A Look at Pascal’s Wager and Expected-Value Reasoning (Column 408)
More than once in the past I’ve dealt with expected-value considerations and with the utility function. Here I wanted to apply these ideas to the well-known argument called “Pascal’s Wager,” which, in my view, many who discuss it miss its sting. It isn’t as stupid as it sounds to some people, but it’s also not as persuasive as it sounds to others. [1]
Pascal and His Wager
Blaise Pascal was the one who wrote the famously short letter: “I didn’t have time to write you a shorter letter.” [2] He was a mathematician and statistician, a physicist, theologian, and very well-known French philosopher of the seventeenth century. Although he lived only thirty-nine years (we should remember that this was a reasonable age relative to the life expectancy of his time), he is regarded as one of the great geniuses humanity has produced. Pascal influenced quite a few fields of inquiry and thought, both theoretical and practical. At the age of thirty-one Pascal had a powerful religious experience, and as a result he began to engage in theology; in that context he formulated his famous argument, cast as a wager. Essentially, it is a statistical consideration based on expected-value calculation.
Pascal’s Wager attempts to combine two significant components of his life—probability on the one hand and theology on the other—yet it’s hard to be impressed by the result. This is further proof, if any were needed, that mathematical ability does not necessarily imply philosophical or theological depth, and as we shall see, it is not correct to expect rational decision-making thanks to that ability. The surprise is that the flaw in his argument is not in the areas where he was an amateur, but precisely in the field in which he was an expert—indeed, one of its founding fathers: probability.
Pascal’s Argument
Pascal presented the wager as an alternative to a proof of God’s existence. He tried to bypass the need for proofs and to motivate us to believe or to observe commandments even without being convinced of God’s existence—by means of a probabilistic argument. As noted, the argument is problematic, but it is nevertheless worth examining it, if only to see the all-important distinction between probabilistic calculation and philosophical/theological conclusions.
Dawkins presents Pascal’s argument as follows (The God Delusion, p. 155):
Even if the odds that God does not exist are very high, there is an even greater asymmetry when one considers the severity of the punishment if it turns out that the guess is wrong. You would be better off believing in God, because if you are right, there is a chance you will gain eternal bliss; and if you are wrong, it doesn’t matter anyway. Conversely, if you don’t believe in God and it turns out that you are wrong, you will be accursed in hell for eternity; and if you turn out to be right, it won’t change anything. On the face of it, the decision is clear: believe in God.
Pascal, as an avowed probability man, makes here an argument based on expected value (average payoff). Expected value, as every statistician knows, is the product of the utility/payoff and the chance of obtaining it. He calculates the average payoff of observing commandments versus not observing them and concludes that the payoff from observing commandments far exceeds the payoff from not observing them, without having to decide whether God does or does not exist (hence the use of expectations).
How is this calculation done? Suppose we are in a state of even doubt—that is, the chance that God exists is 50%. If I observe the commandments, then if God exists, my payoff is immense: I receive eternal reward in the world to come, i.e., an infinite payoff. By contrast, if God does not exist, I incur a small loss (I lived a somewhat constrained life—certainly in the Christian context where there is no halakhah, without justification). Therefore, the bottom line is that the expected payoff from observing commandments is positive—and indeed infinite. That’s if I decide to observe the commandments. Conversely, if I decided not to observe them, I must again compute my expected payoff: if God exists—I am condemned to eternal hell, i.e., an infinitely negative payoff. If God does not exist—I gained something (I lived freely), but nothing very significant. All told, the expected payoff for not observing commandments is negative to infinity. Comparing the two options shows that the expected payoff of observing commandments is huge, and the expected value of not observing them is hugely negative. The meaning is obvious: it is much more rational and advantageous to observe commandments. I’ll note that the calculation does not change significantly even if I assume (as Dawkins himself does) that the probability that God exists is tiny; the product of that small probability and the reward remains infinite, and likewise for the enormous punishment expected from non-observance.
Thus, the gap in payoffs between observing commandments and not observing them, in the absence of certainty regarding belief, is enormously positive; hence every rational person should observe commandments. Seemingly this argument is valid, since it’s a simple probabilistic calculation (of course, assuming the premises that the cost of non-observance is indeed that terrible and that the value of observance is so immense).
Most who have attacked this argument and mocked it did so for various theological reasons. I hardly know of challenges to its mathematical/probabilistic validity. But as I shall try to show here, in my view the situation is exactly the opposite: at the theological level, one can perhaps defend the argument reasonably; but it has a substantive flaw precisely at the probabilistic level—or at least in a closely related domain (decision-making under uncertainty).
The Attack from the Plurality of Religions
Some attack this argument on the grounds of the multiplicity of religions. Even if God exists, who is He? Christian, Jewish, Muslim, Hindu, or pagan? If the existing God is Christian, then the Jew—even if he meticulously keeps all the commandments of Judaism—will be condemned to eternal punishments in hell, and vice versa. Therefore, one must divide the expected payoff by the number of religions and add gigantic losses to each option. Yet despite this claim, one can still argue that the worst option is atheism, since the number of religions is large and each at least gives a chance for enormous reward and escape from awful punishment. Atheism alone leaves us with very small expected gain or loss, and thus it is the worst option. You may ask: even if you are right, which God should I choose? I would answer: draw lots, or choose the God you are most inclined to believe in (the one to whom you assign the highest probability of existing).
But we must understand that at this point one’s basic beliefs (starting point) already enter the picture. If I am an atheist who sees only a minuscule chance that there is a God, the result of the calculation changes significantly, of course. Already it’s not clear that the result unequivocally favors randomly drawing a faith.
However, I contend that, at least in the Jewish context, God is good and therefore does not reproach His creatures. A person who did the best he could will not be punished (and will likely receive reward commensurate with his efforts and deeds). If this holds for the Christian and Muslim Gods and all the others as well (and in my understanding my reasoning applies to them too), then I must factor in the assumption that God is good; and again the calculations change considerably. I am no longer facing a horrific punishment for mistakes I may make.
In short, just as Pascal’s “lottery” does not bypass a person’s a priori stance as atheist or believer, it also does not bypass a priori considerations regarding who the true God is and whether He is good or harsh (whether He reproaches His creatures or not).
