Movatterモバイル変換

1. Hájek 2012 argues that in fact this is too quick. This articleconsiders a series of increasingly strong senses of“superdominance” (the one considered here is“superdominance+”), and he argues that none of them isstrong enough to confer such a requirement. This, in turn, casts doubton the validity of the first of Pascal’s wagers. However, thedetails get somewhat intricate, and I have chosen not to get into themin this exposition.

2. Those interested in the reconstruction over the years of the textitself should consult Lafuma 1954.

3. Our demarcation of the arguments follows that of Hacking 1972,although we will differ on certain points of detail.

4. Unfortunately, he squanders this insight when he lapses back to theassumption that the probability is 1/2 shortly thereafter: “Andso our proposition is of infinite force, when there is the finite tostake in a game where there are equal risks of gain and of loss, andthe infinite to gain.”

5. Consider a gamble that pays you \(W\) units (for ‘win’)if \(X\), and nothing otherwise. We are seeking the fair price \(f\),at which you should be indifferent between playing the gamble and not;you should be prepared to pay that much, but no more. Expected utilitytheory tells us thatthe fair price is the gamble’s expectedutility:

where \(P(X)\) is the chance of gain (and so \(1 - P(X)\) is thechance of loss).

Let’s derive this from Pascal’s answer, quoted hereagain:

… the uncertainty of the gain is proportioned to the certaintyof the stake according to the proportion of the chances of gain andloss …

You will pay \(f\) with certainty: this is “the certainty of thestake”. It is uncertain whether you win, but if you do your gainis \(W - f\): this is “the uncertainty of the gain”. Thisis proportioned to the certainty of the stake, \(f\)—thatis,

according to

That is,

Simple algebra gives

and hence

The fair price is the gamble’s expected utility!

6. In the basic version of decision theory that we have presented,states are assumed to be independent of actions. Evidential decisiontheory generalizes this. It replaces in its expectation calculationfor a given action the unconditional probabilities of states by theconditional probabilities of the states, given the action—seeJeffrey 1983. Now perhaps what you do is not independent of whetherGod exists. For instance, maybe God helps people wager for Him, sothat

Still, the expected utility calculations are as before, provided thefirst conditional probability is positive and finite: infinite forwagering for God, finite for wagering against God.

Causal decision theory replaces evidential decision theory’sconditional probabilities with probabilities that capture the degreeof causal relevance of an action to each state. There are variousversions of causal decision theory—see Lewis 1981. Using somesuch version would presumably not significantly affect matters here.We would just replace the assumption that your probability is positiveand finite with the same assumption about whatever probability is usedinstead.

7. After all, infinite utilities run afoul of the Archimedean, orcontinuity axiom that is commonly assumed in decision theory:

If you prefer \(A\) to \(B\), and prefer \(B\) to \(C\), then there isa gamble between \(A\) and \(C\) (\(A\) with probability \(p\), \(C\)with probability \(1 - p\), for some real-valued \(p\)) that youregard as equally desirable as \(B\) for sure.

For suppose that salvation, say, has infinite utility for you. Youprefer salvation to $1, and prefer $1 to nothing; but there is no suchgamble that rewards you with salvation if you win, and nothing if youlose, that you value at $1. Indeed, assuming that the probability ofwinning remains positive, you prefer the gamble to any finite reward;but if the probability of winning drops to 0, your preferencediscontinuously switches to the finite reward. The objection, then, isthat infinite utilities run afoul of the underpinnings of decisiontheory (by violating a preference axiom), and thus of the theoryitself. Yet that theory is appealed to in Premise 3 of the argument.In short, Premise 1 is in tension with Premise 3.

The issue then becomes whether continuity is a requirement on rationalpreference. Hájek 1997 argues that it is not, and gives furtherpositive arguments for allowing infinite utilities into decisiontheory. Sorensen 1994 likewise argues for “infinite decisiontheory”. For presentations of ‘non-Archimedean’decision theory, see Skalia 1975 and Pivato 2014. For related work oninfinite utilities that is more philosophical, see Cain 1995, Nelson1991, Ng 1995, Vallentyne 1993, Vallentyne 1995, Vallentyne and Kagan1997, and van Liedekerke 1995.

