# Pascal’s Wager and the Space of Possible Gods

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Space is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist, but that’s just peanuts to space.
– Douglas Adams, The Hitchhiker’s Guide to the Galaxy

Julia Galef spoke with Amanda Askell about Pascal’s Wager on Rationally Speaking (podcast, blog). Pascal’s wager being that if you’re among the faithful you shall be infinitely rewarded, if that faith is the right model of the world. Whereas if you’re an atheist you get nowt, if you’re correct about how the world is. So the expected utility of being numbered among the faithful is argued to be higher than that of not being.

Askell makes a number of defences of Pascal’s Wager. I’m going to make two arguments in response. The first is that the appeal to expected utility theory is flawed; in principle because the space of possible gods is not defined and, in the general case, will not support Askell’s proposed defence. The second is that because reward and punishment from a jealous god is asymmetric the calculus could well point the other way. Afterwards I sketch out why the first problem seems pretty hard to avoid.

## Gods of the continuum

Askell’s first defence (about 3 mins 45 into the podcast) relates to a many gods argument. Criticism: the Wager is unconvincing as adherents of other faiths can make similar claims, so being presented with the wager does not help guide our action. Defence: perhaps, but if a god does exist then you have a higher chance of picking the right faith by picking a faith rather than none, and therefore of enjoying infinite utils. Hence you should just pick one. Furthermore if we have any information which makes one deity more likely than another (e.g. we had a good vibe about in a dream, a fortune cookie etc) then the symmetry is broken and we should pick the most likely god to believe in.

The mathematical issue is that this relies on the space of possible gods being finite. In the interest of brevity, this section contains a quick version of a counter argument; with further elaboration below.

Suppose that for every real number, $r$, in the interval $(0,1) \in \mathcal R$ the god $G_r$  holds the number $r$ to be the Real Number. $G_r$ has decreed that on the Day of Observation believers in $r$ shall be risen to spend eternity in heaven. Unbelievers and heretics shall be annihilated. How do we assess the expected utility?

If we are completely ignorant of the Real Number, then a uniform distribution on $(0,1)$ would be appropriate for our model. Let $u(s,r)$ be the utility we receive by believing in $s$ when $r$ is the Real Number. Now compute the expected utility of arbitrarily picking a god $G_s$ to believe in. Here we’re even assuming that there really is a god $G_r$ (though the result would obviously still hold with some mass on an atheist universe):

$$\mathbb E[u(s,r)] = \mathbb E[u(s,r)|r \neq s] P(r \neq s) + \mathbb E[u(s,r) | r=s]P(r=s) = 0 \cdot 1 + \infty \cdot 0 = 0$$

Where the first expectation was equivalent to the integral of a constant, $0$, over the domain of a random variable, $(0,1)$,  since the removal of a set of measure zero $\{r\}$ from the domain has no effect on the integral. While $P(r=s)=0$ since again $\{r\}$ was a set of measure zero. Let $a(r)$ represent the utility of atheism if $r$ is the Real Number, so that the utility of being a non-believer then is:

$$\mathbb E[a(r)] = 0$$

Hence, in a case of uncountably many possible gods, picking a god randomly to believe in has no higher utility than atheism. Ignoring all short run costs and benefits of either policy. It’s worth repeating that this was not a hard atheist argument. We not only supposed a non-zero probability density for every possible god, but even assumed one of the theologies was correct!

Next, Askell’s asymmetry argument. Suppose we have information that we are more likely to find the Real Number in one place than another. Then decision theory directs us to choose the wager for a god $G_r$ which we believe maximises our chances. This defence breaks on a modified example also. For suppose that instead of being uniformly ignorant of the Real Number, we instead believed it to lie in the vicinity of $\frac{1}{2}$. We could model this with a Beta($2,2$) distribution and exactly the same outcome would obtain. For the same reason as above regarding the measure of a particular point in a continuum. Indeed the result should hold up for any continuous density on $(0,1)$ that was non-zero on every element.

## Jealousy on Olympus

4. What about the atheist-loving god? [IC] The basic idea: Suppose there’s a god that sends all non-believers to heaven and all believers to hell. Given the logic of Pascal’s wager, I ought not to believe in God. Answer: If it’s rational for you to think that disbelief in God (or cars, or hands) will maximize your chance of getting into heaven, then that’s what you ought to do under PW. What’s the evidence for the belief-shunning God? Possibly: ‘Divine hiddenness’ plus God making us capable of evidentialism. The evidence against? God making us capable of performing expected utility calculations, all the historical testimonial evidence for belief-loving Gods. I suspect the latter will outweigh the former. But if you’re making this objection you’re already on my side really: we’re now just quibbling about what God wants us to do.

The problem here stems from an assumption that God is not jealous. In addition to, “all the historical testimonial evidence for belief-loving Gods,” there is a good deal of historical testimonial evidence for heresy hating gods. For example let us modify our previous set of gods to become jealous in a first approximation of Dante’s Inferno. There, god really hates heretics (among many other groups). Virtuous atheists sit about in Limbo.

