A Look at Pascal’s Wager and Expected-Value Reasoning (Column 408)
With God’s help
Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.
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.
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A note about the argument about the hypothetical treasure that might be on an island:
There is certainly reason to argue that the probability of finding treasure decreases as the treasure gets bigger.
Assuming it is a physical treasure such as gold or diamonds, its quantity is a. limited by the amount of all those materials on Earth
B. At least from some quantity onwards, it is more likely to find relatively small quantities because huge treasures are found in smaller quantities (many more people have a kilo of gold that they can hide than a ton of gold and only a few, if any, have a treasure of a thousand tons)
Egg chatter, as usual.
When will you deal with the things that really matter?
Tomorrow at 8:30, come prepared.
A. There is something circular here. You propose a decision-making mechanism that is not only based on expectancy, but how do you test whether this mechanism is truly optimal? The way I know to examine decision-making mechanisms is by expectancy... but you don't accept that way of examining either. So there is no solid ground here on which to judge your claim. So there is no factual or mathematical claim here. And there is a psychological claim here about what people nowadays tend to do. Does this seem to you to be a philosophical claim about ‘what is the right thing to do’? Where do you get such a strange “right” if it doesn't lead to some desired result.
B. Doesn't the categorical imperative (which you hold to) actually tell us to make decisions based on what would have happened if there had been many such decisions before us? If the categorical imperative hinges my decision on the general expected profit in many identical decisions of others, it is even more likely that it will hinge my decision on the expected profit in many identical decisions of myself.
C. There does not seem to be a problem here for someone who (aspiringly) makes decisions only according to expected profit (for now, still, at least theoretically, me, for example). I make decisions only according to expected profit, but the expected pleasure for me is not the expected profit in shekels. Although the pleasure function monotonically increases with the amount of money, it has a horizontal asymptote and reaches it quite quickly. For example, from where I am, there is not much difference between one billion and two. In addition, the price of losing a year of life to nothing is very high, more than the annual salary and interest and the total value of the pleasures and knowledge I accumulate in a year. Therefore, going back and forth through the mountains and valleys to search for treasures is simply not optimal. If I were to come into the world in a million incarnations and in each one I would look for immense treasures, then the total pleasure is negative.
I don't know if there is a prohibition on revealing nicknames or if it is because the nickname is not related to the real name of the Lord of the Rings. So I will ask in general: I wonder where all these nicknames come from? Are you a reincarnation of Tolkien and building a mythology from scratch?!
A. I don't see why making decisions based on expectancy seems justified or obvious to you. After all, expectancy is a calculation based on many bets and why does its application to a single bet seem unjustifiable to you?
I am talking about common sense and not about numbers. As I have explained throughout the column, numerical calculation never stands alone, and it always establishes a framework for a discussion that can be examined with common sense. My method is the same. Common sense says not to throw money away, even if there is a tiny chance of earning a huge sum. It simply won't happen, and as everywhere, one does not trust in miracles and does not base one's decisions on tiny probabilities.
B. The categorical imperative is a moral principle, not an economic one. You, in your consequentialist view, believe that morality is the action that will lead to maximum profit. But even in your view, this is not true, since such an action will not lead to maximum profit, contrary to ethical behavior according to the categorical imperative that ultimately leads to profit maximization (as in column 122).
C. I said that the calculations in the column are made without additional assumptions about the utility function. If there are additional assumptions, it is clear that this changes. I explained that my intention here was only to demonstrate why it is not always correct to act according to expectancy, and for this purpose there is no reason to need other utility functions. And in your opinion, it is difficult to understand why you would not dig in your yard with an excavator that costs a million dollars and penetrates the ground in seconds. You would not have to invest a year of effort. I do not believe you make decisions based on the expectancy criterion. For example, in the unfair coin toss, it takes a second? Would you buy a ticket for a million shekels? (Let's assume we are before your asymptote.) Knowingly, Kamina, and I don't believe you.
B. If Eliyahu offers a billion people to pay a thousand shekels, then with a one in a billion chance he will give a billion shekels to every person in the world. Does the categorical imperative require me to participate in the lottery?
C. The demonstration is based on the reader himself admitting in some unknown case that he does not decide based on expectancy. But the reader admits that this is indeed his intuition in this case, but the interpretation of this intuition is different. Not because expectancy is not calculated, but because the expectancy of the perceived profit is not high enough.
