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“Pascal’s Wager” is the name given to an argumentdue to Blaise Pascal for believing, or for at least taking steps tobelieve, in God. The name is somewhat misleading, for in a singlesection of his Pensées, Pascal apparently presents atleast three such arguments, each of which might be called a‘wager’—it is only the final of these that istraditionally referred to as “Pascal’s Wager”. Wefind in it the extraordinary confluence of several important strandsof thought: the justification of theism; probability theory anddecision theory, used here for almost the first time in history;pragmatism; voluntarism (the thesis that belief is a matter of thewill); and the use of the concept of infinity.

We will begin with some brief stage-setting: some historicalbackground, some of the basics of decision theory, and some of theexegetical problems that the Pensées pose. Then we willfollow the text to extract three main arguments. The bulk of theliterature addresses the third of these arguments, as will the bulk ofour discussion here. Some of the more technical and scholarly aspectsof our discussion will be relegated to lengthy footnotes, to whichthere are links for the interested reader. All quotations are from§233 of Pensées (1910, Trotter translation), the‘thought’ whose heading is“Infinite—nothing”.

• 5. Objections to Pascal’s Wager

1. Background

It is important to contrast Pascal’s argument with various putative‘proofs’ of the existence of God that had come before it.Anselm’s ontological argument, Aquinas’ ‘five ways’,Descartes’ ontological and cosmological arguments, and so on, purportto prove that God exists. Pascal isapparently unimpressed by such attempted justifications of theism:“Endeavour … to convince yourself, not by increase of proofs ofGod…” Indeed, he concedes that “we do not know if He is …”.Pascal’s project, then, is radically different: he seeks to provideprudential reasons for believing in God. To put it simply, weshould wager that God exists because it is the best bet. Ryan1994 finds precursors to this line of reasoning in the writings ofPlato, Arnobius, Lactantius, and others; we might add Ghazali to hislist—see Palacios 1920. But what is distinctive is Pascal’sexplicitly decision-theoretic formulation of the reasoning. In fact,Hacking 1975 describes the Wager as “the first well-understoodcontribution to decision theory” (viii). Thus, we should pause brieflyto review some of the basics of that theory.

In any decision problem, the way the world is, and what an agentdoes, together determine an outcome for the agent. We may assignutilities to such outcomes, numbers that represent the degreeto which the agent values them. It is typical to present these numbersin a decision matrix, with the columns corresponding to the variousrelevant states of the world, and the rows corresponding to the variouspossible actions that the agent can perform.

In decisions under uncertainty, nothing more isgiven—in particular, the agent does not assign subjectiveprobabilities to the states of the world. Still, sometimes rationalitydictates a unique decision nonetheless. Consider, for example, a casethat will be particularly relevant here. Suppose that you have twopossible actions, (A_1) and (A_2), and the worst outcomeassociated with (A_1) is at least as good as the best outcome associatedwith (A_2); suppose also that in at least one state of the world,(A_1)’s outcome is strictly better than (A_2)’s. Let us say inthat case that (A_1)superdominates (A_2). Then rationality seems to require you toperform (A_1).^[1]

In decisions under risk, the agent assigns subjectiveprobabilities to the various states of the world. Assume that thestates of the world are independent of what the agent does. A figure ofmerit called the expected utility, or the expectationof a given action can be calculated by a simple formula: for eachstate, multiply the utility that the action produces in that state bythe state’s probability; then, add these numbers. According to decisiontheory, rationality requires you to perform the action of maximumexpected utility (if there is one).

Example. Suppose that the utility of money islinear in number of dollars: you value money at exactly its face value.Suppose that you have the option of paying a dollar to play a game inwhich there is an equal chance of returning nothing, and returningthree dollars. The expectation of the game itself is

[0 times frac{1}{2} + 3 times frac{1}{2} = 1.5,]

so the expectation of paying a dollar for certain, then playing, is

[-1 + 1.5 = 0.5]

Wager Meaning In Urdu

This exceeds the expectation of not playing (namely 0), so you shouldplay. On the other hand, if the game gave an equal chance of returningnothing, and returning two dollars, then its expectation would be:

[0 times frac{1}{2} + 2 times frac{1}{2} = 1.]

Then consistent with decision theory, you could either pay the dollarto play, or refuse to play, for either way your overall expectationwould be 0.

Considerations such as these will play a crucial role in Pascal’sarguments. It should be admitted that there are certain exegeticalproblems in presenting these arguments. Pascal never finished thePensées, but rather left them in the form of notes ofvarious sizes pinned together. Hacking 1972 describes the“Infinite—nothing” as consisting of “two pieces of paper coveredon both sides by handwriting going in all directions, full of erasures,corrections, insertions, and afterthoughts” (24).^[2] This may explain why certain passages are notoriously difficult tointerpret, as we will see. Furthermore, our formulation of thearguments in the parlance of modern Bayesian decision theory mightappear somewhat anachronistic. For example, Pascal did notdistinguish between what we would now call objective andsubjective probability, although it is clear that it is thelatter that is relevant to his arguments. To some extent, “Pascal’sWager” now has a life of its own, and our presentation of it here isperfectly standard. Still, we will closely follow Pascal’s text,supporting our reading of his arguments as much as possible. (See also Golding 1994 for another detailed analysis of Pascal’s reasoning, broken down into more steps than the presentation here.)

