether we are talking about a domain in which the isolated choice is of very little benefit or of no benefit at all.)
Paragraph 1, line 1:
…but the ability to act imposes upon us…
There, line 3:
…surely if we had…
1. In order to make the cost-benefit calculation, one has to calculate the expected value and not focus only on the probability.
Does it make a big practical difference whether Bennett gets in or not?
Of course expected value is not everything, but at the very least it has to be taken into account.
2. As for the cigarette calculations, this is a fallacy we all stumble into: after all, it does not matter if I eat one more slice of cake now—it will not cause significant weight gain. What matters is the cumulative effect. In every situation where persistence is required and only that yields the result, one runs into a similar situation.
1. When the probability is very small, expected value is irrelevant. Cf. the St. Petersburg paradox.
2. The cigarette issue is similar to all the other examples raised here. Admittedly it is not identical. There is a difference, because with cigarettes each cigarette does a little damage and this accumulates, whereas in voting one vote has no effect whatsoever (only a small chance of effect). And there is more to split hairs over here.
1. One in thirty thousand is not such a low probability. Especially since why not add together all the elections over the course of your life and get a much higher probability, just as you do with cigarettes.
Because all the elections you have in your life are about 10–20, while cigarettes you smoke a pack a day, which brings about significant damage with certainty. The fact is that this has never happened until today and apparently will not happen either. And even if it happens once, the expected value is still negligible (in the St. Petersburg paradox too you can get a high result with some small probability).
By the way, this is not about my probability of having an influence but about the probability of one vote having an influence. After all, if there is a tie in the election, then every vote has an influence, not necessarily mine. So your calculation, which refers to the probability regarding your own vote and counts how many times you voted, is not formulated precisely. The more precise question is how many election cycles are needed to reach a result in which a single vote has an effect. You need on the order of 30,000 election cycles to get a situation in which one vote changes one party’s seat count (actually two parties’) by one seat. That is completely negligible.
1. If someone offered you a gamble with a 1 in 30,000 chance of winning 10 million in the lottery for an investment of 10 shekels, wouldn’t you invest?? At what point do you say that you do not take expected value into account?
2. If a person has 15 elections, then the probability is 1 in 2,000 or 3,000. Is that still so negligible?
3. If you say the expected value is negligible because the gain of one seat is negligible, then I have no argument with that; but if someone feels otherwise, then the expected value will not be negligible.
4. There have only been 21 election cycles in this country so far, so the fact that it has not happened proves nothing.
1. No. And not because of the ten shekels; rather, I just wouldn’t bother dealing with it. Only if I were desperate and this were my only chance of being saved from a bitter fate. Beyond that, it does not matter whether we are talking about ten million or a billion or one million, and it does not matter whether the ticket costs ten shekels or a thousand. The question is whether the prize is significant for me and the ticket price tolerable. So this is really not a matter of expected value (the ratio between price and prize does not matter at all). In cases of small probability, expected value is a consideration only when there are many repeated drawings.
2-3. Completely negligible. There is no sharp criterion. There is common sense. If the state were destroyed without that one seat, then of course even with an infinitesimal probability I would go vote (and that is of course not an expected-value consideration; see my remarks in section 1). But someone who thinks one seat means the destruction of the state is of course consistent in terms of his utility function, but the function itself is crooked.
1. What basis is there for saying that when the probability is low, expected value should not be taken into account? And how do you establish what counts as a low probability? Common sense and rationality in this context are in the eye of the beholder.
2. Oslo passed by a single seat, didn’t it?
3. Isn’t one seat worth a budget of ten million? Of course Bennett’s or Zehut’s crossing the electoral threshold can involve budgetary changes like that.
1. Common sense. “Heap” arguments always end up in the eye of the beholder, and still there is common sense. By the same logic you could say it is worth paying a million shekels for a small chance to win one million and one. Who determines that this is not worthwhile? To each his own utility function.
2. True. There were other one-seat votes as well. But it almost never happens, and especially not in a place where your own individual vote was decisive and decisive for the right party.
3. Sometimes yes, but the budget does not necessarily go where you want, and certainly not all of it. An expected-value calculation that compares this to a ticket for winning ten million is ridiculous. Especially since even regarding ten million I said I would not buy a ticket.
1. Why are you looking only at elections? If over the course of life you have many decisions with positive expected value and low probability, if I look at the expected value in the end I will profit. Your policy of not taking expected value into account will lead you never to do it, and perhaps there are many similar cases throughout life.
2. There are extreme cases where common sense indicates what to do, and even there one has to explain—I would not want to lose a million shekels even if the expected value were positive; nor would I fly back from abroad in order to vote. But 15 minutes of investment sounds reasonable.
3. The budget is much bigger than that; an influence of 10 million on the budget so that it will accord with my values (for Shas’s Ma’ayanei Hayeshua) is entirely reasonable. The big difference is that it does not go into my pocket, but I am altruistic.
1. By that logic, you should not step onto the road. There is a very small chance of being hurt there (it seems to me that the probability is significantly greater than the probability of influencing an election). Your argument is mistaken also in the substantive sense. One may combine only utilities of the same type, which add up to one another. Otherwise you should be taking into account steps intended to influence elections together with steps intended to improve the weather and the mood of the lettuce in the territories.
2-3. As I already wrote, regarding the utility function I see no point in arguing. If from your perspective this is significant—then of course you should go vote.
1. The cost of not stepping onto the road is far too high; that too must be taken into account, so the expected value of not stepping onto the road is negative.
I claim that one really should add the growth of the lettuce too, because you are the one lumping them together and saying that consistently one should not relate to expected value at a probability of 1 in 30,000, and I am saying that one should.