So let’s suppose that each god, $G_r$ declares that on the Day of Outcomes believers in the Real Number $r$ shall be infinitely rewarded ($+\infty$). Heretics (adherents to $s \neq r$) shall be consigned to watch I’m a Celebrity! Get Me Out of Here on loop eternally ($- \infty$). While atheists shall be left in Limbo ($0$).

What’s the expected utility of different belief policies for these jealous gods? If we assume all numbers are symmetric then a uniform distribution is appropriate. Well, for the atheist it goes unchanged. For randomly picking a god $V_r$?

$$\mathbb E[u(s,r)] = \mathbb E[u(s,r)|r \neq s] P(r \neq s) + \mathbb E[u(s,r) | r=s]P(r=s) = -\infty \cdot 1 + \infty \cdot 0 = -\infty$$

Because the probability of picking exactly the right god is zero, whereas there is a continuum of ways to end up watching reruns of I’m a Celebrity! until long after the stars go out and the universe turns cold. So, while this might be characterised as quibbling about what god wants us to do, it really matters if we’re taking the wager seriously.

## Discrete gods

The root cause of the first issue is that we don’t actually know what the space of all possible theologies looks like. So, for a start, we can’t actually evaluate the expectation as supposed by Askell. Really, I should just object that the space is not well defined, and then shoot down any attempt to define it satisfactorily However, if we suppose that we’re satisfied with the definition of the space we can look again at the mechanics of Askell’s many gods defence. We’re asked to choose one belief from among the union, $U$, of a (presumably) countable set of theologies, $G$, and a set containing atheism, $A$. It is assumed that a PMF exists over this union such if we restrict our choice of beliefs to the subset $G$, the redistribution of probability mass from $A$ to $G$ increases the probability mass of each god in $G$. The problem is that if any individual item in $G$ has no mass to begin with such a rescaling achieves nothing. So, this issue is a generalisation of the false dichotomy: we’re asked to choose an option from a finite set when the actual set of choices is actually much larger.

In case the example I chose with gods labelled by real numbers seems too artificial, we can consider a more ‘realistic’ scenario that gets us to the same place. Suppose a religion consists of a set of beliefs. What is the space of all possible beliefs? Well as a first stab, we could say that it is the union of all sentences in all languages. There exist plenty of formal languages with a countably infinite number of possible sentences. For example, the number of correct C++ programs is at least countably infinite. Therefore the union of all sentences in all languages is at least countably infinite.

Next, let’s say that a theology is a set of beliefs. Then the space of all possible theologies therefore has a cardinality like the powerset of all possible beliefs. Moreover, the powerset of a countably infinite set is uncountable. An uncountable set has no probability mass function that is nonzero for all of its elements. Hence no probability mass function exists for the set of all possible theologies (defined in this manner) which is non-zero over all of its elements. Therefore redistributing whatever mass we gave to atheism over possible theologies has no effect on their mass and Askell’s defence fails again as in the case of the real numbers.

## Conclusion

We have backed the proposed defences of Pascal’s wager based on expected utility theory into something of a corner. The wriggle room that I can see left is to either:

Assert that the space of all possible theologies is countable, requiring by extension an assertion that the space of all possible beliefs is finite. However this has a zero prior for me, since, for example, I already have a several beliefs about every real number $r$, e.g. that a larger real number $s$ exists.

Assert that some arbitrary subset of possible theologies deserves to be treated as atoms with positive mass (e.g. the historically observed religions). But this is just begging the question (what about the historically unobserved but possible religions).

Assert that gods will reward beliefs that are ‘close enough’ to being correct in some sense to be defined. But without a proper framework (e.g. some weird  filtration) it will not be possible to evaluate the expected utility of the decision. However, it seems that this would resolve as deciding which belief sets have a higher likelihood of reward. This would seem to look something like smaller belief sets having a higher likelihood than larger ones by virtue of being closer to a larger number of other belief sets. Which would lead us again to the same sample space, but with a non-uniform distribution over it, and still no mass function.

Neither is it obvious that the historical religions fit the ‘close enough’ model c.f. the Inferno. Indeed we also have to think about which belief sets would be ‘close enough’ to satisfy deities that we can have no knowledge of, e.g. all religions occurring in the (possibly infinite) universe outside our light cone. We should then reason towards a set of beliefs maximally likely to be rewarded. It’s not clear that such a process would end up anywhere different than secular reasoning about morality would lead us to; particularly if we take the Inferno‘s treatment of virtuous pagans as a model. I’d be interested to see someone try.

Also, everything I’ve said here is agnostic with respect to what ‘god’ actually means. E.g. the simulation hypothesis being correct, and the extra-galactic grad student rewarding all simulants with some particular belief set.

We’ve made it through to the end without griping about infinite expected utils too!

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