In a million shekel currency, you also have to price the suffering in terms of the loss. If you price the perceived suffering and the perceived pleasure correctly,
then I think that expectancy really predicts the decision very well.
Nicknames I don't know. Like starting over without baggage from the past. Non-existent names allow for a convenient search on the site so that only relevant results will come up. I don't know exactly.
B. There is no moral obligation to care about people (not in need) for money. Certainly not to invest your own money in it.
C. You are emptying the concept of expectancy. Any criterion on earth will fit into any definition of expectancy as long as you put all the parameters in there. Put the small chance of profit as a sorrow as a component of expectancy and everything will come out the same.
C. But I'm not saying this ad hoc. I'm talking about the feelings of pleasure and pain, and I think it's just a matter of knowing that's how they work. A "risk-averse individual" is any of us who isn't trained to be a professional broker. To compensate for the feeling of losing a million shekels that I have in my hand, you need a profit expectation of much more than a million.
I still don't see what you consider a consideration that isn't based on expectancy. Anything can be translated that way.
But what did I say? I just said that expectancy is not measured by shekels in the bank, but by perceived suffering and pleasure. It is clear that you also calculate this way, and only this position makes sense to deny. And really, I assume what is wanted, and all observed behavior only indicates to me that that person has a certain function of pleasure as a function of shekels, so that he does indeed choose according to expectancy.
And I asked a question and was not answered.
I thought now that Data is not talking about the expected profit at all, but about a single concrete profit. What you are defining is not expected profit (because it does not depend on the law of large numbers and making many bets. It is the profit from participating in a single bet).
So, you also agree that expected profit is not the criterion in such cases.
And you also do not assume what is sought, but rather define (and do not assert). If anything, this is a tautology and not an assumption of what is sought.
I said that I do assume the desired and I will interpret any observation as meaning that the person does indeed decide according to expectancy, but that his pleasure function is unique. Therefore, there is indeed no ordinary example that will refute it.
If I know the person's pleasure function, perhaps through advanced neurological research, and yet he does not act to maximize it in expectancy, this will be proof that he does not decide only according to expectancy (and whether he will get rid of the commandments as a matter of conscience). Therefore, this is not a definition but an assumption of the desired. Just as I assume that a free person did what he wanted to do at that moment.
[By the term decision according to expectancy, I am using the teaching of deciding in a single case according to the expectancy of the profit from a single game among an infinite number of games. I did not exactly understand what you argued against this].
Let me ask it this way. Suppose you were to be God and you were good and merciful. Wouldn't you program all people to decide based on expectancy? Such programming would lead to more happiness in the long run for humanity. (I hope we don't need to rely on freedom of choice here.)
This is indeed an irrefutable theory. 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 about mere profit from the game, since expected profit is defined by the average over a large number of games. You are talking about the pleasure derived from each game individually and not on an average per game.
I still don't understand what I said. What position did you attack if not the position I presented? Did you attack the position that tries to maximize the expected benefit of shekels in the bank? Or did you attack the expected benefit but you have certain assumptions about the benefit function as a function of shekels? I don't think I said anything new.
I didn't understand why I can't talk about the average benefit in a single game as calculated by the expected benefit.
B. And who did not hunt and God is not with him, etc. I came across this: https://ibb.co/JFYvK87. Hence a question for a small clarification in your method:
A person is offered two options, one option that one person will die and a second option that with a one in a billion chance that two billion will die (if a person does not choose either option, then all of humanity is killed). Do you think that from a moral perspective, one should choose the second option? Does the categorical imperative here command me to choose the first option because if it were presented to a multitude of people at the same time, then of course each one individually would have to choose the first option? And if it really is presented to a multitude of people at the same time, then each one individually should choose the first option?
Good question. In terms 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 the first choice you decide on the certain death of a person (a bit like the difference between a right and a wrong). The example in the drawing you linked to speaks of very practical chances, and there the categorical imperative is ostensibly the determining factor. But with a chance of 1 in a billion, there is the consideration that this is an expectation that will not be realized. And after all, in practice not the whole world will find itself in such a situation. Therefore, in terms of the laws of preservation of life and murder, it seems that option 2 is certainly preferable because there you do not murder anyone (the chance is indeed zero). If in practice this is an experiment that is presented to many people, then it seems to me that there is no preference because the result will be similar, and then perhaps the consideration of expectancy (which is no longer really an expectancy) is the determining factor. And even here I am not entirely sure.
