Pascal's Wager

Pascal's Wager (also known as Pascal's Gambit) is Blaise Pascal's application of decision theory to the belief in God. It is one of three 'wagers' which appear in his Pensées, a collection of notes for an unfinished treatise on Christian apologetics. Pascal argues that it is always a better "bet" to believe in God, because the expected value to be gained from believing in God is always greater than the expected value resulting from non-belief. Note that this is not an argument for the existence of God, but rather one for the belief in God. Pascal specifically aimed the argument at such persons who were not convinced by traditional arguments for the existence of God. With his wager he sought to demonstrate that believing in God is more advantageous than not believing, and hoped that this would convert those who rejected previous theological arguments. Applications of this argument can be found in other religious philosophies, such as Hinduism, and especially Buddhism (see below). A well known example is the C.S. Lewis quote: "Christianity, if false, is of no importance, and if true, of infinite importance. The only thing it cannot be is moderately important."

The incompleteness of his argument is the origin of the term Pascal's Flaw.

Explanation

The Wager can be stated by the following explanation provided by Pascal in the Pensées--

God is, or He is not. But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up...Which will you choose then? Let us see. Since you must choose, let us see which interests you least. 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; 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. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.

To Pascal, God was the Christian God of the Bible. The Bible provides information about the Christian God but not proof for God . Should you believe in this God? In his Wager, Pascal provides an analytical process for a person to evaluate his options in regard to belief in the Christian God. The person who has no more information than that which he finds in the Bible would find himself facing the following possibilities:

You may believe in God, and if God exists, you go to heaven: your gain is infinite.
You may believe in God, and if God doesn't exist, your loss is finite and therefore negligible.
You may not believe in God, and if God doesn't exist, your gain is finite and therefore negligible.
You may not believe in God, and if God exists, you will go to hell: your loss is infinite.

From these possibilities, and the principles of statistics, Pascal deduced that it would be better to believe in God unconditionally. It is an application of game theory to itemize options and payoffs and is valid within its assumptions. After starting with only that information found in the Bible, the person can then be given additional information to encompass all the knowledge that man has assessed and on re-evaluating his position, he will reach the same conclusion; it would be better to believe in the God of the Bible unconditionally.

The following table shows the values that Pascal assigned to each possible outcome:

	God exists (G)	God does not exist (~G)
Belief in God (B)	+∞ (heaven)	0
Non-belief in God (~B)	−∞ (hell)	0

Given the values that Pascal proposes, the option of believing in God (B) dominates the option of not believing in God (~B). In other words, the value gained by choosing B is always greater than or equal to that of choosing ~B.

At first, Pascal assigns equal probability to each of the two possibilities. He argues that "reason is no more shocked in choosing one rather than the other," due to our ignorance. However, he later points out that the probabilities make no difference to the argument, since any non-zero chance multiplied by infinity yields an infinite expected value.

Pascal's Wager is similar in structure to the Precautionary principle, and has similar strengths and weaknesses.

Criticisms of Pascal's wager

Before entering into the criticisms of the Wager, it is only fair to note, as is less widely known, that the wager was never intended to be a basis or reason for faith. The wager is found in an apologetic (his Pensées) aimed at those who didn't consider the question of God worth considering. The wager had the express intention of showing the "happy agnostic" the value and probable necessity of considering the question of God.

Pascal has been severely criticized, for example by Voltaire. Some criticisms are summarized below:

Assumes God rewards belief

Pascal's Wager suffers from the logical fallacy of the false dilemma, relying on the assumption that the only possibilities are:

the Christian God exists and punishes or rewards as stated in Christian theology, or
God does not exist.

The wager does not account for the possibility that there is a God (or gods) who, rather than behaving as stated in certain parts of the Bible, instead rewards skepticism and punishes blind faith, or rewards honest reasoning and punishes feigned faith, or does not punish belief or disbelief at all.

Also, in the many human societies where belief in a particular religion is rewarded by economic and social benefits, the wager permits the hidden addition of such benefits to the wager, and many memeticists (those who work with concept of the meme) believe that this forms much of the unconscious core of religious belief.