There is the further problem of dividing theInfinite-nothing into separate arguments. We will locate threearguments that each conclude that rationality requires you to wager forGod, although they interleave in the text.^[3] Finally, there is some disagreement over just what “wagering for God”involves—is it believing in God, or merelyengendering belief? We will conclude with a discussion of what Pascalmeant by this.

2. The Argument from Superdominance

Pascal maintains that we are incapable of knowing whether God exists ornot, yet we must “wager” one way or the other. Reason cannot settlewhich way we should incline, but a consideration of the relevantoutcomes supposedly can. Here is the first key passage:

“God is, or He is not.” But to which side shall we incline?Reason can decide nothing here. There is an infinite chaos whichseparated us. A game is being played at the extremity of this infinitedistance where heads or tails will turn up… Which will you choosethen? Let us see. Since you must choose, let us see which interests youleast. You have two things to lose, the true and the good; and twothings to stake, your reason and your will, your knowledge and yourhappiness; and your nature has two things to shun, error and misery.Your reason is no more shocked in choosing one rather than the other,since you must of necessity choose… But your happiness? Let us weighthe gain and the loss in wagering that God is… If you gain, you gainall; if you lose, you lose nothing. Wager, then, without hesitationthat He is.

There are exegetical problems already here, partly because Pascalappears to contradict himself. He speaks of “the true” assomething that you can “lose”, and “error” assomething “to shun”. Yet he goes on to claim that if youlose the wager that God is, then “you losenothing”. Surely in that case you “lose the true”,which is just to say that you have made an error. Pascal believes, ofcourse, that the existence of God is “the true”—but that is not something that he can appeal to in thisargument. Moreover, it is not because “you must of necessitychoose” that “your reason is no more shocked in choosingone rather than the other”. Rather, by Pascal’s own account, itis because “[r]eason can decide nothing here”. (If itcould, then it might well be shocked—namely, if you chose in away contrary to it.)

Following McClennen 1994, Pascal’s argument seems to be bestcaptured as presenting the following decision matrix:

God exists	God does not exist
Wager for God	Gain all	Status quo
Wager against God	Misery	Status quo

Wagering for God superdominates wagering against God: the worstoutcome associated with wagering for God (status quo) is at least asgood as the best outcome associated with wagering against God (statusquo); and if God exists, the result of wagering for God is strictlybetter than the result of wagering against God. (The fact that theresult is much better does not matter yet.) Pascal draws theconclusion at this point that you should wager forGod.

Without any assumption about your probability assignment to God’sexistence, the argument is invalid. Rationality does notrequire you to wager for God if you assign probability 0 to Godexisting, as a strict atheist might. And Pascal does not explicitly rule this possibility outuntil a later passage, when he assumes that you assign positiveprobability to God’s existence; yet this argument is presented as if itis self-contained. His claim that “[r]eason can decide nothing here”may suggest that Pascal regards this as a decision under uncertainty,which is to assume that you do not assign probability at allto God’s existence. If that is a further premise, then the argument is apparentlyvalid; but that premise contradicts his subsequent assumption that youassign positive probability. See McClennen for a reading of thisargument as a decision under uncertainty.

Pascal appears to be aware of a further objection to this argument,for he immediately imagines an opponent replying:

“That is very fine. Yes, I must wager; but I may perhapswager too much.”

The thought seems to be that if I wager for God, and God does notexist, then I really do lose something. In fact, Pascal himself speaksof stakingsomething when one wagers for God, whichpresumably one loses if God does not exist. (We have already mentioned‘the true’ as one such thing; Pascal also seems to regardone’s worldly life as another.) In that case, the matrix is mistakenin presenting the two outcomes under ‘God does not exist’as if they were the same, and we do not have a case of superdominanceafter all.

Pascal addresses this at once in his second argument, which we willdiscuss only briefly, as it can be thought of as just a prelude to themain argument.

3. The Argument From Expectation

He continues:

Let us see. Since there is an equal risk of gain and ofloss, if you had only to gain two lives, instead of one, you mightstill wager. But if there were three lives to gain, you would have toplay (since you are under the necessity of playing), and you would beimprudent, when you are forced to play, not to chance your life to gainthree at a game where there is an equal risk of loss and gain. Butthere is an eternity of life and happiness.

