Pascal’s Wager is one of the great classic arguments for belief in God, and one of the most famous arguments in all of philosophy. Other theological arguments — ontological, cosmological, and teleological — aim to establish that God’s existence is necessary or probable. Pascal’s Wager is instead a prudential or “pragmatic” argument. The conclusion is a recommended action: you ought to believe (or to strive to believe) in God, even if you think that the probability of God’s existence is very low.
It can be rational to buy a lottery ticket with a low chance of winning, provided the prize is of enormous value to you. If the prize is infinite, then the “expected value” of the ticket is infinite, no matter how small the chance of winning. The most familiar formulation of Pascal’s Wager runs as follows: so long as the probability of an infinite afterlife reward for belief in God is greater than zero, “wagering for God” has infinite expected value and is therefore superior to “wagering against God” (which has at best finite expected value). In short, wagering is a good gamble. Here, wagering for God means taking steps that are conducive to religious belief. Pascal’s advice is to “take holy water, have masses said and so on”. Contemporary aids might include meditation sessions, lecture tours and new age music.
Pascal’s Wager is unique among theological arguments for the degree to which it evokes both fascination and disgust. The fascination, for some, derives from the fact that the argument brings together big ideas: infinity, God, salvation, prudential and evidential reasoning, and an analytical framework (decision theory) that provides tools for thinking about a large array of philosophical problems. The reasoning is intriguing because, on most versions of the argument, it proceeds without any assessment of the evidence (which Pascal calls “futile proofs”) for or against God’s existence.
Disgust is also a common reaction. Many find the argument contemptible. The reasoning is alleged to be parochial, inauthentic, immoral and mathematically dodgy.
First, it is parochial: would God be so petty as to reward only a narrow class of believers? Why should we regard such an exclusivist God as worthy of worship? This concern is closely related to the notorious many-gods objection, noted long ago by Diderot: if infinite expectation attaches to Pascal’s Christian god, then it also attaches to any rival deity who offers an infinite reward, and we are left with no clear recommendation.
Second, the reasoning is inauthentic, because it is mercenary and self-serving rather than oriented towards the piety and benevolence that are supposed to constitute the core of religious commitment. We might sympathise with William James’ remark that belief “after such a mechanical calculation would lack the inner soul of faith’s reality; and if we were ourselves in the place of the Deity, we should probably take particular pleasure in cutting off believers of this pattern from their infinite reward.”
Third, to believe in God because of the potential reward, without any regard to the evidence, appears to violate the ethics of belief. It looks like a spectacular case of wishful thinking.
Fourth, aspects of the value calculation appear mathematically suspect. How do we know that the probability of God’s existence is a positive (real) number, rather than infinitesimally small? How can the assignment of infinite value to the afterlife be meaningful — does Pascal have a naïve understanding of infinity, and even with contemporary mathematics, can we make sense of infinite value? What is the mathematical basis for the calculation of infinite expected value?
Before we explore the positive and negative elements of the Wager, it will be helpful first to provide some context for Pascal’s argument and then to describe the reasoning more clearly. The Wager occupies one short section of the Pensées, a meandering meditation on the “infinite wisdom and infinite folly of religion” and the “greatness and wretchedness” of human nature. The work as a whole, and the Wager in particular, are designed to appeal both to reason and to the passions. Pascal wants to shock his readers out of their “supernatural torpor,” to dislodge the hard shell of hostility or indifference to religion that he fully expects to encounter. Certainly, he considers the reasoning of the Wager to be impeccable. But given his larger project, his expectations for the Wager are limited: “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.” He expects his readers to acknowledge the cogency of the argument, but he doesn’t expect an instant embrace of religion and he doesn’t claim that the Wager is all that they need for authentic religious faith. We do well to remember this when we consider the Wager in isolation from the rest of the Pensées, as has now become standard.
