action is the solution (to the sleeping beauty problem and the simulation hypothesis)

Someone must have said this before (didn’t really check the literature in depth), however here is a lens that dissolves a lot of the confusion: probabilities are not just degrees of belief, but also guides to action. We should not just ask what to believe, but also how to act, and the answer to the second question changes dramatically depending on the consequences of our actions.

For the sleeping beauty problem, it dissolves the mystery kinda completely. I initially added the simulation hypothesis as a joke at the beginning, but in fact, one reaches quite surprising—and reassuring!—conclusions.

the sleeping beauty problem

Alice is put to sleep, and a coin is tossed. If the results is heads, Alice will be woken up on Tuesday. If the result is tails, she will be woken up on Monday, her memory wiped, and then woken up on Tuesday again. Each time she is woken up, she is asked to bet on the result of the coin toss.

Imagine you are sleeping beauty and you wake up. Would you bet on heads or on tails?

There are two widely recognised positions here:

• halfer: the coin is fair, therefore there is a 50-50 chance of it having fallen on either face, so it doesn’t matter on what you bet.
• thirder there are three scenarios in which Alice is woken up, and only 1 of those times the coins would be heads, so there is a 1/3 chance the coin is heads, you should bet tails.

One could—and, indeed, does—debate about this endlessly. But we really don’t have to. What you bet on depends on what you want.

the solution: probabilities as guides to action

There are a lot of things to be said about probability (see also the previous blog entry), but the main thing is that probabilities are not things that float out in the aether, but they are numbers that help us decide what to do. In our case, whether Alice should guess heads or tails has as much to do with the probability that the coin landed heads as with what she wants to achieve.

A basic principle in betting and microeconomics is that, when we consider different courses of action, we should maximise the expected value of the consequences of our actions. This can be justified in several ways by appeals to axioms of rationality. You don’t have to use it, but basically everyone agrees that it’s the rational thing to do.

Alice gets no information when she wakes up. She already knew she was going to wake up! Even when the coin lands tails and she gets woken on two days, she will not remember on Tuesday having been woken up on Monday. Basically, she might as well can decide on the betting strategy before going to sleep.

Let’s define the expected utility of a strategy $S$ as

${\displaystyle U(\mathrm{S})=U(\mathrm{S}\mathrm{\mid }H)p_H+U(\mathrm{S}\mathrm{\mid }T)p_T,}$

where $p_H$ and $p_T$ are the probabilities for the coin landing heads or tails and $U(S|H)$ and $U(S|T)$ is the utility of picking the strategy $S$ in either case. Since we have thrown a lot of coins in the past, we have seen that the probability to use for things involving coins is

${\displaystyle p_T=p_H=\frac{1}{2},}$

so that the utility of each strategy is simply determined by the utility of the outcome of the payoff:

${\displaystyle U(\mathrm{S})=\frac{1}{2}[U(\mathrm{S}\mathrm{\mid }H)+U(\mathrm{S}\mathrm{\mid }T)]\mathrm{.}}$

That’s it. When considering whether to say “heads” or “tails” when she wakes up, Alice should ask what she will get in return of being right or wrong. Modifying the payoff structure will dictate a change in her behaviour.

There is no platonically right or wrong answer without specifying the payoff.

Note that this completely dissolves the question of trying to understand what is the meaning of “What is the probability of heads, given that Alice has woken up.” This question is ill-posed. There is nothing to learn by simply waking up, as all awakenings are identical. Instead, one should ask “What should Alice say when she wakes up, if she wants to maximise her utility”.

different payoffs, different choices

For example, if they tell her: “we’ll count the number of times you are right, and we give you a reward each time” she might do one thing, and if they tell her: “if you are right once, then we give you $x$, if you are wrong once, we give you $0$” then she might pick another one. Let’s see this concretely.

