“Decision theory is concerned with the reasoning underlying an agents choice” (Steele, Stefansson, 2015). In other words, decision theory is the mathematical study of probabilities that an individual analyses to make a decision of what to do when we are faced with a matter of uncertainty. To help make such decisions, we use a closely related concept known as expected value. The expected value represents every possible value of a given variable multiplied by its respective probability of occurring, all totalled together. One example of measuring the expected value is the rolling of a six-sided die. The maths to calculate the expected value for this measured event breaks down as follows:
1*(1⁄6) + 2*(1⁄6) + 3*(1⁄6) + 4*(1⁄6) + 5*(1⁄6) + 6*(1⁄6) ≈ 3.5
The consecutive numbers one to six represent the possible values of our die (our variable) which is independently multiplied by its probability of occurring; (1/6). With such mathematical probabilities it is easy for an agent to rationally predict outcomes, in this case, we get an expected value of 3.5. However, in circumstances when dealing with variables that do not involve numbers, decision theory as a strategy is greeted with great uncertainty. This is apparent when we are questioned whether we should worship a God or not. Traditionally, arguments of the ontological, teleological and cosmological kind, helped rationalise religious belief by showing God’s existence is probable. On the other hand, Blaise Pascal, a 17th century french philosopher and mathematician, in his incomplete book ‘The Pensees’ sorted to use decision theory to take a pragmatic approach to mandate that we should worship God, by showing it is rational to believe in God through what is called Pascals Wager. We will take the wager as follows, assuming it has three premises. The first of these premises commands that rationality must give some positive probability to God’s existence (Hajek, 2003, 27), irrelevant how small we take this value (i.e. for the sake of the example below 0.001, which leaves a probability of 0.999 that God does not exist).
The second premise, a 2×2 decision matrix, shows the expected values of wagering for or against God, in the two possibilities; God does exist (G) and God does not exist (¬G). Say we wager for God’s existence, i.e. believe in God and (G), we will receive infinite utility from eternal reward in an afterlife. In contrast, if (¬G) we may receive some finite positive utility (f1) from leading a religious lifestyle, since such a lifestyle brings at the least solace. As a result, our expected value for belief in God and worshipping God is (using the hypothetical probabilities of Gods existence from above):
((∞)*(0.001))+((f1)*(0.999)) = ∞
On the other hand, if we decide to wager against God and (G), we will receive an infinite disutility as we are sent to an eternal damnation from leading a secular lifestyle. However if (¬G), we will receive some finite positive utility (f2) from leading a life of hedonism and enjoying worldly pleasures, e.g. pre-marital sex and drugs. We can calculate our expected value for wagering against God:
((-∞)*(0.001))+((f2)*(0.999)) = -∞
Finally, our third premise requires us to employ the Expectation Rule (Jordan, 1998, 420), where rationality insists you should perform the act which produces the maximum expected value. As a result, Pascal concludes that we should therefore wager a belief in God, worshipping him, since its expected value (∞) is far greater than that of not believing in God (-∞) (Hajek, 2003, 28). From this, we can therefore deduce that decision theory mandates that we should worship God, since it is the most rational option.
Prima facie, Pascals Wager seems to be the most sensible path to choose, as he shows that it is pragmatically rational to believe in God. However Pascals strong religious experience, lead him to convert to Catholicism and in turn he created and based his wager on the Christian God. As a result, an immediate fault of the wager follows: there are perhaps millions of different Gods that are worshipped across humanity, driving the question which God we should worship. This fault is relevant because otherwise an Imam may be able to take on the same reasoning seen in the wager about his own God, Allah, rather than the Christian God Pascal maintains. This brings us to the Many-Gods objection against Pascals Wager, shown in example of another 2×2 decision matrix. Let us assume we worship either Yahweh, the God in Judaism, or Allah, the God in Islam in the possibilities that either Allah exists or Yahweh exists. We can also assume that there is an infinite utility to be gained from worshipping the correct one these two different Gods (from an afterlife) and an infinite disutility from an eternal damnation to those who chose to worship the wrong God. As a result, if we worship Yahweh and Yahweh does exist then we will receive an infinite reward (∞), however if we worship Yahweh and Allah exists instead, then we will receive an infinite punishment (-∞). Vica versa if we decide to worship Allah. Subsequently, our expected value of worshipping either Yahweh or Allah will achieve a value of zero from the addition of ∞ and -∞. Dealing with this expected value of zero, it provides no help in rationally deciding which God we should worship. This means we continue to run a threat of an eternal damnation in the afterlife even when using decision theory as a mandatory guide to worshipping God. Pascals Wager therefore seems to dictate that we should worship God since it is the most optimal decision but it fails to draw a conclusion on which God is the correct God to worship. In other words, Pascal has created a wager in favour of theism but not a particular religion. As a result, it achieves no progress since the risk between worshipping God or not is the same as just choosing a God to worship. Consequently, decision theory should not mandate that we worship God.
A second objection, the St. Petersburg paradox, expresses a technical concern for the notion of an infinite expected value. The paradox, taking only the calculated expected value into consideration, shows how the possibility of being able to gain an infinite expected value through some variable can mean very little to participants. The paradox, through a game of chance offers a single player a series of fair coin tosses. The aim of this game is to toss and land as many consecutive heads but once a tails appears, the game terminates and the player takes whatever is in the pot. The initial stake starts at £2 and increases exponentially, doubling as each progressive head is thrown. For example, if the player throws 3 consecutive heads followed by a tails on the fourth throw, s/he will win £8. The question is now posed as to what amount should the player be willing to pay to enter such a game. The probability of winning £2 is 1/2, the probability of winning £4 is 1/4, winning £8 is 1/8 and so on. The expected value of this game is calculated as follows, with the assumption you can throw an infinite number of consecutive heads:
E= 2*(1/2)+4*(1/4)+8*(1/8)+16*(1/16)+…
= 1+1+1+1+…
= ∞
It follows that you should be willing to pay any finite sum of money to play this game, if we only consider the expected value as is done in Pascals Wager. Yet rationality dictates that we do not, in fact we would only pay a small amount of money to play (i.e. £5), since although there is an infinite expected value to be gained, it is accompanied with an infinite risk. The conclusion is that decision theory is a defective guide when dealing with infinite values. Even if we confine this game of chance to a finite number of coin tosses (e.g. 100,001), we would still be unwilling to pay any large finite sum of money to participate (e.g. £99,000). As a result, we should not adopt decision theory as a basis for religious commitment, therefore it does not mandate that we should worship God.