The issue of certain death, which is somewhat similar to the Q, can seemingly be neutralized if we assume that the first possibility is that with a half chance two people will die (and not with a 1 chance one person will die). Do you agree that this is neutralized? Or is absolute certainty not needed for it to be similar to the Q?
Can you clarify what the side (or side) is that even if the experiment is presented to many people, you are not completely sure that you should follow the expectation? I didn't think you would give any room for such a possibility.
It's a little less complicated and there's still a difference (it's not a binary division. After all, murder isn't deterministic either: Maybe there will be a ban on guns? Maybe something else will happen in the end? Therefore, something more likely is more complicated).
Regarding when the experiment is accessible to many people, even in such a situation, I, as an individual, am still doing something that will definitely lead to the death of a person, and the other side will not lead to death at all. Only the final (consequential) weighing brings equality. Like in the troll's dilemma between the fat man who is killed with his hands and five who are killed anyway (not spared). Why is it just a sideshow? Because when this experiment is presented to all people together, I analyze it from a general social perspective and not from my perspective, and then the decision should be made collectively and not personally.
If I understand correctly, from my perspective, the categorical imperative only works for the material (to avoid wasting water, for example) but cannot work for the absolute and permit something that is forbidden without it (to bring about the certain death of a person). This is quite an innovation within the framework of the categorical and branching imperative.
I don't know if I subscribe to such a blanket definition. I would formulate that he is not the only player on the field. It is one consideration out of several. Even in our examples it is not clear what is the cola and what is the hardware.
I understand. Wherever something is just a matter of materiality, the model will be that it is one consideration out of several or are there other options?
I don't understand your insistence on the matter of gravity. In the law of souls, we always go for gravity. Our entire discussion here revolves around the question of what is gravity here: action? result? certainty or probability?
I really don't know what I said.
To the point. He threw a knife that cut with a probability of 40% (or 70%) and cut, is it defective? Perhaps here too the lack of probability impairs the degree of doing and doing? (Then it is possible that there may be a situation that passes the no but is not defective)
A. I don't understand the question. After all, in a simple and ordinary bet (toss a coin, if heads I win a shekel and if tails I lose a shekel), you also admit that in a single bet you go by the expectation. What the "probability" really means without reaching the limit in an infinite number of cases is really not clear from a "philosophical" point of view (by the way, I don't know if it's less clear than the "slope of the function at a point"). But you're working with it too. It's just that you suddenly have a jump in low numbers.
The expectation is calculated by many experiments, and applied to a single case as some criterion for the value of the bet. When it is not a common outcome, this criterion makes no sense.
It is possible that the atheists Pascal knew were the type who think there is a reasonable chance that there is a God. Let's say 20-30 percent. That is also the type of atheists I know.
I addressed that.
A. In principle, I agree, but the way Pascal presents things, you need more than 99.9 percent certainty that there is no way to be willing to not gamble with such a huge profit and a loss like burning forever.. Therefore, if you are not convinced to the nearest 100 that there is no way - the gamble is definitely good in terms of reasonable decision-making.
B. An earlier version of Pascal's gamble is in the Rambam in the Morah Bezhamim but on the negative side - according to him, transferring to Molech is just transferring without burning, and the priests would guarantee the protection of all the children at the low price of transferring only - a person who acts according to cost-benefit considerations may be tempted on the basis of Pascal's gamble, and therefore, in order to change the weight of the gamble in the eyes of the masses - the Torah promises that whoever does this will die a martyr.
C. And here is the big thing that needs to be considered: beyond the probability that the crowd gives to the question of whether or not God exists - there is also another factor to this probability - and that is that most people do not attribute a high probability to their assessment of such questions that are at the top of the world, and therefore even if it seems to a person that there is no God (or that the king is nonsense) he may prefer the utilitarian calculation that seems more certain in itself, than his assessment of the probability. In other words - add to the example of the die that a person in most cases does not have a good assessment of the probability of it falling on the specific face in the bet, but he is much more convinced that he knows what will happen if he is indeed right or wrong in the bet.