8. It should be pointed out that the rival Gods must award infiniteutility for salvation in order to create a problem—otherwisethey will be trumped by the ones that do. (It seems that Kali and Odinthus drop out of consideration, for example.) And to be damaging tothe Wager, the alternative hypotheses about how salvation is achievedshould be mutually exclusive. If there is some common core to thetheistic hypotheses, and it suffices to (strive to) believe that inorder to be saved, then there is no problem. For instance, it will notmatter that you do not know what God’s favorite real number is,if it turns out that you are saved as long as your belief is adequatein other respects. So it is crucial that salvation hinges on gettingthe details of the belief right. What, then, should we believe? Tosettle this question, it seems we get nowhere with Pascal-stylepractical reasoning.

One response is that we are therefore in a position somewhat like thatof Buridan’s ass, unable to settle which course of action isbest; and that like the ass we are better off doing something ratherthan nothing, and in this case that means choosing one of the theistichypotheses, and hoping we choose the right one. So it might still berationally required to be a theist. See Jordan 1994a for a version ofthis “ecumenical” response. There are at least twocounter-responses. Firstly, the assumption that there are alternativeGods who offer infinite rewards really plays no role in themany-Gods objection. All that matters is that there are sources ofinfinite reward besides Pascal’s God. These sources could evenbe inanimate—as it might be, supreme pleasure machines, whichoffer infinite utility irrespective of one’s beliefs. Secondly,one of the alternative Gods might punish those who wager for him, andreward those who don’t—see Martin’s 1983“perverse master”.

9. Here are the third and fourth problems for Premise 2.

3. Infinitesimal probability for God’s existence:
One might reply that you can rationallyassign infinitesimal probability to God’s existence—seee.g. Oppy 1990, 2018,  Hájek 2003, Bartha 2016, andWenmackers 2018. The argument might run, for example, that there areinfinitely many possible Gods to consider (see our discussion of themany Gods objection), and for some infinite subset of them thatincludes Pascal’s God, rationality does not favor any one overthe rest. Treating them even-handedly then requires assigninginfinitesimal probability to each. Or again, a Bayesian might say thatyou could coherently assign to God’s existence an infinitesimalprobability, provided that you also assign a probability toGod’s non-existence that falls short of 1 by the sameinfinitesimal.

It is remarkable that Pascal anticipated the notion of infinitesimalprobability, when he says: “if there were an infinity ofchances, of which one only would be for you, you would still be rightin wagering one [life] … if there were an infinity of aninfinitely happy life to gain.” But what he says here is farfrom obvious. If \(\infty\) is a legitimate utility value, thenoffhand it would seem that \(1/\infty\) is a legitimate probabilityvalue, and indeed it seems to be the very one that he is considering.However, then we have:

\[ \mathrm{E}(\text{wager for God}) =\infty \times \frac{1}{\infty} + f_1 \times \left(1 -\frac{1}{\infty}\right) \approx 1 + f_1 \]

And it is not clear that this should exceed \(f_3\).

All of this treats \(\infty\) as if it is a number, subject toordinary arithmetic operations, such as taking reciprocals,multiplying and adding. Perhaps, for example, \(\infty \times(1/\infty)\) is not defined, much as \(\infty - \infty\) is not. Butthat is just another way in which a probability of \((1/\infty)\)might thwart Pascal’s reasoning. We will say more below aboutinfinite numbers for which such arithmetic operations areunproblematic.