Let’s introduce the function $u(r)$, which represents the utility of being right upon $r$ awakenings. Let $S_H$ and $S_T$ denote the strategies of betting on heads or tails, repsectively. Then we have

${\displaystyle \begin{array}{rl}{\displaystyle U({\mathrm{S}}_H)}& {\displaystyle =\frac{1}{2}[u(1)+u(0)]}\\ {\displaystyle U({\mathrm{S}}_T)}& {\displaystyle =\frac{1}{2}[u(0)+u(2)]}\end{array}}$

Let’s assume that there is no penalty in being wrong, so that $u(0)=0$. Then only the values of $u(1)$ and $u(2)$ matter for her choice. If it matters that she gets it right at least upon one awakening (so if $u(1)=u(2)$), then the two strategies are equally good, it does not matter what she says. But if getting it right twice is better than getting it right once ($u(2)>u(1)$) then she should go for guessing tails.

We can also make things a little more general. For example, suppose that maybe $p_Hp_T$, and that Alice gets woken up only once on heads and $n$ times on tails. Let’s also introduce the (dis)utility $w(n)$ of being wrong $n$ times. If Alice commits to $_H$, if the coin lands heads she will be right once, if the coin is tails she gets it wrong $n$ times,

${\displaystyle U({\mathrm{S}}_H)=p_{H}u(1)+p_{T}w(n)\mathrm{.}}$

If she goes for $_T$ then she will be right $n$ times it the coin lands tails and wrong $1$ times if it’s heads, giving

${\displaystyle u({\mathrm{S}}_T)=p_{H}w(1)+p_{T}u(n)\mathrm{.}}$

You can see how things can get quite complex, but still totally workable.

Ok, now that we solved the sleeping beauty problem, let’s move on to the simulation hypothesis.

the simulation hypothesis

Are we in a simulation? Should we believe we are in a simulation? Endless things have been said about this, but since you made it this far, you know we should ask not so much “what should we believe?” but, rather: “how should we act?”

Let’s introduce two strategies:

• $S_$: behave as if we are in the base reality;
• $S_$: behave as if we are in a simulated reality.

We are going to do like before, and maximise our expected utility. We’ll entertain the possibility, with probability $p_n$, that there are $n=0,1,2,$ simulated realities. In the case of the coin, we have plenty of experience and we are convinced that there is a best choice ($p_H=1/2$). This time, $p_n$ is much harder to determine.¹ We should probably use the laws of physics to decide how plausible $n$ simulations are, but in the end there is plenty that we don’t know. But as we will see, it will not matter too much.

For now, let’s start computing.

If we act like we are in base reality, we will be right in base reality and wrong in all the simulations. We have to weight this by our belief that there are $n$ simulations:

${\displaystyle U({\mathrm{S}}_{\text{base}})=\sum _{n=0}^{\mathrm{\infty }}{p}_n(u(1\mathrm{\mid }\text{base})+w(n\mathrm{\mid }\text{sim}))\mathrm{.}}$

We can simplify this formula by recognising that probabilities sum to 1, so that $_{n=0}^p_n u(1)=u(1)$. Let us also set $w(0)=0,$ meaning that if no one’s wrong, no one is paying the price. We get

${\displaystyle U({\mathrm{S}}_{\text{base}})=u(1\mathrm{\mid }\text{base})+\sum _{n=1}^{\mathrm{\infty }}{p}_{n}w(n\mathrm{\mid }\text{sim})\mathrm{.}}$

Similarly, if we decide to act as if we live in a simulation, we will be wrong in the base, and right in all simulations:

${\displaystyle U({\mathrm{S}}_{\text{sim}})=w(1\mathrm{\mid }\text{base})+\sum _{n=1}^{\mathrm{\infty }}{p}_{n}u(n\mathrm{\mid }\text{sim})\mathrm{.}}$

Here we have our utilities. Pick some $p_n$, pick some $u$ and $w$ and you will know what to do.

making some choices

Ok, I agree, those two formulas don’t actually tells us much. But let’s make a little more assumptions, and see how far they take us.

