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Reverse Pascal's Wager

by Francois Tremblay



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.

 —Blaise Pascal, French Philosopher and Mathematician

Now since Pascal does not evaluate theism on truth but on utility and gains, we will examine the question under this angle.

If strong atheism is true, of course the gains on both side are the same. Let’s see the corresponding matrix (the ~ is used for negation symbol).

  There is a god There is no god
  P(G)=0 P(~G)=1
theist gain=0 gain=0
atheist gain=0 gain=0

Averages: theist 0, atheist 0.

Since being strong atheist is a trivial case, let us analyze weak atheism. Let’s say a weak atheist considers that the chance of a god existing is P.
Now let us attribute a gain of I for Heaven and -I to Hell.

  There is a god There is no god
  P(G)=P P(~G)=1-P
theist gain=IP gain=0
atheist gain=-IP gain=0

Averages: theist 0, atheist 0.

The theist still has the better average of gains. Now let us assume, as most contemporary christians do, that there is no Hell. Obviously the atheist’s case looks much better.

  There is a god There is no god
  P(G)=P P(~G)=1-P
theist gain=IP gain=0
atheist gain=0 gain=0

Averages: theist IP, atheist 0.

For any given I and P (providing that I and P are superior to zero), we would always arrive at a much higher result for the theist. Does this mean that theism, in a utilitarian sense, is better?

More then one concept of god exists, and with the same proof, which is faith. Indeed a potential infinite number of gods can exist, that we have not discovered yet. Let us assume, for this finite example, that there are N possible gods. Let us also assume that the theist believes in the god X.

  god X exists god ~X exists There is no god
  P(X)=P/N P(G~X)=(NP-P)/N P(~G)=1-P
theist gain=IP/N gain=0 gain=0
atheist gain=0 gain=0 gain=0

If there is a Hell associated with each of these gods:

  god X exists god ~X exists There is no god
  P(X)=P/N P(G~X)=(NP-P)/N P(~G)=1-P
theist gain=IP/N gain=-I(NP-P)/N gain=0
atheist gain=-IP/N gain=-I(NP-P)/N gain=0

Averages: theist ((2-N)IP)/N, atheist -IP.

As said before, in reality there is an infinity of possible gods with the same probability. There can also be different kinds of gods (for example, a god which favours atheists).

We can easily see that as N becomes bigger, the atheist’s gain does not change but the theist’s gain rapidly becomes negative. Indeed, a bit of elementary algebra reveals that when N nears infinity, ((2-N)IP)/N is equal to -IP, exactly the same gain as the atheist.

This is also intuitively seen, as the theist becomes less and less likely of going to heaven, and Hell fills up the whole P (since the theist goes in the hell of the gods other then X), which is the atheist’s situation.

Now, since the theist’s advantage over the atheist becomes practically nil when we take all the infinite possibilities of theistic gods, and may even be negated by other kinds of gods, we cannot conclude that weak atheism is less useful then theism.

Note that I have not included the loss of a lot of time in the theists’ life in rituals, intellectual worth, etc. Then the average gain at infinity would indeed be inferior to the atheist’s.

There is also a side problem which is the best heaven/worst hell behaviour.

If there is a Hell, the logical goal would be to choose the religion with the worst hell, to make sure that one does not fall in it. This is because no god has a higher probability then another, hence the choice must be made on other properties.

The same can be applied for the case of heaven, i.e. that we should then choose the religion with the best heaven. Islam definitively wins on this one (who wouldn’t want houris?).

Last updated: January 1, 2005