Pascal’s wager fails

Blaise Pascal, in his Wager Argument, avoids the trend of attempting to prove God’s existence, instead arguing that it is rational to strive to believe that God exists. If humans act rationally and desire maximum utility, then, according to Pascal’s argument, it follows that they must wager for God.

In this paper, I will first explain why Pascal rejects previous attempts to demonstrate God’s existence. I will then define what it means to wager for God, expected utility and rational choice. Then, I will then outline his Wager Argument and explain why one must be compelled to wager for God. Finally, I will present objections to Pascal’s argument, a possible response Pascal would give to these objections, and each reply will be followed by my own reply.

Pascal doesn’t attempt to prove God’s existence because he believed the proofs didn’t work. He believed the proofs in one way or another had a fatal error. For example, the Ontological argument (an argument Descartes endorsed) was later refuted by the Islandargument. These attempts to prove God only proved arrogance by philosophers who attempted to comprehend such a supreme being such as God, Pascal said. In fact, such a God was “infinitely incomprehensible,” Pascal said. “We are therefore incapable of knowing either what he is or whether he is.”

Even if such a proof were true, it would only prove a dry, abstract, omni-god of the philosophers, not the true God, the scripture God of Christian faith. Pascal believed true faith did not require proofs and reasons. So instead of attempting to prove the existence of God, Pascal instead attempted to prove people should strive to believe that God exists.

Before I get into Pascal’s actual wager, I believe it is best to explain expected utility and rational choice. To do this, I will start with an everyday example:

Suppose you are faced with the following situation about how to wager on the outcome of the draw of a card:


‘Ace’ comes up

Probability is (.083)

2-King comes up

Probability is (.917)

You wagered on an Ace Payoff: $10 million Payoff: Zero
You wagered on 2-K Payoff: Zero Payoff: $10

Assume the value of money is directly proportional to the dollar amount – thus, $10 would be exactly twice as much as $5. Also, note that expected utility is defined as payoff multiplied by chance of that outcome occurring for each outcome, added together.

If you wagered for an ace, the expected utility is: 10,000,000 * .083 * 1 = 830,000.
If you wagered for everything else, the expected utility is: 10 * .917 * 10 = 110.04

Given this assumption and this definition, it would be rational to wager the card drawn will be an ace because expected utility is highest with this option.

Now that we have defined expected utility and rational choice, I believe we are almost ready to move on to Pascal’s wager argument. But before we do, I feel it is important to define “Wagering for God.”

“Wagering for God” means you should believe in the God of scripture. Although Pascal concedes one can’t turn believe on and off like a switch, he believes you should strive you believe by acting like a believer. Beliefs, according to Pascal, can be ingrained through repetition – if you go to Church every Sunday, then, eventually, you will end up believing in God.

And now, the Wager Argument:

1)      Either God exists or he does not exist, and you must wager for God or against God.

2)      The payoff structure is as follows:

  1. If you wager for God and he does exist: Infinite payoff (eternal happiness)
  2. If you wager for God and he does not exist: Some +/- finite payoff.
  3. If you wager against God and he does exist: Some +/- finite payoff.
  4. If you wager against God and he does not exist: Some  +/- finite payoff.

3)      The probability you should assign to God existing should be some positive, non-infinitesimal amount.

4)      Rationality requires you to perform the act with the maximum expected utility.

5)      Rationality requires that you wager for God.

If you wagered for God, the expected utility is: Positive infinity * Probability ascribed to God = Positive infinity.

If you wagered against God, the expected utility is: Finite number * Probability ascribed to God not existing = Finite +/- number.

Much like the previous example with the cards, it is in the best interests to wager for the situation with the highest utility.

So long as you ascribe a positive non-infinitesimal probability to God existing, the possibility of gaining positive infinity utility that comes with wagering for God outweighs any finite number that can be brought on by wagering against God. Therefore, anyone who takes Pascal’s Wager and follows the laws of rationality and expected utility must wager for God.

There are several objections to Pascal’s wager, but I will only go into two of the most fatal. I will first give the objection, then give a chance for Pascal to reply, and finally I will present my reasoned analysis.

Objection No. 1 – Is the notion of an infinite payoff suspect?

It seems that an infinite payoff throws a wrench what wager we should choose. Take for example the following situation:

You can wager for God A or God B. Both Gods have positive infinite payoffs. There is a 99 percent chance of God A existing and there is a 1 percent chance of God B existing.

If you wagered for God A, the expected utility is: Positive infinity * .99 = Positive infinity.

If you wagered for God B, the expected utility is: Positive infinity * .1 = Positive infinity.