What, then, is the argument? Ian Hacking finds three versions in the text. William James, in his famous essay, “The Will to Believe,” offers his own distinctive formulation. Several contemporary versions exist (Alan Hájek identifies ten), designed to address one or more perceived weaknesses in the original argument. All work with something like the following decision table:
(p) (1- p)
God exists God does not exist
Wager for God ` U f1
Wager against God f2 f3
The rows represent the two possible acts open to you. The columns represent the two relevant possible states of the world. Each act-state combination is assigned a utility value that reflects your preferences. The utility U of salvation is very large (possibly infinite), while the utilities f1, f2 and f3 are all finite and much smaller than U. Finally, p is your subjective probability that God exists, and it is assumed that p > 0. The expected utility (EU) of each act is a weighted average that combines probability and utility: EU(Wager for God) = pU + (1-p)f1, while EU(Wager against God) = pf2 + (1-p)f3. Provided U is large enough, the rational choice is to Wager for God.
For present purposes, it suffices to distinguish two versions of the Wager: continuous and discontinuous. In the continuous versions, the large utility of the afterlife can be “cancelled” by a small enough probability p, resulting in a finite expected utility for wagering for God. The simplest way to do this is with the finite Wager: assign a large but finite value to U. The reasoning is then sensitive to the exact values of U, p, andthe other utilities. One popular strategy, adopted by William James and contemporary philosophers such as Jeff Jordan, is to stress the benefits of religious belief even if God does not exist: if f1 > f3, then Wager for God comes out ahead for any p > 0 (since we are still supposing that U is much larger than any of f1, f2 or f3). Empirical studies of religious belief and happiness become relevant. As Paul Saka notes, however, the situation is not simple. Some studies suggest that religious belief contributes positively to well-being at the individual level but negatively at the societal level.
In the discontinuous versions of the Wager, the large utility U of the afterlife cannot be cancelled by a small probability p for God’s existence. The simplest way to do this is with the canonical Wager: let U = ∞ and allow only real (finite) probability values for p. (Pascal almost certainly has this version in mind — he even considers and rejects the idea that God’s existence has infinitesimal probability.) The expected utility of Wager for God is then infinite for any p > 0, and the argument succeeds regardless of the values f1, f2 and f3. But the axioms of standard decision theory rule out infinite utility! So for this version of Pascal’s Wager, one has to assume that nonstandard decision theory (which allows infinite utility) makes sense. The main problems for this approach are how to make sense of the infinite utility of salvation while ruling out infinitesimal probability for God’s existence. Alan Hájek notes a number of additional problems in his Stanford Encyclopedia entry (plato.stanford.edu/entries/pascal-wager/).
Let’s restrict our attention to the discontinuous version. There are many ways to resist the argument. Perhaps your subjective probability p that God exists is 0, or infinitesimally small. Perhaps there is no way to make sense of infinite utility, and even if there is, you would assign only finite value to the afterlife. Even if you grant both that p > 0 and that the utility of salvation is infinite, perhaps you recognise the force of the many-gods objection: there is no way to choose between competing deities who promise an infinite reward. Whether or not you reject the Wager for these or other reasons, however, the argument has great philosophical interest and should not be underestimated. In the remainder of this essay, I shall indicate three important ideas that allow us to reformulate, and appreciate the significance of, the Wager. These ideas might also help to mitigate repugnance for the argument and perhaps even to inspire admiration.
The first idea is this: we actually need a nonstandard decision theory that is capable of modelling discontinuous preferences. Pascal’s Wager is one example, but decisions about the environment also require trade-offs between objectives with large but measurable economic value and the conservation of “priceless” parts of the natural world: rivers, elephants, and entire ecosystems. Advocates of an “ecosystem services approach” insist on the assignment of determinate, finite values to all parts of nature. A direct appeal to the social and economic benefits of conservation can lead to environmental protection in some cases. But critics argue that environmental ethics must make room for non-negotiable obligations and constraints on human activity, and they reject any framework that always allows trade-offs. The expansion of ordinary decision theory to include infinite values is one way to represent such obligations and constraints. If decision theory is to provide a neutral framework for evaluating debates about the environment, it is crucial not to rule out this position a priori.