His hypothetically speaking of “two lives” and“three lives” may strike one as odd. It is helpful to bearin mind Pascal’s interest in gambling (which after all provided theinitial motivation for his study of probability) and to take thegambling model quite seriously here. Indeed, the Wager is permeatedwith gambling metaphors: “game”, “stake”,“heads or tails”, “cards” and, of course,“wager”. Now, recall our calculation of the expectationsof the two dollar and three dollar gambles. Pascal apparently assumesnow that utility is linear in number of lives, that wageringfor God costs “one life”, and then reasons analogously tothe way we did in our expectation calculations above! This is, as it were, a warm-up. Since wagering forGod is rationally required even in the hypothetical case in which oneof the prizes is three lives, then all the more it is rationallyrequired in the actual case, in which one of the prizes isan eternity of life (salvation).

So Pascal has now made two striking assumptions:

1. The probability of God’s existence is 1/2.
2. Wagering for God brings infinite reward if Godexists.

Morris 1994 is sympathetic to (1), while Hacking 1972 finds it “amonstrous premiss”. One way to defend it is via the classicalinterpretation of probability, according to which all possibilities aregiven equal weight. The interpretation seems attractive for various gambling games, which by design involve an evidential symmetry with respect to their outcomes; and Pascal even likens God’s existence to a coin toss, evidentially speaking. However, unless more is said, the interpretationyields implausible, and even contradictory results. (You have aone-in-a-million chance of winning the lottery; but either you win thelottery or you don’t, so each of these possibilities has probability1/2?!) Pascal’s argument for (1) is presumably that “[r]eason candecide nothing here”. (In the lottery ticket case, reason can decidesomething.) But it is not clear that complete ignoranceshould be modeled as sharp indifference. Morris imagines, rather, anagent who does have evidence for and against the existence of God, butit is equally balanced. In any case, it (is) clear that thereare people in Pascal’s audience who do not assign probability 1/2 toGod’s existence. This argument, then, does not speak to them.

However, Pascal realizes that the value of 1/2 actually plays noreal role in the argument, thanks to (2). This brings us to the third,and by far the most important, of his arguments.

4. The Argument From Generalized Expectations: “Pascal’s Wager”

We continue the quotation.

But there is an eternity of life and happiness. And thisbeing so, if there were an infinity of chances, of which one only wouldbe for you, you would still be right in wagering one to win two, andyou would act stupidly, being obliged to play, by refusing to stake onelife against three at a game in which out of an infinity of chancesthere is one for you, if there were an infinity of an infinitely happylife to gain. But there is here an infinity of an infinitely happy lifeto gain, a chance of gain against a finite number of chances of loss,and what you stake is finite. It is all divided; wherever the infiniteis and there is not an infinity of chances of loss against that ofgain, there is no time to hesitate, you must give all…

Again this passage is difficult to understand completely. Pascal’s talkof winning two, or three, lives is a little misleading. By his owndecision theoretic lights, you would not act stupidly “byrefusing to stake one life against three at a game in which out of aninfinity of chances there is one for you”—in fact, you should notstake more than an infinitesimal amount in that case (an amount that isbigger than 0, but smaller than every positive real number). The point,rather, is that the prospective prize is “an infinity of an infinitelyhappy life”. In short, if God exists, then wagering for God results ininfinite utility.

What about the utilities for the other possible outcomes? There issome dispute over the utility of “misery”. Hacking interprets this as“damnation”, and Pascal does later speak of “hell” as the outcome inthis case. Martin 1983 among others assigns this a value ofnegative infinity. Sobel 1996, on the other hand, is oneauthor who takes this value to be finite. There is some textual supportfor this reading: “The justice of God must be vast like His compassion.Now justice to the outcast is less vast … than mercy towards theelect”. As for the utilities of the outcomes associated with God’snon-existence, Pascal tells us that “what you stake is finite”. Thissuggests that whatever these values are, they are finite.

Pascal’s guiding insight is that the argument from expectation goesthrough equally well whatever your probability for God’sexistence is, provided that it is non-zero and finite(non-infinitesimal)—“a chance of gain against a finite numberof chances of loss”.^[4]

Pascal’s assumptions about utilities and probabilities are now inplace. In another landmark moment in this passage, he next presents aformulation of expected utility theory. When gambling, “everyplayer stakes a certainty to gain an uncertainty, and yet he stakes afinite certainty to gain a finite uncertainty, without transgressingagainst reason”. How much, then, should a player be prepared tostake without transgressing against reason? Here is Pascal’s answer:“… the uncertainty of the gain is proportioned to thecertainty of the stake according to the proportion of the chances ofgain and loss …” It takes some work to show that thisyields expected utility theory’s answer exactly, but it is work wellworth doing for its historical importance.^[5](The interested reader can see this work done at footnote 5.)