A. I'm not really sure about that. It seems to me that you're introducing expectancy considerations through the back door. I don't think most people will consider a tiny chance even if the price is enormous.
B. This is a normal profit guarantee. I don't think it's right to see this as an early version of Pascal's wager, at least no more than any calculation of expected profit that a person makes with every decision he makes.
C. It's simple, because the question of what will happen if he is right or wrong is his data and not the result of his consideration. Jewish tradition says what will happen to you if she is right. But the decision whether she is right or wrong is yours. In order for a person to also include his considerations in the component of what will happen given that he is right or wrong, he should enter the tradition and find his own interpretation for himself. Most people don't do that.
A. 1 in a thousand is 99.9 percent certainty - if it's tiny in relation to the price, I don't think so. A lottery of 1 in a thousand with a chance of a million profit or loss wouldn't you take? It's true that if you go down to 99.999 then maybe not, but most people don't have that
B. It is a gamble: it's not just “You should serve Molech” but even if it's nonsense - you won't lose anything. Therefore, the punishment given by the Torah is the death of the sons. Anyone who is convinced that serving Molech is right - the punishment from the Torah will not convince him, it is only convincing to those who serve Molech by virtue of Pascal's wager - they can protect the children but the price is low even if Molech is a lie.
According to the Rambam's method, the punishment is determined not only according to the severity of the act, but also according to the size of the interest in committing the offense and the interest of the masses in Israel to serve Molech is literally Pascal's wager. Therefore, the Torah made his punishment more severe than other matters of the law.
The error in Pascal's argument is fundamental and I have already explained it here before:
A priori, the chance that there is a God who rewards commandments and punishes transgressions is equal to the chance that there is a God who punishes commandments and rewards transgressions.
Therefore, the entire argument falls apart. And there is no probabilistic advantage here and the expectation remains zero.
Well, if you can throw common sense away and raise doubts to infinity, then why not? You can base your claim on any evidence or logical verse about God. As soon as there is a verse that says ‘God is X’ you claim that ‘a priori the chance that there is a God who is X is equal to the chance that there is a God who is not X’.
The question is whether it makes sense to most people to assume (intuition, sound logic, you name it) that God rewards for obedience. The answer is yes.
You are welcome to go to those mathematicians who waste their lives in vain trying to prove unsolved conjectures and teach them that mathematical proof is not necessary. We have discovered a new way to prove things, common sense.
The process of proof is: if a statement makes sense to most people, it is proven to be true.
Common sense is misleading. To most people, the uncertainty principle seems illusory, but it exists. To most people, it seems that a heavier object will reach the ground first, but this is wrong.
To most people, if they are shown a drawing of two arrows of the same size, in certain situations they will appear to be different in size.
To most people, the whole is greater than its parts, but in infinite groups it is possible for a partial group to have exactly the same “size” (power) as the whole group.
To most people in the ancient world, it seemed that there were several gods, each responsible for something. Today it seems illusory.
Common sense is misleading and only analytical thinking can overcome it.
Rabbi, if I understood you correctly in rejecting Dawkins' pragmatism, it follows that:
For example, a person who does not believe at all cannot become a believer or keep the commandments (because then they have no value) by virtue of a utilitarian decision; but a person who is in doubt about the existence of God (all of us, actually), he needs to make a decision (factual, whether to believe or not), but even if he has not yet made a decision, he can keep the commandments by virtue of a utilitarian decision and then they will have value (although it is "not for its own sake" but MM is again not "without faith" because he is satisfied) but he can never decide to believe/become a believer by virtue of a utilitarian decision but by virtue of a factual decision.
Is all of this true?
Yes. Except for my comment about setting a decision threshold, but I ignored that in the column.
Hello Rabbi,
I wanted to complicate the answer and say that there is a chance that there is a God who will give me an infinite reward for not serving Him in any way, and that He will punish precisely those who serve Him in a certain way. Because of the possibility that such a God exists, it is possible to say that a person will remain an atheist. No?
There is no such option because no one claims there is. You can invent countless options and that doesn't mean anything. This is Russell's teapot and the experiment on the treasure hidden in the sea islands that I mentioned in the column.
