This is the first post in my series on Anthropics. The next one is Conservation of Expected Evidence and Random Sampling in Anthropics

Introduction

There is a curious Motte-and-Bailey dynamic going on with anthropic reasoning. The Motte is that it's just about a specific scoring rule and a corresponding way to define probability estimates. The Bailey is that we have psychic powers to blackmail reality by creating copies of our minds or using amnesia drugs. I'm going to explore these dynamics using two version of the famous Sleeping Beauty mind experiment: classical with the amnesia drug, and incubator, where either one or two Beauties are created. Simultaneously, I'll showcase the dissimilarities between them.

As far as I know people, who are Thirders in classical Sleeping Beauty, are also Thirders in the incubator version for mostly the same reasons. The SIA school of anthropics considers 1/3 to be the correct answer in both problems. And yet these problems are quite different. Adam Elga's proof for thirdism definitely doesn't work for the incubator version. Reasons for answering 1/2 are also different: Double Halfer in one and Halfer in the other.

Classic version of Sleeping Beauty

Let's start from the original version of the experiment

Here is a simple implementation in Python

def classic(heads_chance=0.5):
    days = ['Monday']
    if random.random() >= heads_chance: # result of the coin toss
        coin = 'Tails'
        days += ['Tuesday']
    else:
        coin = 'Heads'
    return days, coin

First, let's clear out what I mean by different scoring rule producing either 1/3 or 1/2.

coin_guess = []
for n in range(100000):
    days, coin = classic()
    for d in days:            
        coin_guess.append(coin == 'Heads')

coin_guess.count(True)/len(coin_guess) # 0.3322852689217815

coin_guess = {}
for i in range(100000):
    days, coin = classic()
    for d in days:            
        coin_guess[i] = (coin == 'Heads')
        
coin_guess = list(coin_guess.values())
coin_guess.count(True)/len(coin_guess) # 0.50167

That is, depending on whether we count per day or per experiment we get different answers, upon running the experiment multiple times. This is completely unsurprising.

Personally, I've always answered 1/2 in Sleeping Beauty and found the Thirders approach silly. Clearly, my credence for Heads should represent the probability of the coin actually landing Heads per toss, not per my guess! For example, in the Monty Hall problem my credence of winning on a door switch is 2/3 not because now I make two guesses instead of one, but because I actually win two out of three times this way in a repeated experiment. If my credence for a coin being Heads is 1/3 and in reality, if we do the same experiment a hundred times, in about fifty of them the coin is heads - then it's an obvious example of a map not representing the territory - which is quite understandable considering the usage of the amnesia drug.

Yet, even I can appreciate that the Thirders position points to something. If we specifically care about subjective experience, than indeed two thirds of them are going to be when the coin is Tails. As long as we do not claim that this fact gives an ability to predict the result of the coin toss better than chance, then we are just using different definitions, while agreeing on everything. We can translate from Thirder language to mine and back without any problem. Whatever betting schema is proposed, all other things being equal, we will agree to the same bets.

This is what I call the Anthropic Motte. The Bailey is the claim that the coin is actually more likely to be Tails when I participate in the experiment myself. That is, my awakening on Monday or Tuesday gives me evidence that lawfully update me to thinking that the coin landed Tails with 2/3 probability in this experiment, not just on average awakening.

This Bailey has originated from anthropic theories, talking about "updating on existence" such as SIA and SSA and will be more evident in the incubator version of the Sleeping Beauty. But first let's have a look at the Adam Elga's paper Self-locating belief and the Sleeping Beauty problem with a proof for Thirdism, which, I believe has also contributed to the Bailey and the confusion between it and the Motte.

On Adam Elga's proof for Thirdism

I find a lot of issues with it, following from what the author acknowledges himself:

Notice that the difference between your being in T1 [Tails and Monday] and your being in T2[Tails and Tuesday] is not a difference in which possible world is actual, but rather a difference in your temporal location within the world

Basically, you can't lawfully use probability theory in this setting assuming that Tails and Monday, Tails and Tuesday and Heads and Monday are three exhaustive and exclusive outcomes which probabilities are supposed to be added to 1. T1 are causally connected to T2, they happen to the same person, even if this person doesn't remember it.

According to Elga:

However

$P (T u e s d a y | T a i l s) = 1 / 2$

And so

$P (T a i l s) = 2 / 3$

Notice that it's just the probability of Tails, not probability of Tails conditional on awakening - the model Adam Elga uses doesn't account for such things at all, so all the justifications based on the updating on awakening are inappropriate here.