Let us now gather together all of these points into a singleargument. We can think of Pascal’s Wager as having three premises: thefirst concerns the decision matrix of rewards, the second concerns theprobability that you should give to God’s existence, and the third is amaxim about rational decision-making. Specifically:

1. Either God exists or God does not exist, and you can either wagerfor God or wager against God. The utilities of the relevant possibleoutcomes are as follows, where (f_1, f_2), and(f_3) are numbers whose values are not specified beyond therequirement that they be finite:

   God exists	God does not exist
   Wager for God	(infty)	(f_1)
   Wager against God	(f_2)	(f_3)

2. Rationality requires the probability that you assign to Godexisting to be positive, and not infinitesimal.
3. Rationality requires you to perform the act of maximum expectedutility (when there is one).
4. Conclusion 1. Rationality requires you to wager forGod.
5. Conclusion 2. You should wager for God.

We have a decision under risk, with probabilities assigned to the ways the world could be, and utilities assigned to the outcomes. In particular, we represent the infinite utility associated with salvation as ‘(infty)’. We assume that the real line is extended to include the element ‘(infty)’, and that the basic arithmetical operations are extended as follows:

For all real numbers (r): (infty + r = infty).
For all real numbers (r): (infty times r = infty) if (r gt 0).

The first conclusion seems tofollow from the usual calculations of expected utility(where (p) is your positive, non-infinitesimal probability forGod’s existence):

[mathrm{E}(text{wager for God}) = infty times p + f_1 times(1 - p) = infty]

That is, your expected utility of belief in God is infinite—asPascal puts it, “our proposition is of infinite force”. Onthe other hand, your expected utility of wagering against God is

[mathrm{E}(text{wager against God}) = f_2 times p + f_3 times(1 - p)]

This is finite.^[6] By premise 3, rationality requiresyou to perform the act of maximum expected utility. Therefore,rationality requires you to wager for God.

We now survey some of the main objections to the argument.

5. Objections to Pascal’s Wager

5.1 Premise 1: The Decision Matrix

Here the objections are manifold. Most of them can be statedquickly, but we will give special attention to what has generally beenregarded as the most important of them, ‘the many Godsobjection’ (see also the link to footnote 7).

1. Different matrices for different people. The argumentassumes that the same decision matrix applies to everybody. However,perhaps the relevant rewards are different for different people.Perhaps, for example, there is a predestined infinite reward for theChosen, whatever they do, and finite utility for the rest, as Mackie1982 suggests. Or maybe the prospect of salvation appeals more to somepeople than to others, as Swinburne 1969 has noted.

Wager Meaning Betting

Even granting that a single (2 times 2) matrix applies toeverybody, one might dispute the values that enter into it. Thisbrings us to the next two objections.

2. The utility of salvation could not be infinite. One mightargue that the very notion of infinite utility is suspect—seefor example Jeffrey 1983 and McClennen 1994.^[7] Hence, theobjection continues, whatever the utility of salvation might be, itmust be finite. Strict finitists, who are suspicious of the notion ofinfinity in general, will agree—see Dummett 1978 and Wright1987. Or perhaps the notion of infinite utility makes sense, but aninfinite reward could only be finitely appreciated by a humanbeing.

3. There should be more than one infinity in the matrix. Thereare also critics of the Wager who, far from objecting to infiniteutilities, want to see more of them in the matrix. Forexample, it might be thought that a forgiving God would bestowinfinite utility upon wagerers-for and wagerers-againstalike—Rescher 1985 is one author who entertains thispossibility. Or it might be thought that, on the contrary, wageringagainst an existent God results in negative infiniteutility. (As we have noted, some authors read Pascal himself as sayingas much.) Either way, (f_2) is not really finite at all, but(infty) or (-infty) as the case may be. And perhaps (f_1) and(f_3) could be (infty) or (-infty). Suppose, for instance,that God does not exist, but that we are reincarnated adinfinitum, and that the total utility we receive is an infinitesum that diverges to infinity or to negative infinity.

4. The matrix should have more rows. Perhaps there is morethan one way to wager for God, and the rewards that God bestows varyaccordingly. For instance, God might not reward infinitely those whostrive to believe in Him only for the very mercenary reasons thatPascal gives, as James 1956 has observed. One could also imaginedistinguishing belief based on faith from belief based on evidentialreasons, and posit different rewards in each case.

5. The matrix should have more columns: the many Godsobjection. If Pascal is really right that reason can decide nothinghere, then it would seem that various other theistic hypotheses arealso live options. Pascal presumably had in mind the Catholicconception of God—let us suppose that this is the God whoeither ‘exists’ or ‘does not exist’. Byexcluded middle, this is a partition. The objection, then, is that thepartition is not sufficiently fine-grained, and the ‘(Catholic)God does not exist’ column really subdivides into variousother theistic hypotheses. The objection could equally runthat Pascal’s argument ‘proves too much’: by parallelreasoning we can ‘show’ that rationality requires believingin various incompatible theistic hypotheses. As Diderot (1746) putsthe point: “An Imam could reason just as well this way”.^[8]

Since then, the point has been presented again and refined in variousways. Mackie 1982 writes, “the church within which alone salvation isto be found is not necessarily the Church of Rome, but perhaps that ofthe Anabaptists or the Mormons or the Muslim Sunnis or the worshippersof Kali or of Odin” (203). Cargile 1966 shows just how easy it is tomultiply theistic hypotheses: for each real number (x), considerthe God who prefers contemplating (x) more than any otheractivity. It seems, then, that such ‘alternative gods’ area dime a dozen—or (aleph_1), for that matter.