Dawkins’ Attack: Pragmatism
Dawkins attacks the argument as follows (ibid.):
Belief is not something one can decide to perform as a policy. At the very least it is not something I can decide to do as an act of will. I can decide to go to church, and I can decide to recite the Nicene creed, and I can decide to swear on a heap of Bibles that I believe every word written in them. But none of this will make me truly believe if I do not believe. Pascal’s Wager could serve—if at all—as an argument in favor of pretending to believe in God. And you’d better hope the God you claim to believe in isn’t of the omniscient type, because He’ll immediately detect your pretense.
At a basic level Dawkins seems right. A non-believer cannot make himself a believer by a utilitarian decision (to gain a payoff). The reason is that belief is a factual state. If I believe in God, it means I have been convinced that He exists. Whether this benefits me or not should not change my beliefs. Subordinating the true to the useful is pragmatism in its base and foolish sense. [3]
For the same reason, as I’ve written more than once, there can be no formal authority regarding factual claims. You cannot obligate me to adopt a factual claim merely because someone said so. If I’m convinced he is right (because he is very wise)—fine; then I have been convinced. But if it is a demand stemming from someone’s formal authority—an individual or an institution—then I am required to change my views about facts without being convinced. That simply cannot be done. As long as I am not convinced, it is not what I think. You cannot demand that I think something I do not think. You can, of course, demand that I act in a way that I consider incorrect (this is exactly the meaning of formal authority), and this is precisely what Dawkins argues here. A utilitarian wager can lead me to behave in a certain way (even if I don’t think it is correct), but not to think in a certain way.
Rejecting the Pragmatic Attack
One may deflect the attack by arguing that Pascal’s Wager is not meant to persuade me to believe, but to keep the commandments. On that point, even Dawkins agrees that the expected-value calculation makes sense and that expected value has force.
However, an assumption is lurking here—that the observance of commandments has intrinsic value even without a background of belief. That is, a person can be an atheist in his views and still observe commandments and thereby merit full reward and escape full punishment. In my view this is highly problematic. In an article I wrote, I argued that there is no meaning to the technical plane of commandment-observance. Observing commandments does not mean performing a certain set of actions but acting out of commitment to the divine command. Again, I think this is true for all religions—or at the very least I am unwilling to take into account the possibility of a religion that does not relate in this way (its plausibility to me is zero).
But there is another flaw in Dawkins’ argument. He implicitly assumes that belief or non-belief is binary: either you fully believe or you fully don’t. Yet in the second chapter of his book, he himself lists seven different levels on the spectrum of certainty vis-à-vis belief or atheism (and he places himself at level 6; he, too, understands there is no proof that God does not exist). It is very likely that most people are somewhere in the middle. Now the question arises: what should someone do who has some degree of doubt about belief? Pascal argues that even in such a state he should observe the commandments, at almost any level of doubt (unless the strength of the doubt is so low that multiplying it by the enormous utility still yields a small number). In his view, the expected payoff of observing commandments remains huge. In such a situation, considerations of costs and benefits can indeed be relevant.
Of course one may wonder whether there is value to commandments kept by someone who is in doubt. Here, the reasonable answer is decidedly positive. After all, who has no doubts whatsoever in life?! If only those convinced beyond doubt of God’s existence are fit to keep commandments, and everyone else’s deeds are devoid of religious value, then I suspect the set is almost empty. Even those who declare they are certain believers—I don’t believe them (either they are lying, or they are boastful without basis, or they do not understand what certainty is). We have no certainty in any domain (precisely why we resort to probabilistic expectations), and therefore in no domain do people adopt only claims or behaviors that are completely certain to them. We all act under some degree of uncertainty in every sphere of life, and therefore we almost always have to make our decisions under uncertainty. There is no reason and/or justification to exclude the religious realm from this. It seems to me that the attack from pragmatism (ad-pragmatium) does not hold water.
One might perhaps argue that even if a person is in doubt, he must make a decision. If someone has decided that the probability of God’s existence is not high enough, for him, to decide that God exists—then he is an atheist, and the commandments such a person keeps have no value. Conversely, if someone decides that the probability suffices, for him, to commit to God, then his commandments have value. The sufficient threshold varies from person to person as in every domain, but still the value of commandments depends on the decision the individual himself makes according to criteria he accepts.
This is a possible argument, and personally I’m quite inclined to think it is correct; but the attack on Pascal that rests on it already looks rather weak. It’s a consideration, and Pascal can claim he doesn’t accept it. He will say that for him this subjective threshold is meaningless, and that anyone who observes commandments—at least as long as he is not at level 7 on Dawkins’ scale (a complete and absolute atheist, even more than Dawkins himself)—then his commandments do have value. This is a tenable position, and the critique of it depends on reasons to and fro; it is hardly the knockout it is usually presented to be.
The Probabilistic Challenge: A First Look
I already mentioned that most attacks on Pascal’s argument deal with it on the theological plane (attacking its theological assumptions). I claim that these attacks don’t hold water, and they certainly aren’t crushing. The fundamental flaw in Pascal’s argument lies precisely in his home waters—that is, in the probabilistic plane. A serious look at Pascal’s computation points to a misunderstanding of probability. In Pascal’s case this is forgivable, despite his being one of the field’s fathers, since these misunderstandings were discussed only many years after his death. But from a contemporary man of science, like Dawkins, I would expect better understanding.
Pascal’s expected-value calculation, given the premises he assumes, is reasonable. I have no criticism of it at the technical level. My criticism concerns another assumption Pascal tacitly makes—namely, that the rational behavior required of us is to act by maximizing expected payoff (the average gain), i.e., that the calculation he makes is indeed the relevant criterion for deciding whether or not to observe commandments.
