Pascal's Wager

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Pascal’s wager The argumentOne conclusion:from miracles testimony is one, but not the only,Hume’s source of principle evidence which about we testimony. should use when forming a belief. Testimony is relevant because it has a (relatively) high probability 1. There haveof being been true. miracles. But, like any evidence, this can be overriddenWe should by not other believe sources that of M evidence happened on the basis of the 2. If there (like,have forbeen example, miracles, contrary God testimony)exists. which have give a high probability to the negation ______testimony unless the following is the case: of the proposition in question. C. God exists. The probability of the testimony being false < the probability of M occurring. 2.2 Testimony about miracles

Last time Wewe nowdiscussed need to Hume’s apply these principle general about points testimony: about testimony one should and never evidence believe to the testimony case of about some event unless themiracles. probability One of conclusion that testimony seems being to follow false immediately: is less than the probability of the event occurring. This raised the question: how do we determine the relevant probabilities? “That no testimony is suﬃcient to establish a miracle, unless the testimony Hume’s idea seemsbe of suchto be a this. kind, When that its we falsehood are trying would to ﬁgure be more out miraculous,the probability than of the some fact, event happening in certain which it endeavors to establish . . . ” (77) circumstances, we ask: in the past, how frequently as that event been observed to occur in those circumstances? Our answer to this question will give us the probability of the relevant event. The problem for the believer in miracles is that miracles, being departures from the laws of nature, seem to be exactly the sorts of events which we should not expect to happen. This, HumeAs thinks, Hume putsis enough it: to show us that we ought never to believe testimony regarding miraculous events:

“A miracle is a violation of the laws of nature; and as a ﬁrm and unalterable experience has established these laws, the proof against a miracle, from the very nature of the fact, is as entire as any argument from experience can possibly be imagined . . . There must be a uniform experience against every miraculous event, otherwise the event would not merit that appellation.” (76- 7)

Hume’s pointThe is implied that miracles question are is: couldalways testimony departures ever from provide the strong ordinary enough laws evidence of nature. to overrideBut the ordinary laws of nature are regularitiesour massive which evidence have been in favor observed of nature’s to hold following 100% itsof usualthe time. course Of (which course, is we also have evidence not observed testimony to be correctagainst 100% theof the occurrence time. Hence, of the the miracle)? probability of testimony regarding a miracle being false will always be greater than the probability of the miraculous event; and then it follows from Hume’s principle about testimony that we should never accept the testimony. 2.3 The relevance of religious diversity

In II, Hume adds another reason to be skeptical about testimony about miracles, when he§ writes

“there is no testimony for any [miracles] . . . that is not opposed by an inﬁ- nite number of witnesses; so that not only the miracle destroys the credit of testimony, but the testimony destroys itself. To make this better understood, let us consider, that, in matters of religion, whatever is diﬀerent is contrary . . . Every miracle, therefore, pretended to have been wrought in any of these religions (and all of them abound in miracles) . . . has the same force . . . to overthrow every other system.” (81)

Is this best construed as a separate argument against miracles, or as part of the argument sketched above? If the latter, how does it ﬁt in?

3 The argument from miracles Hume’s principle about testimony.

1. There have been miracles. We should not believe that M happened on the basis of the 2. If there have been miracles, God exists. ______testimony unless the following is the case: C. God exists. The probability of the testimony being false < the probability of M occurring.

Pascal situates the question of miracles within (one part of) the Christian tradition. But Hume’s idea seems to be this. When we are trying to ﬁgure out the probability of some event happening in certain the question we want to answer is more general: can miracles play this kind of central circumstances, we ask: in the past, how frequently as that event been observed to occur in those circumstances? Our answer roleto this in question justifying will religious give us beliefthe probability of any sort? of the relevant event. We will focus on the question of whether miracles can justify the religious beliefs of people Hume’s pointwho is that have miracles not themselves are always witnessed departures miracles. from the ordinary laws of nature. But the ordinary laws of nature are regularities which have been observed to hold 100% of the time. Of course, we have not observed testimony to be correct 100% of the time. Hence, the probability of testimony regarding a miracle being false will always be greater than the probability of 2the miraculous Hume’s argument event; and againstthen it follows belief from in miraclesHume’s principle about testimony that we should never accept the testimony. Hume thinks that they cannot, and indeed that no rational person would base belief in Hence Hume’sGod conclusion: on testimony that miracles have occurred. He says:

“. . . therefore we may establish it as a maxim, that no human testimony can have such force as to prove a miracle, and make it a just foundation for any system of religion.” (88)

This is Hume’s conclusion. We now need to understand his argument for it, which begins with some premises about the role of perceptual evidence and testimony in the forming of beliefs.

2.1 Testimony and evidence

Hume’s ﬁrst claim is that we should base belief on the available evidence:

“A wise man, therefore, proportions his belief to the evidence. . . . He weighs the opposite experiments: He considers which side is supported by the greater number of experiments: To that side he inclines, with doubt and hesitation; and when at last he ﬁxes his judgement, the evidence exceeds not what we properly call probability.” (73-4)

The general moral seems to be correct: when deciding whether to believe or disbelieve some proposition, we should weigh the evidence for and against it to see whether it makes the proposition or its negation more probable.