Assumes a Christian God

The "many-gods" argument points out that we can find indefinitely many other possibilities offering eternal bliss and threatening eternal torment. For example, non-Christian gods might exist, and punish Christian believers for their failure to believe in them. Or some powerful entity might decide to punish those who believe in a god while rewarding non-believers:

Suppose there is a god who is watching us and choosing which souls of the deceased to bring to heaven, and this god really does want only the morally good to populate heaven. He will probably select from only those who made a significant and responsible effort to discover the truth. For all others are untrustworthy, being cognitively or morally inferior, or both. They will also be less likely ever to discover and commit to true beliefs about right and wrong. That is, if they have a significant and trustworthy concern for doing right and avoiding wrong, it follows necessarily that they must have a significant and trustworthy concern for knowing right and wrong. Since this knowledge requires knowledge about many fundamental facts of the universe (such as whether there is a god), it follows necessarily that such people must have a significant and trustworthy concern for always seeking out, testing, and confirming that their beliefs about such things are probably correct. Therefore, only such people can be sufficiently moral and trustworthy to deserve a place in heaven--unless god wishes to fill heaven with the morally lazy, irresponsible, or untrustworthy.^*(Richard Carrier)

Even if (contrary to Pascal's original argument) we can assign greater probability to one of the possible outcomes, it makes no mathematical difference. As the previous section mentions, any non-zero probability multiplied by infinity yields an infinite expected value.

In this way, Pascal's Wager can be used to deduce that it is advisable to believe in any or all of a variety of gods; however, the belief systems of most religions require exclusive belief in the god of that religion, so the wager is simply not valid when applied to such religions. This is the solecism known as the argument from inconsistent revelations, though not all religions (for example, Hinduism and other Pantheistic religions) suffer from it. There is also the Jewish faith to consider, which expects a non-Jew only to obey the Noahide Laws in order to receive reward in afterlife. In addition, some religions do not require a focus on a deity, such as Buddhism. A "many-gods" version of Pascal's Wager is reported by the 13th century Persian chronicler Ibn Rustah to have been taken by a king in the Caucasus, who observed Muslim, Jewish and Christian rites equally, declaring that "I have decided to hedge my bets".

Measure theory

The wager assumes that God is possible, and hence there is a non-zero probability of him existing. But this does not work all the time. For instance, in a measure theory conception of probability, one can have infinitely (uncountably) many possibilities, each of which has probability zero. (E.g., choosing a random real number between 0 and 1, all numbers cannot have positive probability or the probabilities sum to more than 1.) See also almost surely.

It is not clear what is meant when "probability" or "chance" is said in the context of something possibly existing, but probability cannot be used as defined in mathematics to justify the wager as is, since God being possible does not mean that God's existence has positive probability.

Statistical arguments

The wager fails to mention any costs relating to belief. It is argued that there may be both direct costs (time, health, wealth) and opportunity costs. Most modern religions require their followers to spend time attending religious services at temples and to donate money to these temples when possible so that they can be maintained. As a result, if a person believes in a God that does not exist, then that person has lost time and money that could have been used for some other purpose. There may be opportunity costs for those who choose to believe: for example, scientific theories such as evolution that appear to some to contradict scripture could theoretically enable a non-believer to discover things and accomplish things the creationist could not. It is also argued that belief incurs a cost by not allowing the believing person to participate in and enjoy actions forbidden by dogma. Many devout people make more noticeable sacrifices for their religious beliefs. For example, Jehovah's Witnesses do not accept blood transfusions. If a Jehovah's Witness' death could have been prevented by a blood transfusion, and there is no God, then the Jehovah's Witness has lost his or her life needlessly.

The wager assumes a non-zero chance that God exists. This makes it ineffective on strong atheism which assigns the chance that God exists to zero, making choosing to believe or not believe provide an equal reward (0). Others have argued that the utility of salvation cannot be infinite, either via strict finitists or belief that an infinite utility could only be finitely enjoyed by finite humans.

Many-way tie

Given that the choice of wagering has an infinite return, then under a mixed strategy the return is also infinite. Flipping a coin and taking the wager based on the result would then have an infinite return, as would the chance that after rejecting the wager you end up taking it after all. The choice would then not be between zero reward (or negative infinite) and infinite reward, but rather between different infinite rewards.