Researchers of rationality have already noted that it is not always correct to act according to expected value. For example, suppose someone comes and tells us that on a certain island in the Pacific there is a huge treasure. Should we immediately go dig there? Suppose that in our assessment the chance is one in a million, but he tells us that the treasure is worth countless billions of dollars, so that the expected payoff surely justifies the effort. Is every rational person really supposed to outfit a ship for several million dollars and go dig there? The expected payoff of such a step is positive (multiply the probability by the expected gain and subtract the cost of the expedition—the result will be hugely positive). In fact, even without someone coming to tell me there’s a treasure there, the claim that there is some chance, however slim, makes sense. If we multiply that chance by the imagined treasure’s colossal value, we reach an expected payoff that certainly seems worth working for. Is that enough to set out? [4]
So what exactly is wrong here? Why wouldn’t any of us do this? It turns out that decision-making in certain situations should not be based solely on expected value but on the entire distribution. Consider someone offering you an offer you can’t refuse. We have a biased coin. The probability it lands on “heads” is 1 in a billion, and “tails” is the complement (almost 1). Now you are offered a ticket for the following game: we toss the coin; if it lands tails, you get nothing, but if it lands heads, you receive a billion billions of dollars. The expected value (average payoff) of such a game is, of course, a billion dollars. How much are you willing to pay for a ticket to play? I assume most of you won’t pay more than a few dollars, if at all. Why?
Because although the expected payoff is enormous, the chance of getting that payoff is tiny. There is almost no chance you will receive anything in this game. True, if you do receive something, it will be a gigantic sum, but in practice you are throwing away the amount you paid to participate (because it will never happen). We gamble with our lives every day when we drive a car or cross a street (even at a crosswalk). How do we do this? Because although the wager is for the entire pot (our lives), the probability of harm is very small. The expected loss is very large, but the chance it will occur is minuscule. Such a risk is one every rational person takes.
In Column 252 I illustrated the problem via the St. Petersburg paradox. It is a wager whose expected value is infinite, and yet there is no rational sense in paying more than a few shekels for a ticket to participate. See the details of the calculation there.
The Probabilistic Challenge: What Is Expected Value?
Why is it, in such cases, that one should not make decisions by the expected-value criterion? Because the wager in which we participate is conducted once. The law of large numbers says that if we run infinitely many trials, the gain per game that remains for us is the expected payoff. [5] For example, if we toss a (this time fair) coin—heads and tails each with probability ½—many times, where on heads we receive 100 ₪ and on tails we pay 50 ₪, the expected payoff is a gain of 25 ₪. What does this mean? In a single game we will either receive 100 or lose 50, but clearly we won’t earn 25. 25 ₪ is not even a possible result in a single bet. The meaning of expected value is that if we repeat the wager many times, in the end we will be left with a gain of 25 ₪ times the number of games—that is, the average gain per game is 25 ₪. Lest you err: this is not the cumulative gain from all the games but the gain for one game. But still, that number reflects what will happen after many trials. In a single trial, that result has little meaning.
If we repeat this calculation for the coin toss described above, the expected value we computed there is a billion dollars. This means that the profit remaining after many games will be a billion dollars times the number of games, i.e., the gain for each individual game is a billion dollars. But that is not, in fact, the gain you will have if you play a single game. In a single game the chance of ending up with 0 is almost 1. There is no real chance of winning, and therefore it is not worth investing even a penny in such a game. Here you have, in a nutshell, why the expected-value criterion has very limited meaning in situations where we have only one trial.
Note that this criterion is more relevant when we have a fair coin wager. But it is much less relevant when the distribution of outcomes is asymmetric—that is, when the chance of getting a gain near the expected value is negligible (as with the unfair coin, as opposed to the fair one). When the average outcome is not the expected (typical) outcome, the expected-value criterion has very little meaning.
The Connection to the “Law of Small Numbers”
I have spoken more than once about what I called “the law of small numbers” (see, for example, Column 38). When we examine a small number of cases we can obtain very atypical results (a statistical miracle). The results converge to the average only when we conduct a large number of trials. In the St. Petersburg paradox one needs a truly enormous number of trials to obtain a normal sum. In the unfair-coin case, you need on the order of a billion tosses to get the expected payoff even once (the billion billions). If you are offered to buy tickets for a hundred billion wagers, it is worthwhile to buy (and to pay a fair price for each bet—up to the expected value). But for a single wager—there is no reason to do so.
As I showed in that column, many daily-life deceptions are based on this. The tendency to act and decide by expected value somewhat resembles the representativeness bias and the fixation on a few cases we have encountered or are thinking about, as if they were a representative sample. The law of large numbers says that with small numbers the sample is usually not representative, and one cannot infer from it what will happen in situations with many trials.
The Meaning of Pascal’s Error
So Pascal did not err in his probabilistic calculation. It’s a very simple expected-value computation and not the focus of the argument. His mistake lies in decision-making based on that calculation. Contrary to the impression many people have, a probabilistic consideration never stands alone. It always rests on assumptions, and one must make additional assumptions to make decisions on its basis.
We can formulate it thus: Pascal’s error was to aim his argument at the atheist. He argues that even if, in the atheist’s view, the chance that God exists is tiny, the expected-value calculation should lead him to observe the commandments. But that is not correct. If indeed the chance of receiving the huge benefit or incurring the terrible loss is tiny (because they appear only if God exists), then even if the expected value is huge there is no reason to observe commandments—exactly like the unfair-coin wager above. In the end, the matter depends on our starting point, and Pascal’s argument is addressed to the already-convinced (or at least to those whose prior odds are roughly even).
Even in Pascal’s Wager, if the prior probabilities for the two possibilities are balanced, the rational decision would indeed be to observe the commandments (what he calls: to believe). But if, for someone, the prior probability of God’s existence is very low, then although it remains true that expected value tips toward the theistic side (because the gain is infinite)—that is, there is no error in Pascal’s expected-value calculation—since the chance of actually attaining that expected gain is very low, one should not rely on it when making a rational decision. The rational decision for each of us, in such a case, depends on the a priori probability we assign to the two possibilities, and not only on expected value.
Note: if a person is evenly in doubt between the possibilities that God exists or does not exist, Pascal’s argument is far from absurd for him. Pascal’s mistake was directing his argument at the atheist who assumes a large a priori difference between the probabilities. This is an important consequence of moving from the conventional critiques—which operate on the theological plane—to my critique, which operates on the probabilistic plane.