How does this sort of general principle ﬁt with our practice of basing beliefs on testimony? Hume has a very plausible answer:

“we may observe, that there is no species of reasoning more common, more useful, and even necessary to human life, than that which is derived from the testimony of men, and the reports of eye-witnesses and spectators. . . . I shall not dispute about a word. It will be suﬃcient to observe, that our assurance in any argument of this kind is derived from no other principle than our observation of the veracity of human testimony, and of the usual conformity of facts to the reports of witnesses.” (74)

2 The argument from miracles Hume’s principle about testimony.

Hume’s idea seems to be this. When we are trying to ﬁgure out the probability of some event happening in certain circumstances, we ask: in the past, how frequently as that event been observed to occur in those circumstances? Our answer to this question will give us the probability of the relevant event.

Hume’s point is that miracles are always departures from the ordinary laws of nature. But the ordinary laws of nature are regularities which have been observed to hold 100% of the time. Of course, we have not observed testimony to be correct 100% of the time. Hence, the probability of testimony regarding a miracle being false will always be greater than the probability of the miraculous event; and then it follows from Hume’s principle about testimony that we should never accept the testimony.

On this reading, Hume’s argument rests on some principle of the following sort:

If some event has never been observed to occur before, then the probability of it occurring is 0%.

Let’s call this the zero probability principle.

This, plus Hume’s principle about testimony, is clearly enough to show that one ought never to believe in miracles on the basis of testimony. The argument from miracles Hume’s principle about testimony.

The zero probability principle:

If some event has never been observed to occur before, then the probability of it occurring is 0%.

This, plus Hume’s principle about testimony, is clearly enough to show that one ought never to believe in miracles on the basis of testimony.

Interestingly, it also seems to be enough to establish a stronger claim: one is never justiﬁed in believing in the existence of miracles, even if one is (or takes oneself to be) an eyewitness.

After all, perceptual experiences of the world, like testimony, don’t conform to the facts 100% of the time. So, the probability of a miraculous event M occurring will always, given the above principle about probabilities, be less than the probability of one’s perceptual experience being illusory. Hence, it seems, one would never be justiﬁed in believing in the existence of a miracle, even on the basis of direct perceptual experience.

This might at ﬁrst seem like a good thing for Hume’s argument: it shows not just that one an never believe in miracles on the basis of testimony, but also that one can never believe in them for any reason at all! But in fact this points to a problem for the zero probability principle. The argument from miracles Hume’s principle about testimony.

The zero probability principle:

If some event has never been observed to occur before, then the probability of it occurring is 0%.

Consider the following sort of example:

You are a citizen of Pompeii in AD 79, and there is no written record of the tops of mountains erupting and spewing forth lava. Accordingly, following the zero probability principle, you regard the chances of such a thing happening as 0%. On the other hand, you know that your visual experiences have been mistaken in the past, so you regard the chances of an arbitrary visual experience being illusory as about 1%. Then you have a very surprising visual experience: black clouds and ash shooting out of nearby Mt. Vesuvius. What is it rational for you to believe?

This sort of case seems to show that the zero probability principle is false. Other such examples involve falsification of well-confirmed scientific theories.

So, if Hume’s argument depends on the zero probability principle, it is a failure. Can we come up with another principle, which would avoid these sort of counterexamples while still delivering the result that Hume wants? The argument from miracles Hume’s principle about testimony.

The zero probability principle:

If some event has never been observed to occur before, then the probability of it occurring is 0%.

It seems that we can. All Hume’s argument needs, it would seem, is the following trio of assumptions:

(a) If some event has never been observed to occur before, then the probability of it occurring is at most X%. (b) The probability of a piece of testimony being false is always at least Y%.

(c) Y>X Suppose, for example, that the probability of an event of some type which has never before been observed is at most 1%, and that there is always at least a 10% chance of some testimony being false. If we assume Hume’s principle about testimony, would this be enough to deliver the conclusion that we are never justiﬁed in believing in miracles on the basis of testimony? The argument from miracles Hume’s principle about testimony.

(a) If some event has never been observed to occur before, then the probability of it occurring is at most X%. (b) The probability of a piece of testimony being false is always at least Y%. (c) Y>X

Suppose, for example, that the probability of an event of some type which has never before been observed is at most 1%, and that there is always at least a 10% chance of some testimony being false. If we assume Hume’s principle about testimony, would this be enough to deliver the conclusion that we are never justiﬁed in believing in miracles on the basis of testimony?

Only if we assume that we only ever have testimony from a single witness. Suppose that we have three witnesses, each of whom are 90% reliable, and each independently reports that M has occurred. Then the probability of each witness being wrong is 10%, but the probability of all three being wrong is only 0.1%. This, by the above measure, would be enough to make it rational to believe that M happened.

So it seems that the possibility of multiple witnesses shows that (a)-(c) are not enough to make Hume’s argument against justiﬁed belief in miracles on the basis of testimony work. (This is true no matter what values we give to “X” and “Y”.)

(One cautionary note: it is important to distinguish between having testimony from multiple witnesses and having testimony from a single witness who claims there to have been multiple witnesses.) Hume’s principle about testimony. The argument from miracles 1. There have been miracles. We should not believe that M happened on the basis of the 2. If there have been miracles, God exists. testimony unless the following is the case: ______C. God exists. The probability of the testimony being false < the probability of M occurring.