In response, some authors argue that in such a competition amongvarious possible deities for one’s belief, some are more probable thanothers. Although there may be ties among the expected utilities—all infinite—for believing in various ones among them,their respective probabilities can be used astie-breakers. Schlesinger (1994, 90) offers this principle: “Incases where the mathematical expectations are infinite, the criterion forchoosing the outcome to bet on is its probability”. (Note that thisprinciple is not found in the Wager itself, although it might be regarded asa friendly addition.) Are therereasons, then, for assigning higher probability to some Gods thanothers? Jordan (1994a, 107) suggests that some outlandish theistichypotheses may be dismissed for having “no backing oftradition”. Similarly, Schlesinger maintains that Pascal isaddressing readers who “have a notion of what genuine religionis about” (88), and we might take that to suggest that Cargile’simagined Gods, for example, may be correspondingly assigned lowerprobability than Pascal’s God. Lycan and Schlesinger 1989 give moretheoretical reasons for favoring Pascal’s God over others in one’sprobability assignments. They begin by noting the familiar problem inscience of underdetermination of theory by evidence. Faced with amultiplicity of theories that all fit the observed data equally well,we favor the simplest such theory. They go on to argue that simplicityconsiderations similarly favor a conception of God as“absolutely perfect”, “which is theologically uniquein that it implies all the other predicates traditionally ascribed toGod” (104), and we may add that this conception isPascal’s. Conceptions of rival Gods, by contrast, leave open variousquestions about their nature, the answering of which would detractfrom their simplicity, and thus their probability.

Finally, Bartha 2012 models one’s probability assignments tovarious theistic hypotheses as evolving over time according to a‘deliberational dynamics’ somewhat analogous to thedynamics of evolution by natural selection. So understood, Pascal’sWager is not a single decision, but rather a sequence of decisions inwhich one’s probabilities update sequentially in proportion to howchoiceworthy each God appeared to be in the previous round. (Thisrelies on a sophisticated handling of infinite utilities in terms ofutility ratios given in his 2007; see below.) He argues that a givenprobability assignment is choiceworthy only if it is an equilibrium ofthis deliberational dynamics. He shows that certain assignments arechoiceworthy by this criterion, thus providing a kind of vindicationof Pascal against the many Gods objection.

5.2 Premise 2: The Probability Assigned to God’s Existence

There are four sorts of problem for this premise. The first two arestraightforward; the second two are more technical, and can be found byfollowing the link to footnote 9.

Wager Meaning

1. Undefined probability for God’s existence. Premise 1presupposes that you should have a probability for God’sexistence in the first place. However, perhaps you could rationallyfail to assign it a probability—your probability thatGod exists could remain undefined. We cannot enter here intothe thorny issues concerning the attribution of probabilities toagents. But there is some support for this response even in Pascal’sown text, again at the pivotal claim that “[r]eason can decide nothinghere. There is an infinite chaos which separated us. A game is beingplayed at the extremity of this infinite distance where heads or tailswill turn up…” The thought could be that any probability assignmentis inconsistent with a state of “epistemic nullity” (in Morris’ 1986phrase): to assign a probability at all—even 1/2—toGod’s existence is to feign having evidence that one in fact totallylacks. For unlike a coin that we know to be fair, this metaphorical‘coin’ is ‘infinitely far’ from us, henceapparently completely unknown to us. Perhaps, then, rationalityactually requires us to refrain from assigning a probabilityto God’s existence (in which case at least the Argument fromSuperdominance would apparently be valid). Or perhaps rationality does not requireit, but at least permits it. Either way, the Wager would noteven get off the ground.

2. Zero probability for God’s existence. Strict atheists mayinsist on the rationality of a probability assignment of 0, as Oppy1990 among others points out. For example, they may contend that reasonalone can settle that God does not exist, perhaps by arguingthat the very notion of an omniscient, omnipotent, omnibenevolent beingis contradictory. Or a Bayesian might hold that rationality places noconstraint on probabilistic judgments beyond coherence (or conformityto the probability calculus). Then as long as the strict atheistassigns probability 1 to God’s non-existence alongside his or herassignment of 0 to God’s existence, no norm of rationality has beenviolated.