Whenever we use a probabilistic calculation to make decisions, we must be aware that this use is saturated with assumptions that must be examined—namely, whether they are justified and whether the calculation is relevant to the decision at hand. This is the secret charm of mathematics, formal logic, and probability—and, in fact, of all quantitative and formal reasoning. They look compelling and well-founded, since you can’t argue with numbers. Well, actually, you can. Usually not with the calculation itself, but with its implications and its relevance, and essentially with the assumptions underlying the calculation and/or the formalization (see on logic in Column 50 and 318). [6]
Two Final Notes
A. The a priori chance of God’s existence or non-existence is not the result of a calculation. There is no way to compute it probabilistically, since we have no sample space or event space. In fact I am speaking here of plausibility, not probability. This is a subjective intuitive assessment of plausibility under conditions of almost total uncertainty—essentially the result of philosophical considerations. There is no hard probability calculation here. It turns out that in many cases it is precisely these assessments that tip the scales of decision. And still there is much logic in treating plausibility like probability (see on this in a comment to Column 402 and in my reply there).
B. The description and calculations I have given here were done in a vacuum, where the utility function is simply the sum of money one earns. But there can be situations in which a person has additional considerations, which may change his utility function. For example, a person is threatened: if he does not give the extortionist a billion dollars today, he will die tomorrow. Now he is offered the aforementioned lottery with the unfair coin, which gives him a tiny chance of receiving a billion billions of dollars. Such a person will certainly buy a ticket for any sum he has, since if he does not buy the ticket, he will certainly die tomorrow. Buying the ticket gives him a tiny chance to live, and that is preferable to certain death. His utility function is not just the sum of money but his life, and that changes the picture and his calculation. Or, in a less extreme case: if a person loves risks and has a lot of money. He has no problem paying a million dollars for the thrill of risk and chance, and perhaps also for the anticipation of the lottery’s outcome. He loves betting and has a lot of money to invest in it. Such a person may be willing to invest a million dollars in the unfair-coin lottery above.
But these remarks are marginal. My intention here was only to demonstrate the caution required when relying on expected-value considerations and on probability in general.
[1] The source is at the end of the second chapter of my book God Plays Dice. After writing this column, I noticed that I had already addressed the probabilistic critique in Column 252, but I left this in place because here I deal with the wager in a more general way, which is also the column’s focus.
[2] As with every good aphorism, this one is attributed also to Mark Twain.
[3] A pragmatism scholar once noted to me that there are pragmatic schools of thought that claim utility is an indicator of truth. If some idea is useful—that is, it creates a better world (at least for me)—this is an indication that it has substance. One can make such a claim on a religious basis (God acts so that the good and the true are also the useful), and therefore in principle it is an acceptable claim (even if, in my view, not correct). What I call here “pragmatism” is the approach that sees the useful as the definition of truth and not merely an indicator of it. This approach is rooted in skepticism about our ability to reach truth, leading to replacing it with the useful.
[4] One might try to claim speculatively that the chance of a treasure being there is inversely proportional to its value—but whence do we know this? One could just as well say the opposite: the higher the value of the treasure, the more likely it was hidden well in a protected, remote place (both because the fear of its discovery is a “cost” that is greater, and because such hard work is justified only if the treasure is very valuable). So—off you go to dig!
[5] The law of large numbers is not universal. It does not hold for every distribution, but it turns out that in the vast majority of interesting cases it is valid. Therefore I shall assume it here without further qualification.
[6] At the end of the third chapter of his book, Dawkins deals with Bayesian probability calculations and again reveals a staggering lack of understanding regarding this important point. He sees probability as an objective tool that is not conditioned by intuitive assessments. For him, an evaluation process that starts from subjective intuitions rather than from an objective calculation is GIGO (Garbage In, Garbage Out). That is, if one inputs intuition (in his view, “garbage”) into a probabilistic or statistical calculation, then the result is no more than intuition (i.e., “garbage”). He does not understand that probability is a way to make rational decisions dependent on our prior assumptions (at least when there is uncertainty). Almost never does a probabilistic calculation begin with the result of a calculation; it begins with assumptions that, by their very nature, can certainly be intuitive. For example, the assumption that the die or the coin is fair is an assumption, not the result of a calculation—especially when dealing with real-world events and not a hypothetical problem.
It is important to understand that if Dawkins were right in his approach, then Pascal’s argument would be correct. As I explained above, assuming the scales are a priori balanced, Pascal’s Wager is a valid and rational argument. The reason it does not lead to a rational decision is solely the fact that the scales are not necessarily balanced. Therefore, for someone who assigns a very small chance to receiving the infinitely large expected payoff (the divine reward in the world to come—the “garbage in” of the problem), there is no rational justification to choose the theistic option.
Discussion
Utter nonsense, as usual from you.
When will you deal with the things that really matter?
A. There is something circular here. You propose a decision-making mechanism that is not based solely on expected value, but how do we check whether that mechanism is really optimal? The way I know to evaluate decision-making mechanisms is by expected value… but you do not accept that method of evaluation either. So there is no solid ground here from which your claim can be judged. So this is not a factual or mathematical claim. There is here a psychological claim about what people nowadays tend to do. In your view, is that a philosophical claim about 'what one ought to do'? From where does one quarry such a strange kind of "ought" if it does not lead to some desirable result.
B. Does the categorical imperative (which you do in fact hold by) not basically tell us to make decisions according to what would happen if there were many such decisions before us? If the categorical imperative makes my decision depend on the overall expected profit over many identical decisions by others, then all the more so it would make my decision depend on the expected profit over many identical decisions by myself.
C. It does not seem that there is any difficulty here for someone who (ideally) makes decisions only by expected value (for now, still, at least theoretically—me, for example). I make decisions only by expected value, but for me the expected utility is not the expected monetary gain in shekels. True, the utility function is monotonically increasing in the amount of money, but it has a horizontal asymptote and reaches it quite quickly. For example, from where I stand there is not much difference between one billion and two. In addition, the price of losing a year of life to idleness is very high, more than the annual salary and the interest and the total value of the pleasures and knowledge I gain in a year. Therefore, going roaming through mountains and valleys looking for treasures is simply not optimal. If I came into the world in a million incarnations and in each one I searched for enormous treasures, then the total utility would be negative.
I don't know whether there is any prohibition against exposing nicknames, or whether since the nick is unrelated to the real name we have no issue with it. So I’ll ask in general: I wonder where all these nicks come from? Perhaps you are a reincarnation of Tolkien, building a mythology of your own?!