4. Imprecise (vague) probability for God’s existence:
So far we have presupposed that probability assignments are sharp.However, Pascal’s argument is addressed to us—mere humans.And it is apparently a fact about us that our belief states areirremediably imprecise: we cannot assign probability, precise toindefinitely many decimal places, to all propositions. (Suchimprecision about probabilities is also called“vagueness”.) Or perhaps even ideally rational agents may,or indeed should, assign imprecise probabilities when their relevantevidence is impoverished, as Pascal thinks that it is here. (See Joyce2005 for more on this route to imprecise probabilities in general.)Perhaps, then, rationality permits us to assign imprecise probabilityto God’s existence. If it moreover permits us to assign itprobability that is imprecise over an interval that includes 0, thenthe Wager fails—see Hájek 2000. Indeed, Pascal’sclaim that “[reason] can decide nothing here” might bethought to support a probability assignment to God’s existencethat is imprecise over the entire \([0, 1]\) interval. However, Rinard2018 argues that if it is contingent whether God exists, thenrationality forbids us from assigning God's existence a probabilitythat is imprecise over an interval that includes 0—the impreciseprobability must lie entirely above 0. In that case, Pascal's Wager isjust as successful when aimed at rational agents with impreciseprobabilities as it is when aimed at rational agents with preciseprobabilities.

10. One could also insist that rational choices must be ratifiable(à la Jeffrey 1983 or Sobel 1996), and that the act of maximumexpectation might not be.

A common rationale for maximizing expectation comes from the variouslaws of large numbers. Their content is roughly that under suitablecircumstances, in the limit, one’s average reward tends to theexpectation; and of course one wants to maximize one’s averagereward. But the strong law of large numbers assumes that theexpectation is finite, and since the expectation of wagering for Godis putatively infinite, it clearly cannot be appealed to here. (Seee.g. Feller 1971, 236.) Perhaps an appeal to the weak law of largenumbers, which allows infinite expectation, would suffice. But being alimit theorem, it concerns infinitely long runs of trials. Far fromhaving such a long run here, we have just a single-shot decisionproblem. This is a decision that you do not get to repeat. This is notso troubling, perhaps, when the variance (a measure of the spread ofthe distribution of possible outcomes) is small, so that getting anoutcome close to the expectation is probable; but what about when thevariance is large?

This brings us to yet another problem for Pascal’s thirdpremise. To be sure, the expectation of wagering for God is infinite,if we accept Pascal’s earlier assumptions; but so too is thevariance. Expectation does not seem to be such a good guide tochoiceworthiness when the variance is large—for what one mightend up getting can then be much worse than the expectation—letalone when the variance is infinite. (See Weirich 1984 and Sorensen1994 for versions of this last point.) Indeed, the lower one makes\(f_2\) (or more generally, some highly dispreferred outcome), theless compelling premise 3 seems; and the lower one makes theprobability of salvation (or more generally, of some highly desiredoutcome), the less compelling premise 3 seems. Yet consistent withpremise 1, \(f_2\) could be (almost) as low as one likes, andconsistent with premise 2, the probability of salvation could be(almost) as low as one likes.

11. Recall Schlesinger’s (1994, 90) tie-breaking principle:“In cases where the mathematical expectations are infinite, thecriterion for choosing the outcome to bet on is itsprobability”. Similarly, Golding (1994, 139–140) offersthis principle: “given a decision problem where an infinitevalue is at stake, the option that offers the highest probability ofattaining the infinite value is the rational choice, regardless ofwhat other probabilities or finite values are at stake”. Thisclearly rules out the coin-tossing strategy, the die-tossing strategy,and all the other mixed strategies, since these have lowerprobabilities of your achieving salvation than outright wagering forGod does. Sorensen 1994 objects to Schlesinger’s principle asbeing ad hoc. One way of putting the objection is that the principlelacks the foundational support that maximizing expected utility has.Bartha 2007, 2016 provides it such support. Be that as it may,Pascal does not appeal to the principle in his argument. As it stands,the argument is apparently invalid.

