So here comes another film inspired philosophical puzzle. Well really a less sophisticated version of this problem occurred to me a long time ago while reading Permutation City by my favorite sci-fi author Greg Egan and seeing someone last week totally consumed with a virtual world reminded me about it but everyone knows what I’m talking about when I mention the matrix. However, some people may not be familiar with the Sleeping Beauty problem so I will provide a short summary in the post body.

The point I want to illustrate is that whether or not this is the actual world or a matrix style simulation is an instance of the sleeping beauty problem. In particular the same reasoning that supports the two-thirds solution to the sleeping beauty problems seems to guarantee we are in a repetitive simulation. I don’t know if this puzzle has appeared in the literature yet (it seems kinda obvious to me so I have a hard time believing it hasn’t) but it is new to me (no I haven’t read all the papers I link so it could be there). Since I find the reasoning supporting the two-thirds solution compelling I expect some other solution will ultimately be forthcoming but I expect that solution will be more interesting than the problem itself.

More concretely it certainly seems possible that we are actually in a virtual world like the matrix. It remains possible even if we suppose that our lives are in fact arbitrarily long (the ‘real’ world has conquered aging) and every time we die in the virtual world our memories are wiped and the simulation is restarted. While we might think ‘a priori’ such a world is unlikely since it seems possible we should assign it some non-zero probability. However, since this matrix world repeats the experience we are having infinitely many times it would seem that the same reasoning which allows us to support the two-thirds solution in the sleeping beauty problem requires that we find the matrix world arbitrarily more probable than our own world, i.e., the probability of the matrix world conditionalized on matrix world or real world is one (or arbitrarily close to one). Of course the virtual world aspect isn’t central to the problem. We could repeat the same argument in favor of Nietchzie’s world of eternal recurrence or any situation which involves making us experience our lives so far arbitrarily many times.

The Sleeping Beauty Problem

In the sleeping beauty paradox we drug the hapless heroine into a deep slumber on sunday. At this point we flip a coin. If the coin lands heads we wake her up on monday tell only giving her time to realize she is awake but not informing her of the day or any other information before we render her unconscious and wipe her memory of the event. In this case we do not wake her again till wednesday. If instead the coin lands tails we do the same thing on monday, wake her wipe her memory and put her back to sleep, but instead of letting her sleep will wednesday we do the same again on tuesday. The puzzle is when woken is it rational for sleeping beauty to believe the probability of tails is still 1/2 or should she now believe it is 2/3.

If you are interested in the reasons for one side or the other you can read the philosophical arguments about the sleeping beauty problem. I may write up my thoughts on the problem in more detail later but rather than get into the details here I will just assume Elga’s argument for the two-thirds answer is correct.

In effect Elga’s solution seems to mandate the following means of calculating the credence we should assign to the coin being tails upon waking up. Consider all the ways you could have the experience you just had (being woken up by the experimenter) where the coin lands tails and weight each of those by its probability of occurring. Divide this number by the sum of all the ways you could have this experience weighted by probability and divide. Since their are two ways you could have the experience of waking up when the coin is tails (it is monday or it is tuesday) and each of these has probability 1/2 (if the coin lands tails you are guaranteed to have both) the numerator is one. The only other way you could have the experience of waking up is if the coin landed heads and it is monday the denominator is 3/2. Hence giving the ‘right’ answer of 2/3.

This solution has great intrinsic appeal. If we want to calculate the probability of T given E we usually sum up all the ways T and E could happen weighted by their unconditional probability and divide by the sum of all the ways E could happen weighted by unconditional probability. Though in this case we may not technically be calculating the probability of tails given that you had the experience of waking up since this is the normal means of updating on new information it seems appealing to do the same in the sleeping beauty case.

Sleeping Beauty and Repeating Simulations

However, let us consider what happens if we apply this reasoning to the matrix example. In this case instead of the coin flip we have the choice of world (real or virtual) and instead of being woken up twice our simulation is repeated n times. So instead of H₁, T₁ and T₂ as Elga had we have the following outcomes.

• R₁: The world we experience is real and we are living our one and only life (after death we just cease to exist or go on to some other type of existence).
• V₁: The world we experience is just a simulation and we are on the 1st repetition of the simulation.
• V₂: The world we experience is just a simulation and we are on the 2nd repetition of the simulation.
• V_n: The world we experience is just a simulation and we are on the nth repetition of the simulation.

• V: The world we experience is just a simulation and we are on some repetition of the simulation, i.e., V=V₁v V₂…v V_n.

Since we only want to compare the probabilities of this world being real and being in this particular simulation we will conditionalize our probabilities on VvR, i.e., P(X) denotes the probability of X given ‘V or R’ and hence P(V)+P(R)=1. Importantly P(V) and P(R) do not give our current probabilities for V and R but the ‘a priori’ probabilities for these worlds before it is conditionalized on any experience. As we are simply postulating some possible world where we are thrust into simulators we can assume that all n repetitions of the simulation are guaranteed, i.e., there is no chance the simulation will break or we will die before all repetitions are complete.