A. I do not see why decision-making by expected value seems to you justified or self-evident. After all, expected value is a calculation over many gambles, so why does applying it to a single gamble not seem to you to require justification?
I am talking about common sense, not numbers. As I explained throughout the whole post, the numerical calculation never stands on its own; it always presupposes a framework for the discussion that can be examined by common sense. My method is the same. Common sense says not to throw money in the trash, even if there is a tiny chance of earning an enormous sum. It simply will not happen, and as everywhere else, one does not rely on a miracle and does not build on minuscule probabilities.
B. The categorical imperative is a moral principle, not an economic one. On your consequentialist view, morality is the act that will bring maximal profit. But even on your own view this is not true, since such an act will not bring maximal profit. This is unlike ethical behavior according to the categorical imperative, which in the end does bring profit maximization (as in post 122).
C. I said that the calculations in the post are made without additional assumptions about the utility function. If there are additional assumptions, then of course things change. I explained that my intention here was only to demonstrate why it is not always correct to act according to expected value, and for that purpose there is no reason at all to resort to other utility functions. And even on your own answer, it is difficult to see why you would not dig in your backyard with an excavator that costs a million dollars and penetrates the ground within seconds. You would not need to invest a year of effort. I do not believe you that you make decisions according to the criterion of expected value. For example, with the biased coin toss, does that not take a second? Would you buy a ticket for a million shekels? (Let us assume we are still before your asymptote.) I am speaking knowingly, and I do not believe you.
B. If Elijah offers a billion people to pay a thousand shekels, and then with a probability of one in a billion he will grant a billion shekels to every person in the world—does the categorical imperative require me to participate in the lottery?
C. The demonstration relies on the fact that the reader himself admits, in some such case or other, that he does not decide according to expected value. But the reader admits it because that is indeed his intuition in this case, only the interpretation of that intuition is different. Not because expected value is not being calculated, but because the expected value of the felt gain is not high enough.
With the coin at a million shekels, you also have to price the suffering in the loss. If one properly prices the felt suffering and the felt pleasure,
then I think expected value really does predict the decision very well.
The nicks—I don't know. Sort of starting over without baggage from the past. Nonexistent names allow convenient searching on the site so that only relevant results come up. Not sure exactly.
It is not impossible that the atheists Pascal knew were the type who think there is a reasonable chance that God exists—say 20 or 30 percent. That is also the kind of atheists I know.
B. There is no moral obligation to provide money to people (who are not needy). Certainly not to invest your own money in that.
C. You empty the concept of expected value of all content. Every criterion on earth will fit some definition of expected value as long as you put all the parameters into it. Put the tiny chance of profit in as suffering, as a component in the expectation, and everything will come out identical.
I addressed that.
C. But I am not saying this ad hoc. I am talking about sensations of pleasure and pain, and in my opinion it is simply agreed by everyone that this is how they operate. A "risk-averse individual" is each of us who is not trained to be a professional broker. To compensate for the feeling of losing a million shekels that I have in hand, I need an expected gain far greater than a million.
I still do not see what, in your view, could count as a consideration that is not based on expected value. Everything can be translated that way.
A. In principle I agree, but as Pascal presents things you need more than 99.9 percent certainty that there is no God in order to be prepared not to gamble when the gain is so enormous and the loss is something like burning forever. Therefore, if you are not convinced to almost 100 percent that there is none—the gamble is definitely good in terms of reasonable decision-making.
B. An early version of Pascal's wager appears in Maimonides' Guide of the Perplexed, but on the negative side: according to him, passing one's child to Molech meant only passing him through, without burning him, and the priests would promise protection for all the children at the low price of mere passing through. A person who acts according to cost-benefit considerations might be tempted on the basis of Pascal's wager, and therefore, in order to change the weight of the wager in the eyes of the masses, the Torah promises that one who does this will die childless.
C. And here is the major issue that needs attention: beyond the probability that the masses assign to the question whether there is or is not a God, there is also another coefficient affecting that probability—and it is that most people do not assign high confidence to their own assessment in such exalted questions. Therefore, even if it seems to a person that there is no God (or that Molech is nonsense), he may still prefer the consequentialist calculation, which seems more certain in itself, over his own probability assessment. In other words—add to the die example that in most cases a person has no good estimate of the chance that it will land on the specific face involved in the wager, but he is much more convinced that he knows what will happen if he turns out to be right or wrong in the wager.
A. I am really not sure of that. It seems to me that you are smuggling expected-value considerations in through the back door. In my view, most people will not take a tiny probability into account even if the price is enormous.
B. That is an ordinary promise of profit. I do not think it is correct to see this as an early version of Pascal's wager, at least no more than any expected-profit calculation a person makes in every decision of his life.
C. It is simple, because the question of what will happen if he is right or wrong is his given datum and not the result of his own consideration. The Jewish tradition says what will happen to you if it is correct. But the decision whether it is correct or not—that is yours. For a person to insert his own considerations also into the component of what will happen given that he is right or wrong, he would have to enter the tradition and sift out a personal interpretation for himself. Most people do not do that.
But what did I actually say? I only said that expectation is not measured by shekels in the bank but by felt suffering and pleasure. Surely it is clear that you too calculate that way, and only that position is worth rejecting. And indeed I beg the question, and for me every observed behavior merely indicates that that person has a certain utility function as a function of shekels, such that he really does choose according to expected value.
And I asked a question and was not answered.
I now think that you are not talking about expected profit at all, but about concrete profit in a single case. What you define is not expectation (because it does not depend on the law of large numbers and conducting many gambles. It is the profit from participating in one gamble).
If so, then you too agree that expectation is not the criterion in such cases.
And you are not begging the question either, but defining (not claiming). If anything, it is a tautology, not begging the question.
I said that I am indeed begging the question, and that I will translate every observation as showing that the person does in fact decide according to expected value, only that his utility function is unique. So indeed there is no ordinary example that would refute it.
If I knew the person's utility function, perhaps through advanced neurological research, and nevertheless he did not act to maximize it in expectation, that would be proof that he does not decide only by expected value (and it would require investigation whether he would then be exempt from the commandments like an imbecile/deaf-mute/minor). Therefore it is not a definition but begging the question. Exactly as I assume that a free person did what at that moment he wanted to do.