This, of course, directly contradicts the initial claim that the coin is fair, which is, among other things, used to prove that

$P (T a i l s | M o n d a y) = P (H e a d s | M o n d a y)$ = 1/2

Now, whether the statement above is true or not, is a big deal. And not only because it's crucial for Adam Elga's proof. Among those who give the answer 1/2 in the Sleeping Beauty experiment there are two distinct positions: Halfers who do not agree with this statement and Double Halfers who do. Thankfully this disagreement is easy to solve.

coin_guess = []
for n in range(100000):
    days, coin = classic()
    beauty_knows_monday = (days[0] == 'Monday')
    if beauty_knows_monday :
        coin_guess.append(coin == 'Heads')
        
print(coin_guess.count(True)/len(coin_guess))  # 0.49958

That is, if the Beauty knows that she is awake on Monday it doesn't give her any information whether the coin landed Heads or Tails. This is expected because Beauty was supposed to be awoken on Monday regardless of the coin toss result, moreover she could've been awaken on Monday even before the coin toss was made. If the credence for Heads in such situation was anything but 1/2 it would mean that participation in the experiment gave the Sleeping Beauty psychic powers to predict the result of the fair coin toss better than chance.

To conclude, in the classic Sleeping Beauty experiment, regardless of what you mean by credence that the coin is Heads, 50% of coin tosses are Heads, Sleeping Beauty doesn't have a better way to determine them than chance and knowing that the current day is Monday doesn't help either.

Incubator version of Sleeping Beauty

Now, let's look at the incubator version of Sleeping Beauty, where there is no sleeping drug or memory loss, but either one or two Beauties are created based on the result of the coin toss.

Here is the implementation of this scenario

def incubator(heads_chance=0.5):
    if random.random() >= heads_chance: # result of the coin toss
        coin = 'Tails'
        my_room = 1 if random.random() >= 0.5 else 2 # room sample
        other_room = 2 if my_room == 1 else 1
        return {'my_room': my_room, 'other_room': other_room}, coin
    else:
        coin = 'Heads'
        my_room = 1
        return {'my_room': my_room}, coin

This version of experiment evades the previous critique. Outcomes T1 and T2 are not causally connected, are randomly selected and do not happen to the same participant in different time.

However, Elga's proof won't work in this situation for a different reason.

Suppose someone left a definitive clue in Room 1 that it's indeed Room 1. This time it will be relevant new information as, finding yourself in Room 1 is twice more likely if the coin is Heads.

$P (H e a d s | R o o m 1) = 2 P (T a i l s | R o o m 1)$

A crucial element of Elga's proof isn't satisfied. In this case Halfers' reasoning is correct and Double Halfers' is wrong. We can see it on the repeated experiment

coin_guess = []
for n in range(100000):
    rooms, coin = incubator()
    beauty_knows_room1 = (rooms['my_room'] == 1)
    if beauty_knows_room1:
        coin_guess.append(coin == 'Heads')
 
 print(coin_guess.count(True)/len(coin_guess))  # 0.6688515435956072

So even if you believe that Adam Elga's proof for Thirdism is sound in the classical version of Sleeping Beauty, it's not applicable for the incubator version. It can't be the justification for answering 1/3 in both cases. Likewise the reasons to answer 1/2 are different in both experiments.

Anthropical Motte here is to claim that Beauties care for each other and invoke total utilitarianism. Being right on Tails means that two Beauties are right, while being right on Heads - that only one. Such attempt to construct the same scoring rule as with the classical version makes it completely clear that we are not talking about probability of coin being Heads or Tails anymore, but about the expected utility of the answer.

The Bailey is that you in particular are going to find yourself in the Tails world more likely if you were created during the experiment. Not that just on average most people created in such experiments are created when the coin landed tails. You in particular can guess that coin landed tails with 2/3 probability if you were part of the experiment due to the update on your existence. That the chance that there is another version of you in the different room is 2/3.

Final words on Anthropical Motte and Bailey

People do not usually hold both the Anthropical Motte and the Anthropical Bailey positions simultaneously. It's easy to see that if your scoring rule counted guesses every day/per every Beauty and it was 2/3 times likely that the coin landed Tails, then you should be not Thirder but Fifther.

coin_guess = []
for n in range(100000):
    days, coin = classic(heads_chance=0.33)
    for d in days:            
        coin_guess.append(coin == 'Heads')

coin_guess.count(True)/len(coin_guess) # 0.19632250462079567

Instead, people just assume that they are the same position, switching between them back and forth, often without noticing. This is understandable because usually they are! Usually if two thirds of your subjective experiences happen when the coin is Tails, Tails outcome of the coin toss are two times more likely. Usually if something is true on average for 2/3 of the population, it is 2/3 likely true for you as well. Sleeping Beauty cases are just specifically designed as adversarial attacks on this heuristic. Of course some people are confused.

Let's get rid of this confusion. Anthropical Motte and Bailey are two different things. And while the Motte is fine, the Bailey is responsible for, or at least, a serious symptom of much of absurdity of anthropic reasoning, which I'm going to resolve in the series of future posts.

The next post in the series is Conservation of Expected Evidence and Random Sampling in Anthropics.