The second idea is that there is a reasonable way to understand assignments of infinite utility. If we think of utility as an evaluation of subjective experience (happiness, pleasure, etc.), then as John Broome and others have rightly pointed out, absolutely infinite utility as a measure of our experience may seem like a meaningless concept for human beings. If, however, we treat utility as a device for representing the structure of an agent’s preferences, in the tradition of mathematicians and economists such as von Neumann and Morgenstern, then (as I have argued) there is a straightforward way to understand what it means for an agent to assign outcome B infinite utility relative to outcome A. It means that the agent would trade A for any bet that offers a positive chance, however small, of gaining B. We have to drop one standard axiom about rational preferences (known as the Continuity Axiom), but we can still construct a rigorous nonstandard decision theory — for example, using lexicographically ordered utility vectors. This gives us a more general form of decision theory that can be used to model the reasoning in Pascal’s Wager and in environmental ethics.
By treating infinite utility as a representational device, we are in a good position to answer some of our initial objections. In the first place, we can settle some of the doubts about the mathematical rigour of Pascal’s argument, specifically, the concerns about the meaning of infinite utility and the expected utility calculation. But to take this perspective, we have to change our understanding of Pascal’s Wager. To represent an agent as assigning infinite utility to salvation and finite utility to all of the other outcomes in the table is to pre-suppose that she has preferences that require her to take the wager. Although this takes us away from Pascal’s conception of the reasoning, it helps to assuage objections about inauthenticity. Pascal’s Wager becomes a device for rationalising the formation of religious beliefs and preferences, rather than a rhetorical weapon for coercing an unreceptive audience. Extending this point further, nonstandard decision theory has the resources to allow expected utility comparisons for wagers about distinct deities to which we might assign different probabilities. These comparisons reinforce the decision to wager for the deity with the highest subjective probability. This gives us at least a partial response to the many-gods objection, and concerns about exclusivity. Briefly, Pascal’s argument can lead to different results, depending on your initial theological beliefs.
All of this leads to a third and final idea that has emerged in recent discussions of Pascal’s argument: the shift from a static to a dynamic view of the Wager. Rather than treating the Wager as a single decision, we should regard religious beliefs (and in particular, subjective credences about religious matters) as capable of evolving. Just as evolutionary game theory, in the hands of Brian Skyrms and others, has shed light on long-standing puzzles about rationality and social behaviour, a dynamical approach to Pascal’s Wager has the potential to help explain the emergence and stability of forms of religious belief. On this re-interpretation, the point of Pascal’s argument is that if 0 < p < 1 (where p is your credence that God exists), your credences are unstable. The expected utility calculation shows that you have reason to act in ways that will increase the value of p. Stable religious beliefs emerge as equilibrium points. As a simple illustration of these ideas, consider the prevalence of belief in jealous gods: those who reward only their followers. The dynamical model shows that in a competition between a universalist god who rewards everyone and a jealous god, wagering for the latter is the dominant choice. Nice gods finish last: there is just no point in wagering for them.
Although these ideas take us some distance from the author of the Pensées, they offer an interesting way of responding to the initial objection that probabilities should be revised only on the basis of evidence, and never for pragmatic reasons. William Clifford famously espoused a strict version of evidentialism: “It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence.” Clifford’s Principle is a tempting refuge for those who feel acutely uncomfortable with Pascal’s Wager. Philosophical work in recent decades, however, including current ideas about pragmatic encroachment in epistemology, undermines the idea that there are any beliefs that are entirely insulated from prudential consideration. A more promising approach is to follow Elliott Sober’s suggestion that the weakness of Pascal’s Wager is that it is insufficiently prudential. Sober’s point is that Pascal singles out belief in God for prudential scrutiny and revision, while holding fixed a “background theology” that allows only believers to go to heaven. Sober develops a model that allows prudential considerations to affect both belief in God and belief in a theology, and he shows that prudential revision can lead to either stable religious belief or stable atheism.
Pascal’s Wager remains a fascinating argument for the same reasons as ever: the appeal to infinity, the link between prudential considerations and religious belief, and the ingenious use of decision theory. The reasons for discomfort and suspicion also remain, but our understanding of Pascal’s Wager can be greatly enriched by contemporary ideas about infinite utility and the dynamics of rational deliberation. The argument then becomes a test case and a model for powerful analytical methods that take us into new territory in decision theory, philosophy of religion and other areas.