We start off with a brief summary of Pascal's Wager to establish proper foundation for its dissection and analysis.
Pascal's Wager makes the following assumptions:
1) if God exists, He will reward those who believe in Him
2) God's reward is infinite
3) there is nothing to lose in believing in God
4) there is an equal chance of God existing and God not existing
With such assumptions we can derive the following calculus to find the expected value of this wager:
(reward x probability God exists) - (cost x probability God does not exist) = expected value of Pascal's Wager
Let reward be infinite and cost be zero. Even if assumption 4 is not true, the expected value will be infinite as long as the probability of God's existence is not zero. Hence it is rational, from the perspective of decision theory, to believe in God.
I will briefly sum up some noteworthy counter-arguments before offering my own. A common rebuttal criticizes the wager as a false dilemma in that it offers only the following possibilities:
1) a benevolent God exists and rewards according to belief
2) a benevolent God does not exist
This dichotomy precludes the possibilities that God is malevolent, or has an alternative system of reward. But as Pascal structures his analysis from a Christian viewpoint, this argument is moot.
Another criticism is that Pascal neglects to factor in all religions, and any possible religion, into this decision matrix. Since Pascal had obviously discounted the validity of all other religions, this argument does not serve to undermine Pascal's framework.
I seek to find errors in Pascal's Wager within his own established paradigm. After some brief analysis, I formulated the following premises:
1) the cost is not zero
2) the cost is not fixed
3) an infinite expected value automatically invokes the law of diminishing marginal utility
We will examine Premise 1 from a Christian perspective. There are few Christians that would deny the potential drawbacks of a Christian lifestyle. Doctrinal obedience places restrictions on potentially anything from diet to social conduct, depending on one's interpretation of the Bible. Furthermore, belief itself might exact a social cost in the form of religious persecution. While this is no longer as salient as during the founding years of Christianity, it still exists in areas of the Middle East, where conversion to non-Islamic religions provoke violence or even murder.
John Milton explicitly mentions the sacrifices of Christianity in Paradise Lost (I have not read the book. This tidbit of info I owe to Matthew Lu of Free Exchange):
"Of servitude, to serve whom God ordains, / Or Nature; God and Nature bid the same, / When he who rules worthiest, and excels / Them whom he governs. This is servitude."
Belief of God is servitude under God. This is cost.
In addition to the daily costs of Christian belief, we must factor in the opportunity cost: what one may gain if one does not believe in God. The opportunity cost of belief is variable. Since prevalent Christian doctrine makes no demands on the time of conversion, one may become choose belief at any point in one's lifetime to claim salvation.
If one enjoys behavior contrary to Christian doctrine (e.g. sodomy, adultery, and other fun things), then one would delay conversion until the last possible instant so as to minimize the opportunity cost. Therefore, precluding sudden deaths, there is no incentive in Pascal's framework to believe in God until one reaches the deathbed. Following the logic of decision theory, the wager Pascal construed as being supremely imperative essentially begs for procrastination.
Even if we accept both Premises 1 and 2, there is still the unaddressed question of the infinite expected value. No amount of trickery with cost and opportunity cost will produce a tangible effect on something infinite.
We therefore invoke the law of diminishing marginal utility. This law states that quantitative increase in an item of value will eventually reach a point where additional items will yield progressively less value.
Take for example a gift of roses. Although pragmatically wasteful, a bouquet of roses is lovely and symbolically meaningful. Two bouquets might be even better. Three bouquets is pushing it. By the time your romantic interest receives the ten thousandth bouquet of roses, he/she will be sick of them and throwing them away as soon as they are delivered.
We can create a new scenario to fit the form of a wager. Let us suppose you are asked to fund a fledgling company. If the company succeeds, you will earn $1 trillion dollars, and there is a 20% chance of success. However, to get the company started, you must invest your entire life savings of $500,000, which will all be lost if the company fails. If we apply the calculus of decision theory, we will find that the expected value of this wager is $199,999,600,000. That's great--but how many will take this wager? The flaw of using expected value to guide decision making is the assumption that more is always better. Someone who is perfectly satisfied with $500,000 does not need to take a wager that will likely (80% chance) be financially destructive.
Applying the law of diminishing marginal utility to Pascal's Wager, we are no longer so awed by the concept of "infinite" reward. Why would one put up with doctrinal restrictions and institutional conformity in search of a possible reward when one is perfectly satisfied with life as it is?
The answer, of course, is the possibility of Hell; but since Pascal did not factor this into his wager, neither will I.
Some might point out that there are pluses to a religious life: community bonding, moral values, etc. Once again, however, these are not part of Pascal's formulation. Unlike many arguments presented against Pascal's Wager, my arguments is based entirely on decision theory within Pascal's framework. This is a logical criticism of Pascal's Wager, not one against Christian or religious belief in general. The wager, as it stands alone, is weak even within the framework Pascal establishes himself.