So far, we have been assuming that Hume’s principle about testimony is true, and asking what assumption could be added to this principle to make Hume’s argument work. But there is reason to doubt whether the principle about testimony is itself true.

This principle can sound sort of obvious; but it isn’t, as some examples show. First, what do you think that the probability of the truth of testimony from the writers of the South Bend Tribune would be?

Let’s suppose that you think that it is quite a reliable paper, and that its testimony is true 99.9% of the time, so that the probability of its testimony being false is 0.1%.

Now suppose that you read the following in the South Bend Tribune:

“The winning numbers for Powerball this weekend were 1-14-26-33-41-37-4.”

What are the odds of those being the winning numbers for Powerball? Well, the same as the odds of any given combination being correct, which is 1 in 195,249,054. So the probability of the reported event occurring is 0.0000005121663739%.

So, if Hume’s principle about testimony is correct, one is never justiﬁed in believing the lottery results reported in the paper, or on the local news, etc. But this seems wrong: one can gain justiﬁed beliefs about the lottery from your local paper, even if it is the South Bend Tribune.

One might wonder how, if at all, Hume’s principle could be modiﬁed to avoid these counterexamples. The argument from miracles 1. There have been miracles. 2. If there have been miracles, God exists. ______C. God exists.

This is a long way from showing that the argument from miracles is a success: for that purpose, we would have to consider speciﬁc examples of miracles, and the sorts of evidence given for their occurrence.

We would also, as the example of Powerball shows, have to get a bit clearer about when testimony is and is not sufﬁcient for justiﬁed belief.

We would also have to answer the question of when we are justiﬁed in believing that some event which is contrary to the usual natural order has supernatural causes. What our discussion today shows is something much more limited: that one prominent attempt to show that the argument from miracles can’t succeed is, as it stands, unconvincing. So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not “Does God exist?” but “Should we believe in God?” What is distinctive about Pascal’s approach to the latter question is that he thinks that we can answer it without ﬁrst answering the former question.

Here is what he has to say about the question, “Does God exist?”:

“Let us then examine this point, and let us say: ‘Either God is or he is not.’ But to which view shall we be inclined? Reason cannot decide this question. Inﬁnite chaos separates us. At the far end of this inﬁnite distance a coin is being spun which will come down heads or tails. How will you wager? Reason cannot make you choose either, reason cannot prove either wrong.”

Pascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question. But, he notes, this does not remove the necessity of our choosing to believe, or not believe, in God:

“Yes, but you must wager. There is no choice, you are already committed. Which will you choose then? Let us see: since a choice must be made, let us see which oﬀers you the least interest. You have two things to lose: the true and the good; and two things to stake: your reason and your will, your knowledge and your happiness. . . . Since you must necessarily choose, your reason is no more aﬀronted by choosing one rather than the other. . . . But your happiness? Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

This is quite different than the sorts of arguments we have discussed so far for belief in God. Each of those arguments made a case for belief in God on the basis of a case for the truth of that belief; Pascal focuses on the happiness that forming the belief might bring about. “Yes, but you must wager. There is no choice, you are already committed. Which will you choose then? Let us see: since a choice must be made, let us see which oﬀers you the least interest. You have two things to lose: the true and the good; and two things to stake: your reason and your will, your knowledge and your happiness. . . . Since you must necessarily choose, your reason is no more aﬀronted by choosing one rather than the other. . . . But your happiness? Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

But what is Pascal’s argument that belief in God will lead to greater happiness? It seems to be contained in the last two sentences of this passage. Pascal is saying that if you believe in God and God exists (“you win”), you win eternal life (“you win everything”), whereas if God does not exist, it doesn’t matter whether you believe in God (“you lose nothing”).

Pascal was one of the first thinkers to systematically investigate the question of how it is rational to act under certain kinds of uncertainty, a topic now known as “decision theory.” We can use some concepts from decision theory to get a bit more precise about how Pascal’s argument here is supposed to work. “Yes, but you must wager. There is no choice, you are already committed. Which will you choose then? Let us see: since a choice must be made, let us see which offers you the least interest. You have two things to lose: the true and the good; and two things to stake: your reason and your will, your knowledge and your happiness. . . . Since you must necessarily choose, your reason is no more affronted by choosing one rather than the other. . . . But your happiness? Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

We are facing a decision in which we have only two options: belief or nonbelief. And there is one unknown factor which will determine the outcome of our choice: whether or not God exists. So, pairing each possible choice with each possible outcome, there are four possibilities. Pascal is here drawing attention to an interesting feature of the way these choices and outcomes line up in the case of belief in God.