The intuitive choice seems to be to wager for God A. God A is extremely likely to exist while God B has very little chance of existing. Imagine if a robber came into your house, pointed a gun to your head, and pulled 100 cards from his pocket, numbered 1-100. You could choose between two options – guess that a card from the set of 1-99 will be selected (so if 88, or 32, or 12 is selected, you will be right) or guess that the card labeled 100 will be selected. He will then select a random card and if you guess correctly, then you will live. Here is the situation presented visually:

1-99 comes up (.99) 100 comes up (.1)
You guessed 1-99 You live You die
You guessed 100 You die You live

In this situation, any rational person would pick the set of 1-99 because it gives the most chances of surviving this robber’s attack. Because there are 99 chances out of 100 that 1-99 will come up, you should choose 1-99.

However, from our example of God A and God B, the expected utilities are the same. If the expected utilities are the same, it follows that there is no difference wagering for either God.

The situation with the robber is nearly identical to the situation with the two Gods. Both offer a 99 percent chance of one instance occurring and only a 1 percent chance of the other occurring. But in the situation with the robber, who wouldn’t take the 99 chances of living over just one chance? It would be irrational to take on 1/100 odds, yet according to the infinite utility formula shown above in the God situation, this seems like a rational decision to make, unlike the situation with the robber, where it seems almost automatic to take the 1-99 option. The only difference in these two examples is, in the example with the two Gods, the two Gods offer infinite payoff. Since the only real difference between these two situations is infinite payoff, infinity seems to throw the whole expected utility formula out the window.

Pascal may reply to this objection to throw infinity out the window and replace it with some large finite number. Instead of offering infinite payoff, this God would instead offer an extremely large finite payoff, dwarfing any other finite payoff associated with wagering against God.

The situation now would follow:

God exists God does not exist
Wager for God Large positive finite payoff Some -/+ finite payoff
Wager against God Some -/+ finite payoff Some -/+ finite payoff

For this situation let’s fill in some numbers.

If you wager for God and he does exist: 10100 (Commonly referred to as a googleplex.)

If you wager for God and he doesn’t exist: 10

If you wager against God and he does exist: 10

If you wager against God and he doesn’t exist: 10


Probability for God existing: .1

Probability against God existing: 99.9


If you wagered for God, the expected utility is: 10100 * .1 = 1010

If you wagered against God, the expected utility is: 10* .999 = 9.99


From Pascal’s new situation, with an infinite payoff replaced by a large finite payoff, it follows we should wager for God because the expected utility with wagering for God is highest. Thus, even with infinity out the window, Pascal would argue, it still follows that you should wager for God.

But this reply falls short. The probability listed above seems to be suspect. Who would Pascal be to ascribe a probability to the existence of a God who rewards those with a finite payoff of a googleplex? For all we know, the probability could be less than that. To refute this reply, simply drop the probability.

Using the above situation…


Probability for God existing: 10-100

Probability against God existing: 99.99999 (95 more 9’s follow this)


If you wagered for God, the expected utility is: 10100 * 10-100 = 1

If you wagered subjective against God, the expected utility is: 10* .999 = 9.99


The probability of God existing comes down to a number based on disposition. No matter how high Pascal would set the finite payoff at, you could just counter with just a small of a finite number, knocking the expected utility to a level where it is logical to wager against God rather then for him. Thus, Pascal’s argument falls short when trying to explain the suspect notion of an infinite payoff.

Objection No. 2 – Wouldn’t this argument work for any religion that promises an infinite payoff to believers (as many do)? It would seem to work as well for Allah, Kali, Odin, etc… as it does for Pascal’s Catholic God. But you can’t simultaneously believe them all. We can also make up all sorts of possible deities. Perhaps there is a malicious God who rewards those who don’t believe in him. As long as these deities each have a finite, non-infinitesimal probability of existing, then it looks like a parallel argument would work for them too.

Pascal might first reply that while this reductio ad absurdum may point out that this objection may rule out his argument as a whole, it doesn’t show exactly where this argument goes wrong.

Furthermore, it is best to just sit on our hands and do nothing? Even if there exists many Gods, all with infinite payoffs, and we can only pick one God out of this group, we should still pick a God for the chance of an infinite payoff. The only way to achieve this payoff is to wager for a God. Pascal likely believed it was best to wager for a Catholic God because of the influences around him at the time. We can apply this situation to our own lives. If we must choose to wager for a God, it seems easier to first wager for a God that many people we grew up with believed in. If many people around you believe in the Catholic God and then you believe in a malicious God that rewards those who don’t believe in God, you’d be hard pressed to find friends – psychologically, believing in such a God may be more difficult. Thus, Pascal would say, pick a God to wager for based on your own personal beliefs or dispositions.

Though Pascal makes a point for wagering one way or the other, this is not the original aim of his argument. At the beginning, I stated Pascal set out to prove that if humans act rationally and desire maximum utility, then, according to Pascal’s argument, it follows that they must wager for God. But because there is a chance other deities can exist that give an infinite payoff and we can’t believe in them all, Pascal’s argument falls short. This objections proves to be fatal to Pascal’s Wager argument.

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