The Multiverse and Inverse Gamblers

The multiverse objection to fine-tuning does not commit the inverse gambler's fallacy

Defenders of the fine-tuning argument claim that the fact that the world is finely tuned—that the constants are within a very narrow range that permits life, such that if they were tweaked slightly, the world would be an atomless mess—is evidence for design. In response, defenders of the multiverse say that, if there’s a multiverse that contains multiple different laws of physics, then some of the universes would be finely tuned. Thus, fine-tuning can be explained by a multiverse just as easily as a designer. Defenders of the inverse gambler’s fallacy charge claim that this commits the inverse gambler’s fallacy. The inverse gambler’s fallacy arose from thinking about cases where you walk into a casino, see someone get a royal flush, and say “wow, he must have been playing all night.” It’s true that the fact that the person gets a royal flush sometimes is more strongly predicted on the hypothesis that he’s been playing for a while, but the hypothesis that the person gets a royal flush in this particular hand isn’t affected by how many past games they’ve been playing.

The IGF occurs when one infers some background condition (BC) that makes some event (E) more likely, when E is more likely conditional on BC than conditional on the falsity of BC, but the specific instance of E is no more likely on BC than on the falsity of BC.

Defenders of the IGF response to the multiverse claim that the multiverse might explain that there would be a finely tuned world, but it doesn’t explain why this particular world would be finely tuned. It doesn’t explain why our world in particular has finely tuned constants. To see this, imagine that the universes are arranged from left to right, each with a number. The furthest to the left has the number 1, the second furthest to the left has the number 2, etc, while the furthest to the right has the number 100 billion. Suppose our world is world number 6,953. A multiverse predicts that there would be a life-permitting universe, but it doesn’t predict that world number 6,953 would be life-permitting.

I used to find this very puzzling, but I do no longer. To see where this reasoning goes wrong, first imagine another scenario where there are two possibilities. The first is that there is a very small casino where only five people are playing poker and one of them cheats. The second is that there is a large casino where no one is cheating. Suppose that the large casino is large enough that someone is guaranteed to get a royal flush. Suppose additionally you know that if someone cheats in the small casino, they would cheat to get a royal flush.

Now suppose that some button allows you to teleport to the nearest room where someone gets a royal flush. You press the button and—surprise—teleport to a room. It seems like this doesn’t give you evidence for either hypothesis. Both theories predict that there will be a room where someone gets a royal flush, they just have different explanations of why someone gets a royal flush.

However, one could reason as follows. Suppose that the person is cheating. Well then you know that there’s only one room, such that if the cheating hypothesis is correct, cheating would occur in this room. In contrast, if the big casino hypothesis is true, it’s unlikely that this person would be cheating in this room. So the cheating hypothesis gets a big boost.

To see what’s wrong with this view, suppose that Jane gets a royal flush. Additionally, Jane would have been in the large casino. Suppose the large casino would have had 10,000 people, one of whom would have gotten a royal flush (I know this is not what the probabilities would actually be). This means that the large casino hypothesis predicts with 1/10,000 odds that Jane would have gotten a royal flush, because each of the 10,000 people getting a royal flush is equally likely.

Now, it’s true that the Jane is a cheater in the small casino hypothesis predicts with certainty the evidence. But the Jane is a cheater in the small casino hypothesis has a very low prior—conditional on someone being a cheater, assuming each of the 10,000 people are equally likely to be cheaters, its prior is 1 in 10,000. Thus, the two probabilities cancel out and all things considered, the overall probability of the casino hypothesis doesn’t change.

Now that we’ve seen why the casino reasoning, we can also diagnose the flaw in the IGF charge against the multiverse hypothesis. Suppose that there are 10,000 universes that God might create, each of which he’d be equally likely to create. Suppose our prior in the multiverse is .5 and our prior in design is also .5.

We find ourselves in one of the 10,000 universes—say universe 6,851. Does this give us good evidence for design? No! It’s true that the multiverse doesn’t predict that we’d be in this universe, which happens to be universe 6,851. But neither does design. Both predict a 1 in 10,000 chance of us being in this universe. Thus, the multiverse doesn’t understate the evidence—it predicts with just as much probability that we’d exist as theism.

This explains what’s wrong with the probability judgment of the proponent of the IGF charge. But what of the bare charge itself? Does inferring fine-tuning commit the inverse gambler’s fallacy? Why is the multiverse scenario, where you infer a large number of universes from the fact that you got lucky in this one different from inferring a large number of rolls from the fact that you got double sixes in this one? The answer is pretty straightforward and not original to me: if there’s an observer selection effect, such that an observer only becomes aware of some event if it occurs, then inferring a large sample is not the inverse gambler’s fallacy. Because you’d only exist if there was fine-tuning, you’re not equivalent to a person looking upon a random roll—as Bostrom says, you’re instead equivalent to a person who will only be alerted if two sixes are rolled, whenever they are rolled. But if you stand outside the casino, and are only alerted when two sixes get rolled, inferring that there have been many rolls is perfectly good reasoning. Similarly, it’s reasonable to guess, all else equal, that you came from a big family, because more people like you are born in bigger families, by definition.

There is a vast ensemble of somewhat convincing objections to the multiverse reply to fine-tuning. But critics of the multiverse need to finely tune their arguments and weed out the unconvincing ones like the IGF charge.