Does not constitute a true belief

Another common argument against the wager is that a belief in a god or religion based solely on the expectation of a reward (and, conversely, the fear of punishment) is not a valid belief. Basically, this viewpoint argues that a person who comes to a belief must come to this point for other reasons (i.e., a desire to do good; revelation).

William James, in The Will to Believe, summed this argument up like this:

Surely Pascal's own personal belief in masses and holy water had far other springs; and this celebrated page of his is but an argument for others, a last desperate snatch at a weapon against the hardness of the unbelieving heart. We feel that a faith in masses and holy water adopted willfully 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.

In modern times, this criticism is often leveled against evangelistic Christianity, especially those who try to incite fear by portraying such events as the Rapture in popular media. Such a belief is sometimes called "Hell insurance" (or "Heaven insurance"), and is considered heretical by many mainstream Christian denominations.

Assumes one can choose belief

The wager may also be criticised for requiring one to choose one's beliefs. A person who accepted the tenets of the wager might act in a pious and believing way throughout their life; however, they do not have the option to choose to believe in God, as Calvinism believes that God alone makes the choice to accept some and reject others, regardless of what the person does.

Pascal acknowledged that there would be some difficulty for an atheist intellectual persuaded by this argument, in putting it into effect. Belief may not come. But in such a case, he said, one could begin by acting as if it had come — hear a mass, take holy water. Belief might then follow.

There is also the argument that one could "game" the wager in a scenario where the deathbed conversion is possible — as is the case in some streams of Christianity. The person who converts on their deathbed could have failed to have been dutiful in fulfilling their doctrinal obligations, and still gain the happiness associated with the Christian concept of "heaven". The danger here is well known to most Christians, as this is a common theme of sermons in a variety of denominations. The risk of taking this gamble only to die suddenly, and without warning, or to experience the time of tribulation is often portrayed as too great a risk to take. There are also a number of Christians who disagree with the doctrine of salvation by repentance alone and that God also rewards good works. Under this assumption, Pascal's Wager is all the more important.

Decision-theoretic arguments

The above criticisms are addressed explicitly in a generalised decision-theoretic version of Pascal's argument, with probabilities interpreted in the Bayesian sense of expressing degrees of belief, and each option carrying certain utilities or payoffs.

This leads to the following matrix, where a, b, c and d are the utilities arising from each of the four options:

	God exists (G)	God does not exist (~G)
Belief in God (B)	a	b
Non-belief in God (~B)	c	d

The total utility for believing in God is then aP + b( 1 - P ) while the total utility for non-belief is cP + d( 1 - P ), where P is the probability of the existence of God. Belief in God is thus optimal in decision-theoretic terms for all P > 0 if the values for the utilities satisfy the inequalities a > c and b ≥ d. The first inequality is relatively uncontroversial, as it requires simply that one considers a well-founded belief in God to have a higher utility than an ill-founded disbelief in God. However, the second inequality holds only if one regards the benefits of an ill-founded belief in God to be no less than those from a well-founded disbelief in God. This is patently a matter of personal choice. Many people maintain they do indeed get tangible benefits here and now from their belief in God, and that these exceed those that would accrue from not having such a belief (e.g. no requirement for regular observance of religious practices). On the other hand, many agnostics would argue the opposite case. The analysis shows atheists are not absolved from having to assess the utilities through setting P = 0; they must also be confident that d > b.

This requirement for such an assessment of utilities suggests that Pascal's Wager should be regarded as a criterion by which the coherence of one's existing beliefs can be judged, rather than as a method of choosing what to believe.

An application in Buddhist scripture

The decision-theoretic approach to Pascal's Wager appears explicitly in the 6th Century BC Buddhist Kalama Sutta, in which the Buddha argues that regardless of whether the difficult concepts of rebirth and kamma are valid, acting as if they are brings tangible rewards here and now. Also known as a "lesser truth" in Buddhism, a religious concept thought to bear fruit in the here and now rather than dealing with the hereafter, or eternity.

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