To the Rabbi, I understood the answer
Indeed, Pascal's Wager does not deal with claims but with possibilities. Even if no one claims that there is such a god, there is the possibility that there is such a god. (That is, even if it is a zero possibility that no one even thinks about, certainly does not claim it, it is still a possibility.) And if so, there is a problem with Pascal's Wager
In my opinion, he only deals with claims or possibilities that have minimal acceptability.
Micki. Excuse me. I think you are wrong. Pascal's wager deals specifically with the Christian God. That is, the one who will condemn to hell for eternity every person who is not a Christian believer. In addition, he belonged to the Jansenist movement. A movement that claims that only a limited group of believers, who were chosen in a completely fatalistic way from birth, will be saved. While the rest of humanity - their fate from a pre-fatalistic decree to hell. It is true that you are discussing the argument itself in the column and not in the statement. But many historians and thinkers have pointed out that in their opinion Pascal's wager is also directly related to the specific church in which he believed - in their opinion, it is initially addressed to the audience of believing Christians who are in doubt or skepticism. For them, there are only two options - either the Christian God exists. Then the way to salvation is only through him. Or the reality of God does not exist at all. The wager Pascal's does not include belief in Islam or Judaism. In fact, for him, belief in them is probably as risky as atheistic heresy.
Your answers in the column are all from Zionism. But they are answers that fit a different version of Pascal's Wager. Not his original version. The probability argument in the original version is indeed ridiculous and rightly so for anyone who does not think that the Christian faith is certainly much more reasonable than other religious beliefs. And from his point of view, the other religious beliefs and probabilities are equally reasonable - once this is the assumption. The whole argument falls apart. There is no point for a skeptic or an atheist who does not believe in Christianity. To believe in it only because of the fear of hell - because that same fear should rationally prevent him from apostatizing or disbelieving in any other religious belief - 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 completely probable. And those who start from the premise that he probably does not exist. It is better for them. Even if it is only out of fear. To keep the commandments. It is likely that he would be much more forgiving towards a person who tried to search for the truth and was found to be mistaken. (Whether because he failed to reach the consciousness of faith or because he chose a faith that he had rejected). Than towards a person who is an infidel and a complete atheist. But this is an assumption (which I also think is correct, by the way). That does not appear in most traditional religions - neither in Islam, nor in Christianity, nor in Judaism. And when believers of different religions - Muslims, Hindus, and the like - make a version of Pascal's Wager - they do not use this argument either. They start from the premise that there are two possibilities: either their religion is correct (and then, by the way, one must make an effort and believe in it wholeheartedly. It is not unrealistic that there is a movement among the masters that claims that faith is weak On a scale of 4 or 5, it is worth something. Or there is no God at all. - Therefore, it is better to believe in their religion. They will also agree that, assuming that other beliefs are also reasonable, the bet falls.
I addressed all of this.
1. Even if he believed that Christianity was more logical and reasonable, it is not necessary for his argument.
2. Just as I disagree with the accepted approach in Judaism, I will also disagree with Christianity and Islam. In my opinion, God of any religion is not supposed to punish rapists. This is the assumption on which I make the bet.
I understand you. So apparently the dispute between us is a matter of pure semantics. I agree that Pascal's Wager in your version is rational and completely reasonable. I just don't think that in this case it's really correct to call it Pascal's Wager. But it's really a matter of pure language and concepts.
*I just don't think it's really right to call it that in this case*
Correct error
I don't think atheism is the worst option. In my opinion, we need to look at the risk of choosing the wrong religion. Many times choosing the wrong religion will bring greater suffering according to another religion, and therefore the suffering of choosing the wrong religion is greater than the suffering of 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. Although they believe in God, they reject the Messiah and the Son of God, and therefore rebel against Him, and their situation is worse because they recognize 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 be among the main streams because of the support of most streams for Zionism (even if they do not actively support it but treat the state as an existing situation). Therefore, most of them are enemies of Islam.
In any case, the risk increases when choosing the wrong religion.
From what I understand, the law of large numbers refers to a finite expectation. Not an infinite expectation. That is, in a number of trials as large as we want, the profit tends to an expectation that is a finite number. But what happens if the profit is infinite (eternal life with an infinite reward that cannot be quantified in terms of its value). In this case, when the number of trials tends to infinity, the expectation also tends to infinity.
In such a case, I do not think it is possible to conclude according to the classical concept.
I don't see any difference between a finite and infinite lifespan. You can treat it as a very large lifespan (or as large as you like).