Incorrect. Do not step onto the road when you do not have to. Go out only for necessary purposes, and certainly not just casually. I do not think you act that way, and I do not think anyone acts that way.
If you think so, then even by your own method you should avoid roads, since this repeats itself and you have 30,000 days in life.
The rabbi’s argument is fundamentally mistaken. Every small probability eventually gets realized. The question should not be what the probability is that a certain event will happen and then I will have an influence, but rather, given that the event has already happened—for low-probability events happen all the time—what is the probability that I will have an influence, and then the percentage of influence is enormous.
This is parallel to asking why run to the bomb shelter when there is a siren—what are the chances that the missile will fall on me? Very small.
But the missile will fall somewhere, and therefore the question is: by how much do I increase my chances of survival if I run to the shelter? And the answer is: by very, very many percent.
You will say the loss here is infinite, but in politics, a party not passing the electoral threshold is exactly the same kind of loss.
It is possible that my remarks are fundamentally mistaken, but I did not see in your words even a shred of an argument in favor of that claim. There are quite a few errors in the little passage you wrote:
1. “Every small probability eventually gets realized.” This is a remarkable innovation in probability theory. It is worth publishing, because to my knowledge no one yet knows it.
2. “Low-probability events happen all the time.” Indeed, and therefore? Are you willing to bet with me on the result of one hundred dice throws all coming up 5? That happens all the time…
3. You are actually being imprecise in your wording, and apparently what you mean to discuss here is expected gain rather than probability. So first of all, that claim has already come up above and been answered. The expected-value criterion is not relevant to very rare events unless the lottery is repeated very many times (on the order of magnitude of one divided by the probability). That is not the case here.
4. The argument from running to the bomb shelter is plainly irrelevant. The personal probability you gain from running to a shelter is negligible. Unless you publish the innovation from point 1 and convince the world that no one understands what probability is.
In short, one can talk about mistakes in probability, mine or others’, but that requires a minimal understanding of the field. It is quite clear that here that understanding does not exist.
What you wrote is logical, but one must remember that by the same token, when a resident of the Negev hears a siren, the probability that the missile will fall and Iron Dome will not be activated, and that if it does fall it will land exactly on him, is equivalent to—and even smaller than—the probability of influencing the election, and therefore there is no logic in obeying Home Front Command instructions and entering protected areas.
Likewise, most of the things people are afraid of are fears with an extremely tiny probability—for example, people are afraid to walk through certain streets that are prone to terror attacks [Damascus Gate and others], even though the percentage of Jews injured there in a year out of all the Jews who pass through there is much smaller than the percentage that I will have an influence in the election.
People set themselves a reminder on their phone to check that they did not forget their child in the car, even though the percentage of those who forget their child in the car is much smaller than the chance of influencing the election.
And if so, there is no reason for very many of the things people do throughout life out of fear of the small chance that they will be harmed.
Perhaps this indeed is not rational, but that is how almost the whole world acts in dozens of other matters besides elections, and there is no reason this human nature should not operate in elections as well.
[And by the way, if driving on the road were something that happened once every three years and were not so necessary, many people would not do it—rational or not, but that is human nature.]
Eli, there is a fundamental misunderstanding in your remarks of what I wrote in this column.
I claim that they are right to enter the shelter. Not because of fear, but because that is the correct policy on the collective level. If there are people who are afraid, that is an understandable psychological phenomenon, but it has no justification, and I criticize it just as I criticize those who go to vote because they think they have influence.
The story about Rav Shach reminded me of the Rambam’s words about Rabbi Akiva.
“Do not imagine that the King Messiah must perform signs and wonders, introduce new things into the world, resurrect the dead, or do similar things. The matter is not so, for Rabbi Akiva, a great sage among the sages of the Mishnah, was the armor-bearer of King Bar Koziva, and he would say of him that he was the King Messiah. He and all the sages of his generation thought that he was the King Messiah until he was killed because of sins. Once he was killed, it became known to them that he was not. The sages had asked of him neither sign nor wonder…” (Mishneh Torah – Kings 11:3).
That is, Rabbi Akiva proceeds according to the rules of appearance and perception as he sees them, and according to them Bar Kokhba was fit to be the King Messiah, for he was fighting the wars of God, and thus had the presumption of being the Messiah. Afterwards he was killed because of sins and it became clear that Rabbi Akiva had erred—but he did not err in following Bar Kokhba; rather, it simply became clear that Bar Kokhba was not the Messiah.
Or something similar: “the burden of proof is on the claimant” does not prove that justice is on the side of the current possessor; rather, this is the correct practical rule until it becomes clear otherwise.
And I’ll conclude with a joke about the head of a women’s seminary who lectures to her students that Rav Kook explains that Rabbi Akiva thought Bar Kokhba was the King Messiah, and he was mistaken.
– A seminary student cries out in alarm: “Rav Kook was mistaken?!!!!!”
The head of the seminary: “No, Rabbi Akiva was mistaken.”
Seminary student: |sigh of relief| “Ah, whew, okay.”
🙂
Meanwhile, as Ayelet Shaked says, the Holy One, blessed be He, intervened and gave the Right a second chance. If only we take advantage of it…
Best, Sh. Tz.
See the posts on the (excellent, in my opinion) statistics blog “Princess of the Sciences”:
http://www.sci-princess.info/archives/2225
http://www.sci-princess.info/archives/1744
http://www.sci-princess.info/archives/1725
The posts were written when the electoral threshold was lower, but the chance of having an impact is still greater than people think without running simulations.
Moshe