Sapiens: How Shared Myths Change the World

I recently read Yuval Noah Harari’s book Sapiens and loved it. In additional to fascinating and disturbing details about the evolutionary history of Homo sapiens and a wonderful account of human history, he has a really interesting way of talking about the cognitive abilities that make humans distinct from other species. I’ll dive right into this latter topic in this post.

Imagine two people in a prisoner’s dilemma. To try to make it relevant to our ancestral environment, let’s say that they are strangers running into one another, and each see that the other has some resources. There are four possible outcomes. First, they could both cooperate and team up to catch some food that neither would be able to get on their own, and then share the food. Second, they could both defect, attacking each other and both walking away badly injured. And third and fourth, one could cooperate while the other defects, corresponding to one of them stabbing the other in the back and taking their resources. (Let’s suppose that each of the two are currently holding resources of more value than they could obtain by teaming up and hunting.)

Now, the problem is that on standard accounts of rational decision making, the decision that maximizes expected reward for each individual is to defect. That’s bad! The best outcome for everybody is that the two team up and share the loot, and neither walks away injured!

You might just respond “Well, who cares about what our theory of rational decision making says? Humans aren’t rational.” We’ll come back to this in a bit. But for now I’ll say that the problem is not just that our theory of rationality says that we should defect. It’s that this line of reasoning implies that cooperating is an unstable strategy. Imagine a society fully populated with cooperators. Now suppose an individual appears with a mutation that causes them to defect. This defector outperforms the cooperators, because they get to keep stabbing people in the back and stealing their loot and never have to worry about anybody doing the same to them. The result is then that the “gene for defecting” (speaking very metaphorically at this point; the behavior doesn’t necessarily have to be transmitted genetically) spreads like a virus through the population, eventually transforming our society of cooperators to a society of defectors. And everybody’s worse off.

One the other hand, imagine a society full of defectors. What if a cooperator is born into this society? Well, they pretty much right away get stabbed in the back and die out. So a society of defectors stays a society of defectors, and a society of cooperators degenerates into a society of defectors. The technical way of speaking about this is to say that in prisoner’s dilemmas, cooperation is not a Nash equilibrium – a strategy that is stable against mutations when universally adopted. The only Nash equilibrium is universal defection.

Okay, so this is all bad news. We have good game theoretic reasons to expect society to degenerate into a bunch of people stabbing each other in the back. But mysteriously, the record of history has humans coming together to form larger and larger cooperative institutions. What Yuval Noah Harari and many others argue is that the distinctively human force that saves us from these game theoretic traps and creates civilizations is the power of shared myths.

For instance, suppose that the two strangers happened to share a belief in a powerful all-knowing God that punishes defectors in the afterlife and rewards cooperators. Think about how this shifts the reasoning. Now each person thinks “Even if I successfully defect and loot this other person’s resources, I still will have hell to pay in the afterlife. It’s just not worth it to risk incurring God’s wrath! I’ll cooperate.” And thus we get a cooperative equilibrium!

Still you might object “Okay, but what if an atheist is born into this society of God-fearing cooperative people? They’ll begin defecting and successfully spread through the population, right? And then so much for your cooperative equilibrium.”

The superbly powerful thing about these shared myths is the way in which they can restructure society around them. So for instance, it would make sense for a society with the cooperator-punishing God myth to develop social norms around punishing defectors. The mythical punishment becomes an actual real-world punishment by the myth’s adherents. And this is enough to tilt the game-theoretic balance even for atheists.

The point being: The spreading of a powerful shared myth can shift the game theoretic structure of the world, altering the landscape of possible social structures. What’s more, such myths can increase the overall fitness of a society. And we need not rely on group selection arguments here; the presence of the shared myth increases the fitness of every individual.

A deeper point is that the specific way in which the landscape is altered depends on the details of the shared myth. So if we contrast the God myth above to a God that punishes defectors but also punishes mortals who punish defectors, we lose the stability property that we sought. The suggestion being: different ideas alter the game theoretic balance of the world in different ways, and sometimes subtle differences can be hugely important.