This can be illustrated by an analogy with a simple bet. “Yes, but you must wager. There is no choice, you are already committed. Which will you choose then? Let us see: since a choice must be made, let us see which oﬀers you the least interest. You have two things to lose: the true and the good; and two things to stake: your reason and your will, your knowledge and your happiness. . . . Since you must necessarily choose, your reason is no more aﬀronted by choosing one rather than the other. . . . But your happiness? Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

Should you choose heads, or tails? It seems that neither option is better; each gives you This can be illustrated by an analogy with a simple bet. the same odds of winning and losing, and the relevant amounts are the same in each case. Suppose I offer you the chance of choosing heads or tails on a fair coin ﬂip, with the following payoffs:Now if suppose you choose that heads, you areand giventhe coin a slightlycomes up stranger heads, you and win more $10; complicated if you choose bet: heads, if youand the choosecoin comes heads, up andtails, theyou coinwin $5. comes If you up choose heads, tails, you then win if the10; coin if you comes choose up heads, heads, you and get the $2, andcoin if it comes comes up up tails, tails, you you get win$5. $5. However, if you choose§ tails, and the coin comes up heads, you win 0; but if you choose tails, and the coin comes up tails, you win $5. Your We choicescan represent can then the§ bepossibilities represented open as to follows: you with the following table:

Courses of action Possibility 1: Possibility 2: Coin comes up heads Coin comes up tails Choose ‘heads’ Win $10 Win $5 Chose ‘tails’ Win $2 Win $5

Obviously,If given you this should bet, choose should heads. you choose One way heads, to put or the choose reason tails? for this Unlike is as follows: the simpler there is bet, one it possibilityseems on that which here you there are better is an off obvious having answer chosen to heads, this and question: no possibility you should on which choose you are heads. worse off choosingThe reason heads. why This is clear. is to say There that are choosing only two heads relevant dominates ways choosing that things tails. could turn out: the coin could come up heads, or come up tails. If it comes up heads, then you are better oﬀ It seems very plausible that if you are choosing between A and B, and choosing A dominates choosing if you chose ‘heads.’ If it comes up tails, then it doesn’t matter which option you chose. B, it is rational to choose A. One way of putting this scenario is that the worst outcome of choosing ‘heads’ is as good as the best outcome of choosing ‘tails’, and the best outcome of choosing ‘heads’ is better than the best outcome of choosing ‘tails’. When this is true, we say that choosing ‘heads’ superdominates choosing ‘tails.’

It seems clear that if you have just two courses of action, and one superdominates the other, you should choose that one. Superdominance is the ﬁrst important concept from decision theory to keep in mind.

The next important concept is expected utility. The expected utility of a decision is the amount of utility (i.e., reward) that you should expect that decision to yield. Recall the more complicated coin bet above. If you choose ‘heads’, the bet went, there were two possible outcomes: either the coin comes up heads, and you win $10, or the coin comes up tails, and you win $5. Obviously, this is a pretty good bet, since you win either way. Now suppose that you are oﬀered the chance to take this bet on a fair coin toss, but have to pay $7 to make the bet. Supposing that you want to maximize your money, should you take the bet?

The answer here may not seem obvious — it is certainly not as obvious as the fact that you should, if given the choice, choose ‘heads’ rather than ‘tails.’ But this is the kind of question that calculations of expected utility are constructed to answer. To answer the question:

Should I pay $7 to have the chance to bet ‘heads’?

we ask

Which is higher: the expected utility of paying $7 and betting ‘heads’, or the expected utility of not paying, and not betting?

3 “Yes, but you must wager. There is no choice, you are already committed. Which will you choose then? Let us see: since a choice must be made, let us see which offers you the least interest. You have two things to lose: the true and the good; and two things to stake: your reason and your will, your knowledge and your happiness. . . . Since you must necessarily choose, your reason is 3.1no more The affronted argument by choosing from one superdominance rather than the other. . . . But your happiness? Let us weigh up the gain and the loss involved in calling heads that God exists. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.” One version of Pascal’s argument is that the decision to believe in God superdominates Let’sthe say decision that if we notare choosing to believe between in God, actions in A the and above B, sense. He seems to have this in mind when he writes, A dominates B if and only if there is at least one scenario in which A leads to a better outcome than B, and no scenario in which A leads to a worse outcome than B. “. . . if you win, you win everything, if you lose you lose nothing.” One way to read Pascal here is as saying that believing in God dominates nonbelief. If this were true, this would be a veryThis strong indicates argument that, that atit is least rational at to this believe point in God. in the Here’s next, how he he seesmight the be thinking decision about to believethe choice or whether to believe: not believe in God as follows: Courses of action Possibility 1: Possibility 2: God exists God does not exist Believe in God Infinite reward Lose nothing, gain nothing Do not believe in God Infinite loss Lose nothing, gain nothing

How should you respond to the choice of either believing, or not believing, in God? It Ifseems this is the easy: right just way as to inthink the about above the case choice you between should belief choose and non-belief, ‘heads’, so then in believing this case seems you shouldto dominate notchoose believing. belief in God. After all, belief in God superdominates non-belief: the worst case scenario of believing in God is as good as the best case scenario of non-belief, and the Isbest this the case right scenario way to think of believing about the inchoice? God is better than the best case scenario of non-belief.

Pascal, however, seems to recognize that there is an objection to this way of representing the choice of whether or not one should believe in God: one might think that if one decides to believe in God, and it turns out that God does not exist, there has been some loss: you are then worse oﬀ than if you had not believed all along. Why might this be?