[By the term decision by expected value I use the sense of deciding in a single case according to the expected profit from a single game within an infinity of games. I did not quite understand what you claimed against this].
Tomorrow at 08:30, come prepared.
A. 1 in a thousand is 99.9 percent certainty—if that is tiny relative to the price, in my opinion no. A lottery of 1 in a thousand with a chance of a billion in gain or loss—you wouldn't take it? True, if you go down to 99.999 then maybe not, but most people do not have that.
B. It is a wager: it is not only "serve Molech—it is worthwhile for you," but also if it is nonsense, you lose nothing. Therefore the punishment the Torah gave is the death of one's children. Someone convinced that worship of Molech is true will not be persuaded by the Torah's punishment; it persuades only one who serves Molech by force of Pascal's wager—he can protect the children, and the price is low even if Molech is false.
According to Maimonides, the punishment is determined not only by the severity of the act, but also by the size of the incentive to commit the transgression, and the incentive of the masses in Israel to worship Molech is precisely Pascal's wager. Therefore the Torah made its punishment more severe than the other matters of idolatry.
Let me ask it this way. Suppose you are appointed to be God, and you are good and merciful. Would you not program all people to decide according to expected value? Such programming would, overall, lead to more expected pleasures for humanity. (I hope we will not need here to hang things on free will).
The error in Pascal's argument is basic, and I have already explained it here in the past:
A priori, the probability that there exists a God who gives reward for commandments and punishes for transgressions is equal to the probability that there exists a God who punishes for commandments and rewards for transgressions.
Therefore the whole argument collapses. There is no probabilistic advantage here, and the expectation remains zero.
All right then—if one may throw common sense in the trash and raise doubts to infinity, then why not? You can raise your claim about any evidence or any logical proposition whatsoever regarding God. The moment there is a proposition saying 'God is X,' you claim that 'a priori, the probability that there exists a God who is X is equal to the probability that there exists a God who is not X.'
The question is whether it sounds reasonable to most people to assume (intuition, hearsay reasoning, you name it) that God gives reward for commandments. In my humble opinion, yes.
That is indeed a theory that cannot be refuted. Even if you had a measure of his pleasures, you would classify him as insane.
What I said is that you are not talking about expected profit but simply about profit from the game, since expectation is defined by the average over a large number of games. You are talking about utility produced from each game separately, not the average utility per game.
You are invited to go to those mathematicians who waste their lives in vain trying to prove unsolved conjectures and teach them that there is no need for mathematical proof. We have discovered a new way to prove things: common sense.
The course of the proof is: if a statement sounds reasonable to most people, it is proven true.
I still do not understand what I actually said. Which position did you attack, if not the one I presented? Did you attack the position that tries to maximize expected shekels in the bank? Or did you attack expected utility but with certain assumptions about the utility function as a function of shekels? I do not think I said anything new.
I did not understand why I cannot talk about the average utility in a single game as calculated by expectation.
Rabbi, if I understood you correctly in rejecting Dawkins's pragmatism, it comes out that:
A person who does not believe at all can never become a believer or observe commandments by force of a utilitarian decision (because then they have no value); but a person who is in doubt about God's existence (really all of us) has to make a decision (a factual one: whether to believe or not), but even if he has not yet made that decision he can observe commandments on the basis of a utilitarian decision and then they will have value (even though this is "not for its own sake," still it is again not "without faith," because he is in doubt). But he can never decide to believe / become a believer by force of a utilitarian decision, only by force of a factual decision.
Is all this correct?
Hello Rabbi,
I wanted to raise an objection to the answer and say that there is a possibility that there exists a God who will give me infinite reward for not worshipping him in any way, and specifically punish those who do worship a god in some particular way. Because of the possibility that such a God exists, ostensibly one could say that a person should remain an atheist. No?
Yes, except for my remark about setting a decision threshold, but in the post I ignored that.
There is no such option, because nobody claims there is. One can invent countless options, and that has no significance. That is Russell's teapot and the hidden treasure experiment on the islands of the sea that I mentioned in the post.
Rabbi, I understood the answer.
Still, Pascal's wager does not deal with claims but with possibilities. Even if nobody claims that there is such a God, there is the possibility that there is such a God. (That is, even if it is a remote possibility that nobody even thinks about, certainly nobody claims, it is still a possibility.) And if so, there is a problem with Pascal's wager.
A. I do not understand the objection. After all, in a simple ordinary gamble (flip a coin: if heads I gain a shekel and if tails I lose a shekel), even you agree that in a single gamble one goes by expectation. What really is the meaning of "probability" without reaching a limit over infinitely many cases? Philosophically this is indeed not clear (by the way, I do not know whether it is less clear than "the slope of the function at a point"). But in practice you work with it too. Except that with you there is suddenly a jump at low numbers.
In my opinion it deals only with claims, or with possibilities that have minimal plausibility.
Expectation is calculated by many trials, and applied to a single case as some criterion for the value of the gamble. When it is not a common result, there is no logic in that criterion.
I do not think atheism is the worst option. In my opinion one should look at the risk in choosing the wrong religion. Many times choosing the wrong religion will bring greater suffering according to another religion, and therefore the suffering in choosing the wrong religion is greater than the suffering in choosing atheism.
According to quite a few Christians, the Jews are the synagogue of Satan and their punishment is great because they rejected the Messiah. They do believe in God, but reject the Messiah and the Son of God, and therefore rebel against Him, and their state is worse because they acknowledge God (at least partially) and rebel against Him.
According to Islam, the Jews are descendants of apes and pigs, and on the Day of Judgment their punishment is great. And even if there are decent Jews who will be saved, they cannot belong to the mainstream streams because of the support of most streams for Zionism (even if they do not actively support it but relate to the state as an existing reality). Therefore most of them are enemies of Islam.
In any case, the risk increases when choosing the wrong religion.
As I understand it, the law of large numbers refers to a finite expectation, not an infinite one. That is, in a number of trials as large as we like, the profit approaches the expectation, which is a finite number. But what happens if the profit is infinite (eternal life with infinite reward that cannot be quantified in terms of its value)? In that case, when the number of trials tends to infinity, the expectation also tends to infinity.