This is exactly the problem. If you treat the expectation as a number as large as you want, then the profit becomes large until it is equal to the required price. Let's say if the required threshold for which I am supposed to make the choice is M, then I will choose the expectation to be M+1 (after all, this is the definition of striving for 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.
On the 11th of Elul, Tashahhud
Pascal's argument is very simple and logical. If there is an option that involves risk compared to a neutral option - a person will feel it and do what he can to eliminate the possibility of risk.
Therefore, a person takes the vaccine against Corona, even though the chance of getting sick and being seriously injured by Corona is quite small, even without a vaccine; and therefore a person also takes out insurance for damages that the risk of suffering from is small.
In both cases, the calculation is that if nothing happens - you have not lost a significant loss, a little wealth or some minor side effect. But if the less common pessimistic scenario comes true and a person gets infected or is damaged - then if he did not get vaccinated or did not take out insurance and what happened happened, then the person has "eaten it big".
And so Pascal says about God. If it turns out that he exists and you disobeyed his will, then you have eaten it in a big way, and doubt God for sure 🙂
And your sign is that you don't put a line lightly on the fear that there is a God 🙂 It was worthwhile 🙂
With blessings, there is doubt, there is no doubt
Don't you take risks? Do you align yourself with the strictest positions among all the rabbis of our time and the past? If you start to involve too many beliefs, this is what you personally think is right, then you've canceled the whole bet.
On the 11th of Elul, 5th of September
To Amram, peace be upon him,
The Torah did not command to hear all opinions but to reach the decision of the ‘judge who would be in those days, and in a dispute, to follow the majority of the sages, as it is written ‘following many to incline’. When there was a large house in the office of the sheriff – the decision was immediate and quick.
When the people were scattered into exile, the decision became more difficult, but still over the generations, ‘signatures’ arose that led to a broad consensus. Such was the signature of the Mishnah, the Talmud, and the Shul, which led to broad agreements that formed the basis for future generations. The aim is to reach a gathering of methods and a decision according to the 'agreed opinion of the majority of poskim' (see, for example, the words of Rabbi Nissim, Responsa Yain HaTov 2, Yod Si' 11).
Of course, it is impossible to place all our trust solely on the 'Pascalian gamble', which cannot decide which way is just to worship God. But Pascal's explanation gives a strong indication of the search for a 'leader for the capital' who demands that one behave accordingly, and who is found with a whole, critical, and 'hard-nosed' people Who experienced a powerful public divine revelation, and whose steadfast testimony endures over many generations and despite exiles and persecutions, and leads to the miraculous existence of the people of the Torah despite all odds – there is a strong indication here that there is something real here.
With greetings, Yaron Fish”l Ordner
The argument against paralyzing skepticism is Miriam's argument against Amram, who divorced his wife due to the decree ‘every unborn son’. Against this, Miriam argued that just as one must feel that the newborn will be thrown away, one must feel that a daughter will be born and a son will be born that they will be able to hide. The ’Pascalian gamble’ to have children despite the great chance that they will not be viable – is what brought into the world the Savior of Israel, who was miraculously saved despite the slim chance that a miracle would occur.
Paragraph 3, line 1
Of course, we can't throw all our love at…
We had a discussion about this before here: https://mikyab.net/posts/64124#comment-27350
But I will mention here again that the more money a person has, the less useful he is (the marginal usefulness decreases). Therefore, a bet that costs me one dollar and in which I win a billion dollars with a chance of one in 100 million is probably an unprofitable bet in terms of the expected usefulness (although in terms of the usefulness of the profit it does pay off). This should be taken into account in the financial examples. Regarding spiritual profit, the spiritual usefulness does not necessarily decrease the greater it is, and therefore there is an important difference here in relation to the financial examples.
Expected profit has a meaning as if an infinite number of attempts had been made.
And so, even the atheist, every time he fulfills a commandment, he first considers whether there is value in it, that is, whether God exists. And so this bet (if there is a God I will receive a billion and if not I will lose only a free life without debts) is made every day by man, and not just once.
Is this a question for me? I didn't understand it.
I'll just comment on your last sentence. It's not a bet that's made anew every day, but in that same bet you invest another shekel every day. If there is a God, he exists every day, and if there isn't, he doesn't exist every day. These are not different bets.