If this is right, then it looks like the following is a better representation of our choice:

Courses of action Possibility 1: Possibility 2: God exists God does not exist Believe in God Inﬁnite reward Loss Do not believe in God Inﬁnite loss Gain

If this is a better representation of the choice, then it is not true that believing in God superdominates non-belief.

3.2 The argument from expected utility

How, then, should we decide what to do? One method was already suggested earlier: you should see which of the two courses of action has the higher expected utility.

But, to ﬁgure this out, we have to know what probabilities we should assign to the possibilities that God exists, and that God does not exist. Pascal suggests when setting up the argument that there is “an equal chance of gain and loss”, which would put the probabilities of each at 1/2.

We also need to ﬁgure out how to measure the utility of each of the two outcomes, given either of the two choices. Below is one way to do that:

5 3.1 The argument from superdominance

One version of Pascal’s argument is that the decision to believe in God superdominates the decision not to believe in God, in the above sense. He seems to have this in mind when he writes,

“. . . if you win, you win everything, if you lose you lose nothing.”

This indicates that, at least at this point in the next, he sees the decision to believe or not believe in God as follows: Courses of action Possibility 1: Possibility 2: God exists God does not exist Believe in God Inﬁnite reward Lose nothing, gain nothing Do not believe in God Inﬁnite loss Lose nothing, gain nothing

How should you respond to the choice of either believing, or not believing, in God? It Ifseems this is easy:the right just way as to in think the about above the case choice you between should belief choose and ‘heads’,non-belief, so then in thisbelieving case seems you should to dominate notchoose believing. belief in God. After all, belief in God superdominates non-belief: the worst case Isscenario this the right of believing way to think in about God the is choice? as good as the best case scenario of non-belief, and the best case scenario of believing in God is better than the best case scenario of non-belief. However, it is not obvious that belief does dominate nonbelief, since it is not obvious that one really loses Pascal,nothing if however, one believes seems in God to in recognize the scenario that in which there God is andoes objection not exist. toWouldn’t this way one ofbe representingundertaking thereligious choice obligations of whether which or one not might one have should avoided? believe And in isn’t God: having one a might false belief think something that if one bad decidesin itself? to believe in God, and it turns out that God does not exist, there has been some loss: youIf we arethink then about worse a scenario oﬀ than in which if you one hadbelieves not in believed God but God all along.does not Why exist mightas involving this some be? loss -- either because one would not do things which one might like to do, or because having a false belief is in Ifitself this a loss is right, -- then then believing it looks does likenot dominate the following not believing. is a better representation of our choice:

Courses of action Possibility 1: Possibility 2: There is, however, another way to think about Pascal’s argument, which does not involve dominance God exists God does not exist reasoning. This uses another concept from decision theory, namely expected utility. Believe in God Inﬁnite reward Loss Do not believe in God Inﬁnite loss Gain

If this is a better representation of the choice, then it is not true that believing in God superdominates non-belief.

3.2 The argument from expected utility

How, then, should we decide what to do? One method was already suggested earlier: you should see which of the two courses of action has the higher expected utility.

We also need to ﬁgure out how to measure the utility of each of the two outcomes, given either of the two choices. Below is one way to do that:

5 There is, however, another way to think about Pascal’s argument, which does not involve dominance reasoning. This uses another concept from decision theory, namely expected utility.

Recall the example of the coin toss above, in which, if you bet heads and the coin comes up heads, you win $10, and if it comes up tails, you win $5. Suppose I offer you the following deal: I will give you those payoffs on a fair coin ﬂip in exchange for you paying me $7 for the right to play. Should you take the bet?

Here is one way to argue that you should take the bet. There is a 1/2 probability that the coin will come up heads, and a 1/2 probability that it will come up tails. In the ﬁrst case I win $10, and in the second case I win $5. So, in the long run, I’ll win $10 about half the time, and $5 about half the time. So, in the long run, I should expect the amount that I win per coin ﬂip to be the average of these two amounts -- $7.50. So the expected utility of my betting heads is $7.50. So it is rational for me to pay any amount less than the expected utility to play (supposing for simplicity, of course, that my only interest is in maximizing my money, and I have no other way of doing so).

To calculate the expected utility of an action, we assign each outcome of the action a certain probability, thought of as a number between 0 and 1, and a certain value (in the above case, the relevant value is just the money won). In the case of each possible outcome, we then multiply its probability by its value; the expected utility of the action will then be the sum of these results.

In the above case, we had (1/2 * 10) + (1/2 * 5) = 7.5.

The notion of expected utility seems to lead to a simple rule for deciding what to do:

The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility. Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

The rule of expected utility A diﬀerence in kind between this argument and the arguments for the existence of God we haveIt is considered.always rational Pascal to pursue does notthe providecourse of us action any evidence with the forhighest thinking expected that Godutility. exists. He gives us prudential rather than theoretical reasons for forming a belief that God exists. The distinction between these two kinds of reasons.