Furthermore, an assignment of (p = 0) would clearly block the route toPascal’s conclusion, under the usual assumption that

[infty times 0 = 0]

For then the expectation calculations become:

[begin{align*}mathrm{E}(text{wager for God}) &= infty times 0 + f_1 times(1 - 0) &= f_1 &mathrm{E}(text{wager against God}) &= f_2 times 0 + f_3 times(1 - 0) &= f_3end{align*}]

And nothing in the argument implies that (f_1 gt f_3). (Indeed,this inequality is questionable, as even Pascal seems to allow.) Inshort, Pascal’s wager has no pull on strict atheists.^[9]

5.3 Premise 3: Rationality Requires Maximizing Expected Utility

Finally, one could question Pascal’s decision theoretic assumptionthat rationality requires one to perform the act of maximum expectedutility (when there is one). Now perhaps this is an analytictruth, in which case we could grant it to Pascal without furtherdiscussion—perhaps it is constitutive of rationalityto maximize expectation, as some might say. But this premise has metserious objections. The Allais 1953 and Ellsberg 1961 paradoxes, forexample, are said to show that maximizing expectation can lead one toperform intuitively sub-optimal actions. So too the St. Petersburgparadox, in which it is supposedly absurd that one should be preparedto pay any finite amount to play a game with infinite expectation.(That paradox is particularly apposite here.)^[10]

Various refinements of expected utility theory have been suggested asa result of such problems. For example, we might consider expecteddifferences between the pay-offs of options, and prefer oneoption to another if and only if the expected difference of the formerrelative to the latter is positive—see Hájek and Nover 2006,Hájek 2006, Colyvan 2008, and Colyvan & Hájek 2016. Or we might considersuitably defined utility ratios, and prefer one option toanother if and only if the utility ratio of the former relative to thelatter is greater than 1—see Bartha 2007. If we either admitrefinements of traditional expected utility theory, or are pluralisticabout our decision rules, then premise 3 is apparently false as itstands. Nonetheless, the door is opened to some suitable reformulationof it that might serve Pascal’s purposes. Indeed, Bartha argues thathis ratio-based reformulation answers some of the most pressingobjections to the Wager that turn on its invocation of infiniteutility.

Finally, one might distinguish between practicalrationality and theoretical rationality. One could thenconcede that practical rationality requires you to maximize expectedutility, while insisting that theoretical rationality might requiresomething else of you—say, proportioning belief to the amountof evidence available. This objection is especially relevant, sincePascal admits that perhaps you “must renounce reason” in order tofollow his advice. But when these two sides of rationality pull inopposite directions, as they apparently can here, it is not obviousthat practical rationality should take precedence. (For a discussion ofpragmatic, as opposed to theoretical, reasons for belief, see Foley1994.)

5.3 Is the Argument Valid?

A number of authors who have been otherwise critical of the Wagerhave explicitly conceded that the Wager is valid—e.g. Mackie1982, Rescher 1985, Mougin and Sober 1994, and most emphatically,Hacking 1972. That is, these authors agree with Pascal that wageringfor God really is rationally mandated by Pascal’s decision matrix intandem with positive probability for God’s existence, and the decisiontheoretic account of rational action.

However, Duff 1986 and Hájek 2003 argue that the argument isin fact invalid. Their point is that there are strategies besideswagering for God that also have infinite expectation—namely,mixed strategies, whereby you do not wager for or against Godoutright, but rather choose which of these actions to perform on thebasis of the outcome of some chance device. Consider the mixedstrategy: “Toss a fair coin: heads, you wager for God; tails, you wageragainst God”. By Pascal’s lights, with probability 1/2 your expectationwill be infinite, and with probability 1/2 it will be finite. Theexpectation of the entire strategy is:

[frac{1}{2} times infty + frac{1}{2} times [f_2 times p + f_3 times(1 - p)] = infty]

That is, the ‘coin toss’ strategy has the same expectationas outright wagering for God. But the probability 1/2 was incidental tothe result. Any mixed strategy that gives positive and finiteprobability to wagering for God will likewise have infiniteexpectation: “wager for God iff a fair die lands 6”, “wager for God iffyour lottery ticket wins”, “wager for God iff a meteor quantum tunnelsits way through the side of your house”, and so on.

It can be argued that the problem is still worse than this, though, for there is a sensein which anything that you do might be regarded as a mixedstrategy between wagering for God, and wagering against God, withsuitable probability weights given to each. Suppose that you choose toignore the Wager, and to go and have a hamburger instead. Still, youmay well assign positive and finite probability to your winding upwagering for God nonetheless; and this probability multiplied byinfinity again gives infinity. So ignoring the Wager and having ahamburger has the same expectation as outright wagering for God. Evenworse, suppose that you focus all your energy into avoidingbelief in God. Still, you may well assign positive and finiteprobability to your efforts failing, with the result that you wager forGod nonetheless. In that case again, your expectation is infiniteagain. So even if rationality requires you to perform the act ofmaximum expected utility when there is one, here there isn’t one.Rather, there is a many-way tie for first place, as it were. All hell breaks loose: anything you might do is maximally good by expected utility lights!^[11]