Another take-away from this simple example is that shared myths can become embodied within us, both in our behavior and in our physiology. Thus we come back to the “humans aren’t rational” point: The cooperator equilibrium becomes more stable if the God myth somehow becomes hardwired into our brains. These ideas take hold of us and shape us in their image.

Let’s go further into this. In our sophisticated secular society, it’s not too controversial to refer to the belief in all-good and all-knowing gods as a myth. But Yuval Noah Harari goes further. To him, the concept of the shared myth goes much deeper than just our ideas about the supernatural. In fact, most of our native way of viewing the world consists of a network of shared myths and stories that we tell one another.

After all, the universe is just physics. We’re atoms bumping into one another. There are no particles of fairness or human rights, no quantum fields for human meaning or karmic debts. These are all shared myths. Economic systems consist of mostly shared stories that we tell each other, stories about how much a dollar bill is worth and what the stock price of Amazon is. None of these things are really out there in the world. They are in our brains, and they are there for an important reason: they open up the possibility for societal structures that would otherwise be completely impossible. Imagine having a global trade network without the shared myth of the value of money. Or a group of millions of humans living packed together in a city that didn’t all on some level believe in the myths of human value and respect.

Just think about this for a minute. Humans have this remarkable ability to radically change our way of interacting with one another and our environments by just changing the stories that we tell one another. We are able to do this because of two features of our brains. First, we are extraordinarily creative. We can come up with ideas like money and God and law and democracy and whole-heartedly believe in them, to the point that we are willing to sacrifice our lives for them. Second, we are able to communicate these ideas to one another. This allows the ideas to spread and become shared myths. And most remarkably, all of these ideas (capitalism and communism, democracy and fascism) are running on essentially the same hardware! In Harari’s words:

While the behaviour patterns of archaic humans remained fixed for tens of thousands of years, Sapiens could transform their social structures, the nature of their interpersonal relations, their economic activities and a host of other behaviours within a decade or two. Consider a resident of Berlin, born in 1900 and living to the ripe age of one hundred. She spent her childhood in the Hohenzollern Empire of Wilhelm II; her adult years in the Weimar Republic, the Nazi Third Reich and Communist East Germany; and she died a citizen of a democratic and reunited Germany. She had managed to be a part of five very different sociopolitical systems, though her DNA remained exactly the same.

Anthropic reasoning in everyday life

Thought experiment from a past post:

A stranger comes up to you and offers to play the following game with you: “I will roll a pair of dice. If they land snake eyes (i.e. they both land 1), you give me one dollar. Otherwise, if they land anything else, I give you a dollar.”

Do you play this game?

[…]

Now imagine that the stranger is playing the game in the following way: First they find one person and offer to play the game with them. If the dice land snake eyes, then they collect a dollar and stop playing the game. Otherwise, they find ten new people and offer to play the game with them. Same as before: snake eyes, the stranger collects $1 from each and stops playing, otherwise he moves on to 100 new people. Et cetera forever.

When we include this additional information about the other games the stranger is playing, then the thought experiment becomes identical in form to the dice killer thought experiment. Thus updating on the anthropic information that you have been kidnapped gives a 90% chance of snake-eyes, which means you have a 90% chance of losing a dollar and only a 10% chance of gaining a dollar. Apparently you should now not take the offer!

This seems a little weird. Shouldn’t it be irrelevant if the game if being offered to other people? To an anthropic reasoner, the answer is a resounding no. It matters who else is, or might be, playing the game, because it gives us additional information about our place in the population of game-players.

Thus far this is nothing new. But now we take one more step: Just because you don’t know the spatiotemporal distribution of game offers doesn’t mean that you can ignore it!