OurPascal question goes onis then: to spell how out might more Pascal explicitly argue histhat reasoning believing forin God thinking has higher that itexpected is rational utility to than nonbelief?believe in Our God, answer using seems an analogy to be withgiven gambling:in the following passage:

“. . . since there is an equal chance of gain and loss, if you stood to win only two lives for one you could still wager, but supposing you stood to win three? . . . it would be unwise of you, since you are obliged to play, not to risk your life in order to win three lives at a game in which there is an equal chance of winning and losing. . . . But here there is an infinity of happy life to be won, one chance of winning against a finite number of chances of losing, and what you are staking is finite. That leaves no choice; wherever there is infinity, and where there are not infinite chances of losing against that of winning, there is no room for hesitation, you must give everything. And thus, since you are obliged to play, you must be renouncing reason if you hoard your life rather than risk it for an infinite gain, just as likely to occur as a loss amounting to nothing.”

Clearly Pascal thinks that there is some analogy between believing in God and making an even-odds bet in which you stand to win three times as much as you stand to lose; to be What is Pascal’s argument here? How should we perform the relevant expected utility calculations? more precise about what this analogy is supposed to be, we can introduce some concepts from decision theory, the study of the principles which govern rational decision-making.

2 The wager and decision theory

Pascal was one of the ﬁrst thinkers to systematically investigate what we now call ‘decision theory’, and elements of his thought on this topic clearly guide his presentation of the wager.

Suppose that we have two courses of action between which we must choose, and the consequences of each choice depend on some unknown fact. E.g., it might be the case that we have to bet on whether a coin comes up heads or tails, and what the result of our bet is depends on whether the coin actually does come up heads or tails. Imagine ﬁrst a simple bet in which if you guess correctly, you win $1, and if you guess incorrectly, you lose $1. We could represent the choice like this:

Courses of action Possibility 1: Possibility 2: Coin comes up heads Coin comes up tails Choose ‘heads’ Win $1 Lose $1 Chose ‘tails’ Lose $1 Win $1

2 Let us assess the two cases: if you win you win everything, if you lose you lose nothing. Do not hesitate then; wager that he does exist.”

A diﬀerence in kind between this argument and the arguments for the existence of God we have considered. Pascal does not provide us any evidence for thinking that God exists. He givesThe usruleprudential of expectedrather utility than theoretical reasons for forming a belief that God exists. The distinction between these two kinds of reasons. It is always rational to pursue the course of action with the highest expected utility. Pascal goes on to spell out more explicitly his reasoning for thinking that it is rational to believe in God, using an analogy with gambling:

PascalClearly says Pascal two thinks things thatwhich there help is us some here. analogy First, he between emphasizes believing that “there in God is an and equal making chance an of gain and loss”even-odds -- an equal bet in chance which that you God stand exists, to win and three that timesGod does as much not exist. as you This stand means to that lose; we to should be assign eachmore a precise probability about of 1/2. what this analogy is supposed to be, we can introduce some concepts from decision theory, the study of the principles which govern rational decision-making. Second, he says that in this case the amount to be won is inﬁnite. We can represent this by saying that the utility of belief in God if God exists is ∞. 2 The wager and decision theory

Pascal was one of the ﬁrst thinkers to systematically investigate what we now call ‘decision theory’, and elements of his thought on this topic clearly guide his presentation of the wager.

Courses of action Possibility 1: Possibility 2: Coin comes up heads Coin comes up tails Choose ‘heads’ Win $1 Lose $1 Chose ‘tails’ Lose $1 Win $1

2 The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

Pascal says two things which help us here. First, he emphasizes that “there is an equal chance of gain and loss” -- an equal chance that God exists, and that God does not exist. This means that we should assign each a probability of 1/2.

Second, he says that in this case the amount to be won is inﬁnite. We can represent this by saying that the utility of belief in God if God exists is ∞.

Let’s concede the objection made above: if we believe in God, and God does not exist, this involves some loss of utility. This loss will be ﬁnite -- let’s symbolize it by word “LOSS”.

We can then think of the decision as follows:

Possibility 1: God Possibility 2: God does Courses of action exists (Prob. = 0.5) not exist (Prob. = 0.5)

Believe in God ∞ LOSS

Don’t believe 0 0

Supposing that this is the correct assignment of values, what is the expected utility of each action? The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

Supposing that this is the correct assignment of values, what is the expected utility of each action?

Possibility 1: God Possibility 2: God does Courses of action Expected utilities exists (Prob. = 0.5) not exist (Prob. = 0.5)

Believe in God ∞ LOSS .5* ∞ + .5-LOSS= ∞

Don’t believe 0 0 .5*0 +.5*0 = 0

So it looks as though the expected utility of believing in God is inﬁnite, whereas the expected utility of nonbelief is 0. If the rule of expected utility is correct, it follows that it is rational to believe in God -- and it is not a very close call.

We will consider two sorts of objections to this expected utility argument for the rationality of belief in God. One sort is based on the values in the above table: one might dispute Pascal’s claims about the values or probabilities of different outcomes. A second sort of objection is based on the rule of expected utility itself. The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

Possibility 1: God Possibility 2: God does Courses of action Expected utilities exists (Prob. = 0.5) not exist (Prob. = 0.5)

Believe in God ∞ LOSS .5* ∞ + .5-LOSS= ∞

Don’t believe 0 0 .5*0 +.5*0 = 0

Let’s consider a few objections of the ﬁrst sort. First, suppose that the probability that God exists is not 1/2, but some much smaller number -- say, 1/100. Would that affect the argument above?