In such a case, in my opinion, one cannot infer according to the classical conception.
Common sense misleads. To most people the uncertainty principle seems absurd, but it holds. To most people it seems that a heavier object will reach the ground first, but that is wrong.
To most people, if you show them a drawing of two arrows of the same size in certain settings, they will see them as different in size.
To most people it seems that the whole is greater than the rest of its parts, but in infinite sets there may be a proper subset with the same "size" (cardinality) as the whole set.
To most people in the ancient world it seemed that there were several gods, each responsible for something. Today that seems absurd.
Common sense misleads, and only analytical thinking can overcome it.
Michi, with all due respect, I think this is where your mistake lies. Pascal's wager deals specifically with the Christian God, namely the one who will condemn to eternal hell every person who is not a Christian believer. In addition, he belonged to the Jansenist movement, a movement that claims that only a certain group of believers, chosen in a completely deterministic way from birth, will be saved, while the rest of humanity—their fate by an earlier deterministic decree—is hell. True, in the post you discuss the argument itself and not its speaker, but many historians and thinkers have pointed out that in their opinion Pascal's wager is also tied directly to the specific church in which he believed—in their opinion, from the outset he is addressing an audience of believing Christians who are wavering or skeptical, for whom there are only two options: either the Christian God exists, and then the path to salvation is only through him, or the reality of God does not exist at all. Pascal's wager does not include belief in Islam or Judaism. In fact, from his point of view, belief in them is apparently the same degree of risk as atheistic unbelief.
Your answers in the post are all excellent, but they are answers that fit another version of Pascal's wager, not his original version. The probabilistic argument in the original version is indeed ridiculous, and justifiably so, for anyone who does not think the Christian faith is certainly much more reasonable than other religious beliefs; and for anyone for whom the other beliefs and religious probabilities are equally plausible—once that is the assumption, the whole argument falls apart. There is no point for a skeptic or atheist who does not believe in Christianity to believe in it merely because of fear of hell—because that fear, rationally, should also trouble him in his heresy or unbelief regarding any other religious conjecture. In this case Dawkins and other atheists and agnostics who attack the original, Christian version of Pascal's wager are completely right.
You present here a version that says: the existence of God is entirely plausible, and even those who start from the assumption that He probably does not exist—it is worthwhile for them, still, even if only out of fear, to observe commandments. Presumably He would be much more forgiving toward a person who tried to seek the truth and turned out to be mistaken in error (whether because he did not manage to attain the consciousness of faith, or because he chose the wrong faith), than toward a person who is a complete heretic and atheist. But that is an assumption (which I too happen to think is correct) that does not appear in most traditional religions—not in Islam, not in Christianity, and not in Judaism. And when believers of different religions—Muslims, Hindus, and so on—make a version of Pascal's wager, they do not use this claim either. They proceed on the assumption that there are two possibilities: either their religion is true (and then, incidentally, one has to strive and believe in it wholeheartedly; I do not think there is in mainstream currents any movement that claims weak belief at level 4 or 5 is worth anything), or there is no God at all—and therefore it is preferable to believe in their religion. They too would agree that on the assumption that the other beliefs are plausible as well, the wager falls.
I see no difference between finite and infinite expectation. You can relate to it as a very large expectation (or as large as you like).
I addressed all this.
1. Even if he thought Christianity more logical and more plausible, that is not necessary for his argument.
2. Just as in Judaism I disagree with the accepted approach, so too I would disagree in Christianity and Islam. In my opinion, a God of any religion whatsoever ought not punish the coerced. That is the assumption on the basis of which I make the wager.
I understand you. So apparently the disagreement between us is purely one of semantics. I agree that Pascal's wager in your version is completely rational and plausible. I just do not think that in this case it is really correct to call it Pascal's wager. But that really is purely a matter of language and concepts.
.*I just do not think that in this case it is really correct to call it*
Correction of an error
That is exactly the problem. If one relates to the expectation as a number as large as you like, then the profit becomes large until it equals the required price. Say the required threshold for which I am supposed to make the choice is M; then I will choose the expectation to be M+1 (for that is what it means to tend to infinity).
The required inference is also problematic, because in general what holds in the finite usually does not hold in the infinite, and there the rules are different and contradict human intuition.
With God's help, 11 Elul 5780
Pascal's argument is simple and very logical. If there is an option that involves risk as opposed to a neutral option—a person should be concerned and do what he can to rule out the risky option.
That is why a person takes the coronavirus vaccine, even though the chance of contracting corona and being seriously harmed by it is quite small, even without a vaccine; and that is why a person gets insurance even against damages whose risk is small.
In both cases the calculation is that if nothing happens—you have not suffered any significant loss, just a bit of money or some mild side effects. But if the less common pessimistic scenario is realized and a person gets infected or harmed—then if he did not get vaccinated or did not buy insurance and what happened happened, the person has really 'been badly burned.'
And so Pascal says regarding God: if it turns out that He exists and you violated His will—then you have really 'had it,' and 'when in doubt about God, be stringent' 🙂
And your mnemonic: 'Don't take a pass lightly, on the concern that there is a God 🙂 It was worthwhile 🙂
Regards, There-Is-Doubt No-Doubt
You don't take risks? Do you align yourself with the strictest positions among all the rabbis of our time and the past? If you start mixing in too many reasonings about what personally seems right to you, you have nullified the whole wager.
With God's help, 11 Elul 5780
To Amram—greetings,
The Torah did not command us to take all opinions into account, but to arrive at a ruling of 'the judge who shall be in those days,' and in a dispute we follow the majority of the sages, as it is written, 'incline after the majority.' When there was a Great Court in the Chamber of Hewn Stone, the ruling was immediate and swift.
Once the people were scattered into exiles, arriving at rulings became harder, but over the generations there still arose 'sealings' that brought broad consensus. Thus the sealing of the Mishnah, the Talmud, and the Shulchan Arukh, which brought broad agreements that serve as the basis for later generations. The aspiration is to arrive at a gathering of the views and a ruling according to 'the opinion agreed upon by most decisors' (see, for example, the words of R. Yitzhak Nissim, Responsa Yayin HaTov, vol. 2, Yoreh De'ah sec. 11).