Monton 2011 defends Pascal’s Wager against this line ofobjection. He argues that an atheist or agnostic has more than oneopportunity to follow a mixed strategy. Returning to the first exampleof one, suppose that the fair coin lands tails. Monton’s thought isthat your expected utility now changes; it is no longer infinite, butrather that of an atheist or agnostic who has no prospect of theinfinite reward for wagering for God. You are back to where youstarted. But since it was rational for you to follow the mixedstrategy the first time, it is rational for you to follow it againnow—that is, to toss the coin again. And if it lands tailsagain, it is rational for you to toss the coin again … Withprobability 1, the coin will land heads eventually, and from thatpoint on you will wager for God. Similar reasoning applies to wageringfor God just in case an n-sided die lands 1 (say): with probability 1the die will eventually land 1, so if you repeatedly base your mixedstrategy on the die, with probability 1 you will wind up wagering forGod after a finite number of rolls. Robertson 2012 replies that notall such mixed strategies are (probabilistically) guaranteed to leadto your wagering for God in the long run: not ones in which theprobability of wagering for God decreases sufficiently fast onsuccessive trials. Think, for example, of rolling a 4-sided die, thena 9-sided die, and in general an ((n+1)^2)-sided die on the(n)^th trial …, a strategy for which the probability thatyou will eventually wager for God is only 1/2, as Robertsonshows. However, Easwaran and Monton 2012 counter-reply that with acontinuum of times at which the dice can be rolled, the sequence ofrolls that Robertson proposes can be completed in an arbitrarily shortperiod of time. In that case, what should you do next? By Monton’sargument, it seems you should roll a die again. Easwaran and Montonprove that if there are uncountably many times at which one implementsa mixed strategy with non-zero probability of wagering for God, thenwith probability 1, one ends up wagering for God at one of thesetimes. (And they assume, as is standard, that once one wagers for Godthere is no going back.) They concede that imagining uncountably rollsof a die, say, involves an idealization that is surely not physicallyrealizable. But they maintain that you should act in the way that anidealized version of yourself would eventually act, onewho can realize the rolls as described—that is, wagerfor God outright.

There is a further twist on the mixed strategies objection. Torepeat, the objection’s upshot is that even granting Pascal all hispremises, still wagering for God is not rationally required. But wehave seen numerous reasons not to grant all hispremises. Very well then; let’s not. Indeed, let’s suppose that yougive tiny probability p to them all being true,where (p) is positive and finite. So you assignprobability (p) to your decision problem being exactly asPascal claims it to be. But if it is, according to the mixedstrategies objection, all hell breaks loose. Yet again, (p)multiplied by infinity gives infinity. Hence, it seems that eachaction that gets infinite expected utility according to Pascalsimilarly gets infinite expected utility according to you;but by the previous reasoning, that is anything you might do. The fullforce of the objection that hit Pascal now hits you too. There aresome subtleties that we have elided over; for example, if you alsoassign positive and finite probability to a sourceof negative infinite utility, then the expected utilitiesinstead become (infty) – (infty), which is undefined. But thatis just another way for all hell to break loose for you: in that case,you cannot evaluate the choiceworthiness of your possible actions atall. Either way, you face decision-theoretic paralysis. We might callthis Pascal’s Revenge. See Hájek (2015) formore discussion.

5.4 Moral Objections to Wagering for God

Let us grant Pascal’s conclusion for the sake of the argument:rationality requires you to wager for God. It still does not obviouslyfollow that you should wager for God. All that we have grantedis that one norm—the norm of rationality—prescribeswagering for God. For all that has been said, some other normmight prescribe wagering against God. And unless we can show that therationality norm trumps the others, we have not settled what you shoulddo, all things considered.

There are several arguments to the effect that moralityrequires you to wager against God. Pascal himself appears tobe aware of one such argument. He admits that if you do not believe inGod, his recommended course of action “will deaden youracuteness” (This is Trotter’s translation. Pascal’soriginal French wording is “vous abêtira”, whoseliteral translation is even more startling: “will make you abeast”.) One way of putting the argument is that wagering forGod may require you to corrupt yourself, thus violating a Kantian dutyto yourself. Clifford 1877 argues that an individual’s believingsomething on insufficient evidence harms society by promotingcredulity. Penelhum 1971 contends that the putative divine plan isitself immoral, condemning as it does honest non-believers to loss ofeternal happiness, when such unbelief is in no way culpable; and thatto adopt the relevant belief is to be complicit to this immoralplan. See Quinn 1994 for replies to these arguments. For example,against Penelhum he argues that as long as God treats non-believersjustly, there is nothing immoral about him bestowing special favor onbelievers, more perhaps than they deserve. (Note, however, that Pascalleaves open in the Wager whether the payoff for non-believers (is)just; indeed, as far as his argument goes, it may be extremelyunjust.)