It seems not. Let’s suppose that the probability that God exists is some nonzero ﬁnite number m, which can be as low as one likes. Then it seems that we should think of the choice between belief and nonbelief as follows:

Possibility 1: God Possibility 2: God does Courses of action Expected utilities exists (Prob. = m) not exist (Prob. = 1-m)

Believe in God ∞ LOSS m* ∞ + .(1-m)-LOSS= ∞

Don’t believe 0 0 m*0 + (1-m)*0 = 0 The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

Possibility 1: God Possibility 2: God does Courses of action Expected utilities exists (Prob. = m) not exist (Prob. = 1-m)

Believe in God ∞ LOSS m* ∞ + .(1-m)-LOSS= ∞

Don’t believe 0 0 m*0 + (1-m)*0 = 0

This is a real strength of Pascal’s argument: it does not depend on any assumptions about the probability that God exists other than the assumption that it is nonzero. In other words, he is only assuming that we don’t know for sure that God does not exist, which seems to many people -- including many atheists -- to be a reasonable assumption.

It is always rational to pursue the course of action with the highest expected utility.

But another objection might seem more challenging. It seems that Pascal is assuming that, if God exists, there is a 100% chance that believers will get inﬁnite reward. But why assume that? Why not think that if God exists there is some chance that believers will get inﬁnite reward, and some chance that they won’t? How would that affect the above chart? To accommodate this possibility, we would have to add another column to our chart, to represent the two possibilities imagined. Let’s call these possibilities “Rewarding God” and “No reward God”, and let’s suppose that each has a nonzero probability of being true -- respectively, m and n. The resulting chart looks like this:

Possibility 1: Possibility 2: Possibility 3: Courses of Rewarding God No reward God God does not Expected utilities action exists (Prob. = exists (Prob. = exist (Prob. = m) n) 1-(m+n))

m* ∞ + n*0 + (1-(m Believe in God ∞ 0 LOSS +n))*LOSS= ∞

Don’t believe 0 0 0 m*0 + n*0 + (1-(m+n))*0 = 0

As this chart makes clear, adding this complication has no effect on the result. Pascal needn’t assume that God will certainly reward all believers; he need only assume that there is a nonzero chance that God will reward all believers. The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

However, one might think that there is yet another relevant possibility that we are overlooking. After all, isn’t there some chance that God might give eternal reward to believers and nonbelievers alike? This is surely possible. Let’s call this the possibility of “Generous God.” Setting aside the possibility of No reward God, which we have seen to be irrelevant, taking account of the possibility of Generous God has a striking effect on the expected utilities of belief and nonbelief:

Possibility 1: Possibility 2: Possibility 3: Courses of Rewarding God Generous God God does not Expected utilities action exists (Prob. = exists (Prob. = exist (Prob. = m) n) 1-(m+n))

m* ∞ + n* ∞ + (1-(m Believe in God ∞ ∞ LOSS +n))*LOSS= ∞

m*0 + n* ∞ + (1-(m+n))*0 Don’t believe 0 ∞ 0 = ∞

Now, it appears, belief and nonbelief have the same inﬁnite expected utility, which undercuts Pascal’s argument for the rationality of belief in God. The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

Possibility 1: Possibility 2: Possibility 3: Courses of Rewarding God Generous God God does not Expected utilities action exists (Prob. = exists (Prob. = exist (Prob. = m) n) 1-(m+n))

m* ∞ + n* ∞ + (1-(m Believe in God ∞ ∞ LOSS +n))*LOSS= ∞

m*0 + n* ∞ + (1-(m+n))*0 Don’t believe 0 ∞ 0 = ∞

Now, it appears, belief and nonbelief have the same inﬁnite expected utility, which undercuts Pascal’s argument for the rationality of belief in God.

However, Pascal seems to have a reasonable reply to this objection. It seems that the objection turns on the fact that any probability times an infinite utility will yield an infinite expected value. And that means that any two actions which have some chance of bring about an infinite reward will have the same expected utility.

But this is extremely counterintuitive. Suppose we think of a pair of lotteries, EASY and HARD. Each lottery has an inﬁnite payoff, but EASY has a 1/3 chance of winning, whereas HARD has a 1/1,000,000 chance of winning. What is the expected utility of EASY vs. HARD? Which would you be more rational to buy a ticket for?

How might we modify our rule of expected utility to explain this case? Would this help Pascal respond to the case of Generous God? The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

Even if we can solve the case of Generous God in this way, it raises a troubling sort of objection to Pascal’s argument. That argument turns essentially on the possibility of decisions with inﬁnite expected utility. But it is plausible that, in at least some cases which involve inﬁnite expected utility, the rule of expected utility gives us incorrect results. Consider the following bet:

The St. Petersburg

I am going to flip a fair coin until it comes up heads. If the first time it comes up heads is on the 1st toss, I will give you $2. If the first time it comes up heads is on the second toss, I will give you $4. If the first time it comes up heads is on the 3rd toss, I will give you $8. And in general, if the first time the coin comes up heads is on the nth toss, I will give you $2n.

Would you pay $2 to take this bet? How about $4?

Suppose now I raise the price to $10,000. Should you be willing to pay that amount to play the game?