Clearly one cannot cast all our lot only upon the 'Pascalian wager,' which cannot decide which path is the correct one for serving God. But Pascal's reasoning gives a strong indication to search for a 'Master of the palace' who demands of man that he behave accordingly; and once an entire people is found—critical and 'stiff-necked'—who experienced a powerful public divine revelation, and whose stable testimony endures over many generations and despite exiles and persecutions, and leads to the miraculous survival of the people of the Torah against all odds—there is here a strong indication that something real is present.
Regards, Yaron Fishel Ordner
The claim against paralyzing skepticism is Miriam's claim against Amram, who separated from his wife because of the decree 'Every son that is born.' Against this Miriam argued that just as one must fear that the newborn will be thrown into the Nile, one must also fear lest a daughter be born, and lest a son be born whom they will manage to hide. The 'Pascalian wager' to bring children into the world despite the great chance that they would not survive was what brought into the world the savior of Israel, who was saved in a miraculous way despite the slim chance that a miracle would occur.
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Clearly one cannot cast all our lot on…
B. And as for 'and if a man did not lie in wait, but God caused it to come to his hand,' etc.—I came across this: https://ibb.co/JFYvK87. And from here a small clarifying question about your approach:
Someone is offered two options: one option is that one person dies, and the second option is that with a probability of one in a billion, two billion people die (if he chooses neither option, they kill all of humanity). In your opinion, morally speaking, should one choose the second option? Would the categorical imperative here instruct me to choose the first option because if this were presented simultaneously to masses of people, then obviously each individual should choose the first option? And if it really is presented simultaneously to many people, then should each individual choose the first option?
Good question. From the standpoint of the categorical imperative, it seems you are right. But the categorical imperative is not the only player in the arena. There is also the consideration that in choosing the first option you are deciding on the certain death of a person (somewhat similar to the difference between an action and passive inaction). The example in the picture you linked to speaks about very practical probabilities, and there, ostensibly, the categorical imperative is determinative. But with a probability of 1 in a billion there is the consideration that this is an expectation that will not be realized. And after all, in practice the whole world will not find itself in such a situation. Therefore, from the standpoint of the laws of preserving life and murder, option 2 certainly seems preferable, because there you are not murdering anyone (the chance that you are is negligible). If in practice this is an experiment presented to many people, then it seems to me there is no preference, because the result will be similar, and then perhaps the consideration of expected value (which is no longer really expected value) is determinative. And even here I am not completely sure.
The issue of certain death, which is somewhat similar to action, can ostensibly be neutralized if we set it up so that the first option is that with a probability of one-half two people die (rather than with probability 1 one person dies). Do you agree that this neutralizes it? Or is absolute certainty not required for it to resemble an action?
Could you clarify the side (or slight side) on which, even if the experiment is presented to many people, you are still not completely sure one should go by expected value? I did not think you would leave any room for such a possibility.
It is a little less like direct action, but there is still a difference (it is not a binary distinction. After all, even murder is not deterministic: perhaps there will be a jam in the weapon? Perhaps in the end something else will happen? Therefore something more probable is more like direct action).
As for when the experiment is made available to many people—even in such a situation I as an individual am still doing something that certainly brings about the death of a person, whereas the other side will bring about no death at all. Only the final balancing (the consequentialist one) brings the equivalence. As in the trolley dilemma, between the fat man whom one kills with one's own hands and the five who are killed anyway (not saved). Why is this only a slight side? Because when this experiment is presented to all people together, I analyze it from a general social point of view and not from my own point of view, and then the decision should be made collectively and not personally.
If I understand correctly, on the side where things are analyzed from my own point of view, the categorical imperative turns out to operate only stringently (for example, to refrain from wasting water), but cannot operate leniently and permit something that would otherwise be forbidden (to bring about with certainty the death of a person). That is quite an innovation in the parameters of the categorical imperative and what follows from it.
I do not know whether I would sign on to such a sweeping definition. I would formulate it by saying that it is not the only player on the field. It is one consideration among several. Even in our examples it is not clear what counts as the leniency and what counts as the stringency.
I see. In every place where something is only stringent, will the model be that it is one consideration among several, or are there other possibilities?
I do not understand your insistence on the matter of stringency. In matters of life and death one always goes stringently. Our whole discussion here revolves around the question what the stringency is here: action? result? certainty or probability?
As a matter for pilpul: if one threw a knife that cut off a leprous blemish with a probability of 40% (or 70%), and it did cut it off, is he flogged? Perhaps here too the lack of probability impairs the degree of active commission? (And then perhaps there could be a situation where one violates the prohibition but is not flogged.)
Indeed I do not know what I was saying
We already had a discussion about this here in the past: https://mikyab.net/posts/64124#comment-27350
But I will mention again here that the more money a person has, the less utility he gets from it (diminishing marginal utility). Therefore a gamble that costs me one dollar and in which I would win a billion dollars with a probability of one in 100 million is probably not a worthwhile gamble in terms of expected utility (even though in terms of expected monetary gain it is worthwhile). This has to be taken into account in monetary examples. As for spiritual gain, there the spiritual utility does not necessarily decrease the larger it becomes, and therefore there is an important difference here as compared to monetary examples.
Expected profit has significance when infinitely many trials are performed.
If so, then the atheist too, every time he performs a commandment, first considers whether there is value in it—that is, whether God exists. If so, this wager (if there is God I will get a billion, and if not I will lose only a free life without obligations) is conducted by the person every single day, and not only once.
Is this a question to me? I did not understand it.
I will only comment on your last sentence. This is not a wager conducted anew every day; rather, in the very same wager you invest another shekel every day. If there is a God, He exists on all the days, and if not, He does not exist on all the days. These are not different wagers.
A note regarding the argument about the hypothetical treasure that may be on the island:
There is definitely reason to argue that the probability of finding a treasure decreases ככל the treasure is larger.
Assuming this is a physical treasure such as gold or diamonds, its quantity is (a) limited by the total amount of all these materials on Earth;
(b) at least beyond some quantity, it is more likely to find relatively small amounts, for the reason that enormous treasures exist in smaller numbers (far more people have a kilo of gold they could hide than a ton of gold, and a treasure of a thousand tons belongs only to a few people, if any).