Finally, Voltaire protests that there is something unseemly about thewhole Wager. He suggests that Pascal’s calculations, and his appeal toself-interest, are unworthy of the gravity of the subject of theisticbelief. This does not so much support wagering against God, asdismissing all talk of ‘wagerings’ altogether. Schlesinger(1994, 84) canvasses a sharpened formulation of this objection: anappeal to greedy, self-interested motivations is incompatible with“the quest for piety” that is essential to religion. Hereplies that the pleasure of salvation that Pascal’s Wagercountenances is “of the most exalted kind”, and that ifseeking it counts as greed at all, then it is “the manifestationof a noble greed that is to be acclaimed” (85).

6. What Does It Mean to “Wager for God”?

Let us now grant Pascal that, all things considered (rationality andmorality included), you should wager for God. What exactly does thisinvolve?

A number of authors read Pascal as arguing that you shouldbelieve in God—see e.g. Quinn 1994, and Jordan 1994a.But perhaps one cannot simply believe in God at will; and rationalitycannot require the impossible. Pascal is well aware of this objection:“[I] am so made that I cannot believe. What, then, would you have medo?”, says his imaginary interlocutor. However, he contends that onecan take steps to cultivate such belief:

You would like to attain faith, and do not know the way;you would like to cure yourself of unbelief, and ask the remedy for it.Learn of those who have been bound like you, and who now stake alltheir possessions. These are people who know the way which you wouldfollow, and who are cured of an ill of which you would be cured. Followthe way by which they began; by acting as if they believed, taking theholy water, having masses said, etc. …

But to show you that this leads you there, it is this which willlessen the passions, which are your stumbling-blocks.

We find two main pieces of advice to the non-believer here: act like abeliever, and suppress those passions that are obstacles to becoming abeliever. And these are actions that one can perform at will.

Believing in God is presumably one way to wager for God. Thispassage suggests that even the non-believer can wager for God, bystriving to become a believer. Critics may question the psychology ofbelief formation that Pascal presupposes, pointing out that one couldstrive to believe (perhaps by following exactly Pascal’s prescription),yet fail. To this, a follower of Pascal might reply that the act ofgenuine striving already displays a pureness of heart that God wouldfully reward; or even that genuine striving in this case is itself aform of believing.

According to Pascal, ‘wagering for God’ and‘wagering against God’ are contradictories, as there is noavoiding wagering one way or another: “you must wager. It is notoptional.” The decision to wager for or against God is one thatyou make at a time—at (t), say. But of course Pascal doesnot think that you would be infinitely rewarded for wagering for Godmomentarily, then wagering against God thereafter; nor that you wouldbe infinitely rewarded for wagering for God sporadically—only onthe last Thursday of each month, for example. What Pascal intends by‘wagering for God’ is an ongoing action—indeed, onethat continues until your death—that involves your adopting acertain set of practices and living the kind of life that fostersbelief in God. The decision problem for you at (t), then, iswhether you should embark on this course of action; to fail to do sois to wager against God at (t).

7. The Continuing Influence of Pascal’s Wager

Pascal’s Wager vies with Anselm’s Ontological Argument for being the most famous argument in the philosophy of religion. Indeed, the Wager arguably has greater influence nowadays than any other such argument—not just in the service of Christian apologetics, but also in its impact on various lines of thought associated with infinity, decision theory, probability, epistemology, psychology, and even moral philosophy. It has provided a case study for attempts to develop infinite decision theories. In it, Pascal countenanced the notion of infinitesimal probability long before philosophers such as Lewis 1980 and Skyrms 1980 gave it prominence. It continues to put into sharp relief the question of whether there can be pragmatic reasons for belief, and the putative difference between theoretical and practical rationality. It raises subtle issues about the extent to which one’s beliefs can be a matter of the will, and the ethics of belief.

Reasoning reminiscent of Pascal’s Wager, often with an explicitacknowledgment of it, also informs a number of debates in moralphilosophy, both theoretical and applied. Kenny 1985 suggests thatnuclear Armageddon has negative infinite utility, and some might saythe same for the loss of even a single human life. Stich 1978criticizes an argument that he attributes to Mazzocchi, that thereshould be a total ban on recombinant DNA research, since such researchcould lead to the “Andromeda scenario” of creating akiller strain of bacterial culture against which humans are helpless;the ban, moreover, should be enforced if the “Andromeda scenariohas even the smallest possibility of occurring” (191), inMazzocchi’s words. This is plausibly read, then, as anassignment of negative infinite utility to the Andromedascenario. More recently, Colyvan, Cox, and Steele 2010 discussPascal’s Wager-like problems for certain deontological moraltheories, in which violations of duties are assigned negative infiniteutility. Colyvan, Justus and Regan 2011 canvas difficulties associatedwith assigning infinite value to the natural environment. Bartha andDesRoches 2017 respond, with an appeal to relative utilitytheory. Stone 2007 argues that a version of Pascal’s Wagerapplies to sustaining patients who are in a persistent vegetativestate; see Varelius 2013 for a dissenting view.

Pascal’s Wager is a watershed in the philosophy of religion. As we have seen, it is also a great deal more besides.

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