What is the expected utility of playing the game? The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

The St. Petersburg

What is the expected utility of playing the game?

We can think about this using the following table:

First heads First heads First heads First heads First heads Outcome ..... is on toss #1 is on toss #2 is on toss #3 is on toss #4 is on toss #5

Probability $2 $4 $8 $16 $32 .....

Payoff 1/2 1/4 1/8 1/16 1/32 .....

The expected utility of playing = the sum of probability * payoff for each of the inﬁnitely many possible outcomes. So, the expected utility of playing equals the sum of the inﬁnite series

1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+...... The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

The St. Petersburg

First heads First heads First heads First heads First heads Outcome ..... is on toss #1 is on toss #2 is on toss #3 is on toss #4 is on toss #5

Probability $2 $4 $8 $16 $32 .....

Payoff 1/2 1/4 1/8 1/16 1/32 .....

The expected utility of playing = the sum of probability * payoff for each of the inﬁnitely many possible outcomes. So, the expected utility of playing equals the sum of the inﬁnite series

1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+ 1+1+1+1+1+......

But it follows from this result, plus the rule of expected utility, that you would be rational to pay any ﬁnite amount of money to have the chance to play this game once. But this seems clearly mistaken. What is going on here?

Does this show that the rule of expected utility can lead us astray? If so, in what sorts of cases does this happen? Does this result depend essentially on their being inﬁnitely many possible outcomes? Let’s now turn to a pair of quite different objections, which focus on the fact that Pascal is asking us to make decisions about what to believe on the basis of expected utility calculations. The objections are that this is (1) impossible and (2) the wrong way to form beliefs.

If I offerPascal you seems $5 to raise to consider your arm, this you reply can todo hisit. But, argument to illustrate when objection he imagines (1), suppose someone I replying offeredas follows: you $5 to believe that you are not now sitting down. Can you do that (without standing up)? Pascal“. seems . . is there to consider really no this way reply of seeing to his what argument the cards when are? he imagines . . . I am someone being forced replying to wager and I am not free; I am begin held fast and I am so made that I Pascalas considered follows: this objection, and gave the following response: cannot believe. What do you want me to do then?” “. . . is there really no way of seeing what the cards are? . . . I am being forced Pascal’sto reply: wager and I am not free; I am begin held fast and I am so made that I cannot believe. What do you want me to do then?” “That is true, but at least get it into your head that, if you are unable to Pascal’sbelieve, reply: it is because of your passions, since reason impels you to believe and yet you cannot do so. Concentrate then not on convincing yourself by multiplying proofs of God’s existence, but by diminishing your passions. . . . ” “That is true, but at least get it into your head that, if you are unable to believe, it is because of your passions, since reason impels you to believe and yet you cannot do so. Concentrate then not on convincing yourself by What does he have4.2 in mind Rationality here? does not require maximizing expected utility multiplying proofs of God’s existence, but by diminishing your passions. . . . ”

The St. Petersburg paradox; the counterintuitive consequences which result from (i) the requirement that4.2 we Rationality should act does so as not to maximize require maximizing expected expected utility, and utility (ii) the possibility of inﬁnite expected utilities. The St. Petersburg paradox; the counterintuitive consequences which result from (i) the Why the result that we should sometimes fail to maximize expected utility is puzzling. requirement that we should act so as to maximize expected utility, and (ii) the possibility of inﬁnite expected utilities.

Why the result that4.3 we We should should sometimes assign 0 fail probability to maximize to God’s expected existence utility is puzzling.

How this blocks the argument. 4.3 We should assign 0 probability to God’s existence The case against assignment of 0 probability to the possibility that God exists. How this blocks the argument.

The case against assignment4.4 of 0 probability The ‘many to gods’ the objection possibility that God exists.

(For more detail, and a list of relevant further readings, see the excellent entry “Pascal’s Wager” in the Stanford Encyclopedia4.4 The ‘manyof Philosophy gods’ objection by Alan H´ajek,from which much of the above is drawn.) (For more detail, and a list of relevant further readings, see the excellent entry “Pascal’s Wager” in the Stanford Encyclopedia of Philosophy by Alan H´ajek,from which much of the above is drawn.)

7 Let’s now turn to a pair of quite different objections, which focus on the fact that Pascal is asking us to make decisions about what to believe on the basis of expected utility calculations. The objections are that this is (1) impossible and (2) the wrong way to form beliefs. To see how objection (2) might be developed, consider the following claim about how we ought to form beliefs:

The rule of responsible belief formation

It is irrational to form a belief in a claim unless you have more reason to think that it true than that it is false.

It is plausible that, in at least many cases, we hold each other to this sort of rule. If your friend often forms beliefs despite having no reason for thinking that the belief is more likely to be true than false, you would likely take him to be irrational -- as well as untrustworthy. But now recall the rule used in Pascal’s wager:

The rule of expected utility

It is always rational to pursue the course of action with the highest expected utility.

This also seemed plausible; and it led, without the assumption that we have any more reason to think that God exists than not, to the conclusion that it is rational to believe in God. This shows that, if we think of the rule of expected utility as applying to beliefs, these two rules sometimes come into conﬂict. This means that at least one must be, as it stands, false.