Applying the same solution Elga does to this situation we calculate the current credence we should hold for being in situation V to be P(V)*n/(P(R)+n*P(V)). Note that if we hold P(V) and P(R) fixed and allow n to go to infinity our rational credence in V goes to 1. Of course one might have a lower probability for being in a situation where the simulation is repeated n+1 times than one where it is repeated n times but we can avoid this problem by considering a world where the simulation is repeated infinitely (omega) many times. Countable additivity guarantees the credence we should have in this case is just the limit of our prior answer as n goes to infinity. Since we seem happy to consider the possibility that our universe will keep expanding forever it seems perfectly possible that such a situation could exist. Heck one could just pick some system of particles and dynamical laws which repeat at regular intervals forever and postulate psycho-physical laws which makes the motion of these particles create the experiences of your life. So long as we assign non-zero ‘a priori’ probability to such a situation the above argument shows we should believe it is infinitely more likely we are in this repeating simulation than the real world, i.e., the credence we should give V given V or R is 1.

Potential Solutions

Obviously this result seems absurd. It seems totally unreasonable that we should be required to believe the world is infinitely more likely to be a simulation or even that we will repeat our current life after we die just because we are having some experience. So what moves can we make to avoid this result? Our first implication might be to discard the most abstract and non-intuitive principle we used, countable additivity. However, this really does no good as just using finite additivity we should be able to prove the credence we should assign to being in V is greater than any number below 1. Alright so what about giving up the assumption that P(V) isn’t 0? Remember P(V) refers to the conditional probability of V given V or R so this would require believing we have grounds to believe that the real non-recurring world was infinitely more likely than this world filled with simulations. Even if you are willing say this about a world running indefinite simulations of this one are you really prepared to (essentially) rule out a world of eternal recurrence?

A more substantive move might be to reject the idea that ‘a priori’ probabilities exist. After all most philosophers are skeptical of objective prior probabilities so don’t we get rid of this paradox for free when we give up objective priors? Even if true it would still be pretty cool to show that reasonable assumptions actually excluded objective priors. However, nothing in this argument required that these probabilities be objective only that we have some notion of likelihood about various ways the world might have turned out. I have been careful to put ‘a priori’ in scare quotes everywhere because this probability function can be the result of any model we adopt and need not be a consequence of logic alone. If the physicists come up with some theory which gives probabilities for various types of initial conditions (and at least some of these allow for a universe which lasts forever) it seems the problem reoccurs. Do we really want to be philosophically committed to the absence of such a physical theory?

Moreover, denying the existence of this sort of ‘a priori’ probability function leaves us at a loss about whether or not to believe we are in some sort of simulation. If I observed certain very bizarre or intuitively absurd events it would be perfectly rational of me to conclude I was in some type of simulation. For instance if a giant black unix terminal opened up in the sky and what appeared to be computer commands appeared inside with corresponding effects out in the world it seems reasonable I should accept that I’m in a simulation. Yet despite never having seen (or at least remembered) the real world I would judge certain ways the real world might be more likely than others, e.g., fundamental physical objects are particles not toasters or starfish. Moreover, if I was trying to predict what the real world was like I would first judge how probable that sort of world seemed and then modify that judgement by how likely that sort of world was to cause simulated experiences like mine. These judgments may not be a priori in the sense of being a priori truths but if does seem like our basic intuitions of probability give us the sort of probability which lead to all this trouble.

This seems to leave us only two possible options: bite the bullet or abandon the ‘two-thirds’ solution to the Sleeping Beauty problem. As both the limiting frequency and a number of compelling arguments favor the ’2/3′ option I am loathe to give it up. Additionally biting the bullet may not be quite as unpalatable as it seems. It seems to be a valid scientific possibility that the universe will collapse and be recreated in a new big bang. If this happens indefinitely our world effectively is a recurring world and the world of repeated simulation seems to lose any apparent advantage. However, this draws the paradoxes of infinity deeply into our judgments about the world. In particular the sleeping beauty problem asks us to calculate the probability of being in a world where the coin lands heads versus one where it lands tails. Once we assume the world is recurring both possibilities present the same number of experiences of waking, countably many in both cases, nullifying the ‘two-thirds’ solution. Alternatively we might postulate that our world actually consists of countably many (or more) copies (as in many worlds interpretations of quantum mechanics). However, this still leaves the above problem and adds the additional puzzle of what it even means for the same experience to be happening countably many times.

Ultimately I’m pretty much at a loss about this problem. Maybe someone reading this will have some ideas.