Playing games in many worlds - Part I
Would you stake your fortune on a 100 to 1 outsider? Probably not. But what if, somewhere in a parallel universe, the straggling nag does come in first? Would the pleasure you feel in that universe outweigh the pain you feel in the one in which you've lost?
Questions not dissimilar to this one occupy physicists and for entirely respectable reasons. Quantum mechanics suggests that reality is fuzzy, at least at very small scales. Particles can be in a state of superposition, simultaneously possessing properties we would normally deem mutually exclusive. For example, they can be in several places at once (see the previous article for more on this).
How would you bet in a parallel universe?
The big question is why we never see superposition in everyday life. Traditional interpretations of quantum mechanics say that when we make a measurement (such as looking where a particle is) the superposition somehow collapses and only one of the superposed states remains real. We don't know which one that will be but quantum mechanics provides us with probabilities. The method which extracts those probabilities from the maths of quantum mechanics is known as the Born rule.
The many-worlds interpretation takes a different approach. It suggests that when the measurement is made the world splits into separate branches. In each of the branches you see one of the possible outcomes.
But then, what of the Born rule? If every possible outcome happens, it makes no sense to talk about the chance of it happening — that chance is 1. If the many-worlds interpretation is to be taken seriously, then the Born rule needs a new explanation. What do the numbers it attaches to different outcomes, to different branches of the world, mean?
Looking after yourself
Probability is a slippery concept even in the ordinary one-world view of reality. Yet we all use some notion of it, albeit perhaps a hazy one, whenever we ponder a decision. Whether it's buying a house or deciding whether to go on a date, we come up with some sort of assessment of how likely it is that things turn out well. Combining this with some assessment of how much we care how things turn out, we make a decision. And a lot of the time what we are trying to look after when making the decision is the well-being of our future self.
This still applies if you're an inhabitant of a many-worlds world. "If I believe that I live in branching reality and I know that there will be various versions of me after branching, I have the same reasons to care about those various versions of me as I have to care about the one version of me if there wasn't any branching," says David Wallace, a philosopher of physics at the University of Oxford. "So I can ask: what are my preferences between various actions I can choose?" This intriguing idea lies at the heart of an attempt to make sense of the Born rule in the many-worlds context that was pioneered in by the physicist David Deutsch in the 1990s.
How to bet wisely in one world...
It makes sense to phrase all of this in the language of betting. Suppose you're just about to perform a measurement of a quantum system. For example, electrons have an electromagnetic property called spin and when you measure spin it can come out to be pointing up or down. Now someone offers you a bet: if the spin points up you win £10, if it points down you lose £10.
Heads, tails or both?
There is a rational way of approaching this problem in the ordinary one-world view in which there is no branching and you get either spin-up or spin-down but not both. First you should rate how much you like or dislike each possible outcome of the bet (in the technical language, you attach a utility to each outcome). You might simply rate each outcome by its monetary value, so that’s +10 for spin-up and -10 for spin-down. But maybe you resent losing £10 more than you enjoy winning it, so in that case you might rate spin-up as only, say, 5 and spin-down as -10. You also rate the value of not taking the bet. Write and for the utilities you attach to spin-up and spin-down, and for the utility you attach to not taking the bet.
Now suppose you believe that the probability of spin-up is . The probability of spin-down is therefore . The expected utility of taking the bet is
This quantity measures the average amount of utility, or satisfaction, you’d expect to get if you repeated the bet many times. The expected utility of not taking the bet is simply
There is a mathematical result which says that a rational person (someone adhering to certain principles of rationality) should aim to maximise their expected utility. That is, if your expected utility of taking the bet is bigger than the expected utility of not taking it (), then you should take it, otherwise you shouldn’t.
...and how to bet wisely in many worlds
Now let’s go back to the many-worlds situation. You know that the moment you measure the electron spin the world splits into two branches. In one of them you observe spin-up and in the other you observe spin-down. So if you accept the bet, then in one branch you win £10 and in the other you lose £10.
What should you do now? You might calculate how much the bet would gain you on average over the two branches,
and see how that compares to the utility of not taking the bet. This is the same calculation as a person in a non-branching universe would make, in which the probabilities of spin-up and spin-down are both 1/2.
But perhaps you feel sorry for the you in the loosing branch. In that case you might base your decision on the quantity
in which spin-down counts for more than spin-up. You’d be behaving like a person in a non-branching universe, where the probability of spin-up is 1/4 and the probability of spin-down is 3/4.
The question of which weight to attach to each outcome, whether it’s 1/2 to both or 1/4 to one and 3/4 to the other, could of course be very tricky — until you realise that quantum mechanics comes with an inbuilt suggestion for the weights. The Born rule attaches a number to each outcome, say to spin-up and to spin down, which the traditional view point interpreted as a probability. So why not use those numbers for the weights? Then, like a person in a non-branching universe, you’d base your decision on the quantity
You’d accept the bet if is bigger than the expected utility of not accepting it.
The Born rule is named after the physicist Max Born (1882-1970).
In an intriguing result Wallace, building on work by Deutsch, has shown that this is indeed the only rational way to go. "[If you know that you live in a branching universe] and you think that the underlying laws of physics are [those given by] quantum mechanics, then not only is it rationally required to bet according to some probabilities, it's rationally required to bet as if the probabilities were the [numbers given by the Born rule]," explains Wallace. Anything else will contravene one of several principles of rationality (we'll have a closer look at some of these, quite controversial, principles the next article).
The result does not just apply to this simple toy example of spin measurement, it works in a more general setting too. Suppose the universe is just about to split into a number of branches to each of which the Born rule attributes a number. Then if you are faced with a number of possible action (to bet or not to bet) you're rationally compelled to choose the one that maximises your expected utility, calculated using the numbers given by the Born rule.
But why does this "explain" the Born rule in the many-worlds context? This is what we'll look at in the next article.