The problem is that multiplying ∞ by any positive, finiteprobability again yields ∞. Let us call this property of ∞reflexivity under multiplication (by such a probability).Such reflexivity is at once the strength of the Wager (for then Pascaldoes not need to say anything more about your probability ofGod’s existence), and its weakness (for then all the variousmixed strategies get maximal expectation also). One could try to fixthe weakness, while saving as much one can of the strength. This wouldinvolve finding a utility for salvation that is not reflexive undermultiplication, yet which is still sufficiently large to swamp yourprobability, whatever it is, in the expectation calculation.

For instance, if the utility of salvation were enormous, but finite(as it is on Golding’s reading of the Wager, 1994,130–131), then the mixed strategies would yield lowerexpectation than outright wagering for God. Multiplying that utilityby 1/2, 1/6, etc. makes a difference. And the utility could be madeenormous enough to offset any actual person’s probabilityassignment, however small (provided it is positive and finite), sothat the expectation of outright belief is maximal for everybody. Orsuppose that the utility of salvation were an infinite number that isnot reflexive under multiplication. Consider, for example, theinfinite ‘hyperreal’ numbers of non-standard analysis (seeRobinson 1966, Nelson 1987). Or consider the infinite‘surreal’ numbers of Conway 1976. Multiplication of such autility by a positive, finite probability (less than 1) yieldsanother, smaller infinite number. So the expectation of wagering forGod again exceeds that of wagering against God, and also that of eachmixed strategy, whatever your probability is (provided it is positiveand finite). Hájek 2004 suggests the hyperreal and surrealapproaches. Herzberg 2011 and Chen and Rubio 2020 develop them,offering valid reformulations with hyperreal and surreal utilities,respectively. See Wenmackers 2018, §2.2 for a summary ofhyperreal numbers, and the entry on Infinity for summaries of bothkinds of numbers.

These proposals yield valid arguments for wagering for God, wherePascal’s argument was invalid. The trouble is that they do notseem adequately to capture Pascal’s reasoning. He writes:“Unity added to infinity adds nothing to it”. Let us callthis property of infinityreflexivity under addition. We cansee why Pascal would want the utility of salvation to be reflexiveunder addition: salvation is supposed to be the best possible thing.But if that utility is finite, or hyperreal infinite, or surrealinfinite, then adding one to it does make a difference. What iswanted, then, is the seemingly impossible: a representation of thereward of salvation that is reflexive under addition (so that itcannot be bettered), but not reflexive under multiplication bypositive, finite probabilities (so that the mixed strategies can bedistinguished in expectation from outright belief). Rota (2017,footnote 4) replies that one could regard salvation as the best thingpossible for humans as they actually are, even if one could conceiveof greater things. And Wenmackers (2018, 304) points out that“it is not clear that the existence of larger numbers on theutility scale entails the existence of states or rewards correspondingto them”. But still such modern understandings of infinity, inwhich it is not reflexive under addition, are at odds with Pascal'sown view about it, which explicitly denies this (quoted above). Thisprovides some support for representing the infinite utility as wehave, with ∞, to which unity indeed adds nothing: ∞ + 1 =∞ in the extended reals. See Hájek 2012 and especially2018 for further reformulations of Pascal’s Wager that appear tobe valid. Moreover, one of them captures Pascal’s idea ofsalvation being the best thing conceivable: the utility of salvationis again ∞, which is reflexive under addition, while the utilityof damnation is −∞ (the way that some authors understandthe Wager in the first place).

Another approach, apparently faithful to Pascal’s theology,eschews premise 1’s numerical utilities altogether, optinginstead forcomparative judgments of value, and makingsuitable adjustments elsewhere in the argument. For example, Rota 2016offers a reformulation that replaces premise 1’s values of\(\infty\), \(f_1\), \(f_2\), \(f_3\), with O1, O2, O3, O4,respectively, where O1 is much greater than O3, and O2 is eithergreater than, equal to, or only a small amount less than O4. Premise2’s quantification over all positive probabilities is replacedby one over probabilities of \(1/2\) or higher. The conclusionessentially is that practical rationality requires agents with suchprobabilities to wager for God (slightly changing Rota’s wordingto conform with ours).