Now, I would argue that we don’t really care how many times we are right as, in contrast to Alice, there is no way we can pool our winnings across simulations. We should only care about whether we are right or wrong. This is not self-evident, and one could write books about it (Parfit 1986).

But let’s roll with it, and set

${\displaystyle u(n\mathrm{\mid }\text{sim})=u(1\mathrm{\mid }\text{sim}),}$

${\displaystyle w(n\mathrm{\mid }\text{sim})=w(1\mathrm{\mid }\text{sim})\mathrm{.}}$

Our utilities become, simply²

${\displaystyle \begin{array}{rl}{\displaystyle U({\mathrm{S}}_{\text{base}})}& {\displaystyle =u(1\mathrm{\mid }\text{base})+(1-p_0)w(1\mathrm{\mid }\text{sim})}\\ {\displaystyle }\\ {\displaystyle U({\mathrm{S}}_{\text{sim}})}& {\displaystyle =w(1\mathrm{\mid }\text{base})+(1-p_0)u(1\mathrm{\mid }\text{sim})}\end{array}}$

From this point of view, the number of simulations does not matter! The only thing that matters how much we believe that at least one simulation that is indistinguishable (to us) from reality is possible.

I was actually surprised when i derived this! Often one hears arguments say that since there are way more simulated realities than base realities, most likely we are in one of the simulations. Whereas here the number completely drops out. Sure, we made it drop out by hand a couple of formulas above, but what do we care how many copies of us there are? What we care about is whether we are wrong.

Or, more precisely, if you don’t care how many copies of you are wrong, then it does not matter how many copies of you there are.

Matrix as metaphysics

Since we are even talking about it, we must find it at least conceivable that we are in a simulation. So let’s assume that there is at least one simulated reality, so that $p_0=0$. Then:

${\displaystyle \begin{array}{rl}{\displaystyle U({\mathrm{S}}_{\text{base}})}& {\displaystyle =u(1\mathrm{\mid }\text{base})+w(1\mathrm{\mid }\text{sim}),}\\ {\displaystyle }\\ {\displaystyle U({\mathrm{S}}_{\text{sim}})}& {\displaystyle =w(1\mathrm{\mid }\text{base})+u(1\mathrm{\mid }\text{sim})\mathrm{.}}\end{array}}$

Up to you to decide on the utilities i guess. It’s a sort of Pascal’s wager. In fact, i kid you not, there is a whole parallel between classical creationist beliefs and the simulation hypothesis (Chalmers 2016).

a final little surprise

Suppose you do not care whether we are in a simulation or not, all you care about is whether you are right or wrong in guessing about it. Then we should set

${\displaystyle u(1\mathrm{\mid }\text{base})=u(1\mathrm{\mid }\text{sim})=u(1)}$

${\displaystyle w(1\mathrm{\mid }\text{base})=w(1\mathrm{\mid }\text{sim})=w(1),}$

but then, what we have is:

${\displaystyle \begin{array}{rl}{\displaystyle U({\mathrm{S}}_{\text{base}})}& {\displaystyle =u(1)+w(1),}\\ {\displaystyle }\\ {\displaystyle U({\mathrm{S}}_{\text{sim}})}& {\displaystyle =w(1)+u(1)\mathrm{.}}\end{array}}$

The two strategies have exactly the same utility, and you don’t have to worry about which one to pick!

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Bibliography

Remember that sci-hub and libgen exist!

Image credit to OpenAI

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1. Little aside: both here and in the case of sleeping beauty, the probability is intended as a personalist probability, ie a degree of belief, a credence. This is the correct thing to do for decision making, and it allows us to make a decision in one-off scenarios.↩︎

2. We used the fact that ${\sum }_{n=0}^{\mathrm{\infty }}{p}_n=p_0+{\sum }_{n=1}^{\mathrm{\infty }}{p}_n=1${n=0}^p_n = p_0 + {n=1}^p_n=1.↩︎

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