So far the strange implications of anthropic reasoning have been mostly confined to bizarre thought experiments that don’t seem too relevant to the real world. But the implication of this line of reasoning is that anthropic calculations bleed out into ordinary scenarios. If there is some anthropically relevant information that would affect your probabilities, then you need to consider the probability that this information

In other words, if somebody comes up to you and makes you the offer described above, you can’t just calculate the expected value of the game and make your decision. Instead, you have to consider all possible distributions of game offers, calculate the probability of each, and average over the implied probabilities! This is no small order.

For instance, suppose that you have a 50% credence that the game is being offered only one time to one person: you. The other 50% is given to the “dice killer” scenario: that the game is offered in rounds to a group that decuples in size each round, and that this continues until the dice finally land snake-eyes. Presumably you then have to average over the expected value of playing the game for each scenario.

EV_1 = - \$1 \cdot \frac{35}{36} + \$1 \cdot \frac{1}{36} = \$ \frac{34}{36} \approx \$0.94 \\~\\ EV_2 = \$1 \cdot 0.1 + - \$1 \cdot 0.9 = - \$ 0.80 \\~\\ EV = 0.50 \cdot EV_1 + 0.50 \cdot EV_2 \approx \$ .07

In this case, the calculation wasn’t too bad. But that’s because it was highly idealized. In general, representing your knowledge of the possible distributions of games offered seems quite difficult. But the more crucial point is that it is apparently not enough to go about your daily life calculating the expected value of the decisions facing you. You have to also consider who else might be facing the same decisions, and how this influences your chances of winning.

Can anybody think of a real-life example where these considerations change the sign of the expected value calculation?

Pushing anti-anthropic intuitions

A stranger comes up to you and offers to play the following game with you: “I will roll a pair of dice. If they land snake eyes (i.e. they both land 1), you give me one dollar. Otherwise, if they land anything else, I give you a dollar.”

Do you play this game?

Here’s an intuitive response: Yes, of course you should! You have a 35/36 chance of gaining $1, and only a 1/36 chance of losing $1. You’d have to be quite risk averse to refuse those odds.

What if the stranger tells you that they are giving this same bet to many other people? Should that change your calculation?

Intuitively: No, of course not! It doesn’t matter what else the stranger is doing with other people.

What if they tell you that they’ve given this offer to people in the past, and might give the offer to others in the future? Should that change anything?

Once again, it seems intuitively not to matter. The offers given to others simply have nothing to do with you. What matters are your possible outcomes and the probabilities of each of these outcomes. And what other people are doing has nothing to do with either of these.

… Right?

Now imagine that the stranger is playing the game in the following way: First they find one person and offer to play the game with them. If the dice land snake eyes, then they collect a dollar and stop playing the game. Otherwise, they find ten new people and offer to play the game with them. Same as before: snake eyes, the stranger collects $1 from each and stops playing, otherwise he moves on to 100 new people. Et cetera forever.

We now ask the question: How does the average person given the offer do if they take the offer? Well, no matter how many rounds of offers the stranger gives, at least 90% of people end up in his last round. That means that at least 90% of people end up giving over $1 and at most 10% gain $1. This is clearly net negative for those that hand over money!

Think about it this way: Imagine a population of individuals who all take the offer, and compare them to a population that all reject the offer. Which population does better on average?

For the population who takes the offer, the average person loses money. An upper bound on how much they lose is 10% ($1) + 90% (-$1) = -$.80. For the population that reject the offer, nobody gains money or loses It either: the average case is exactly $0. $0 is better than -$.80, so the strategy of rejecting the offer is better, on average!

This thought experiment is very closely related to the dice killer thought experiment. I think of it as a variant that pushes our anti-anthropic-reasoning intuitions. It just seems really wrong to me that if somebody comes up to you and offers you this deal that has a 35/36 chance of paying out you should reject it. The details of who else is being offered the deal seem totally irrelevant.

But of course, all of the previous arguments I’ve made for anthropic reasoning apply here as well. And it is just true that the average person that rejects the offer does better than the average person that accepts it. Perhaps this is just another bullet that we have to bite in our attempt to formalize rationality!

An expected value puzzle

Consider the following game setup:

Each round of the game starts with you putting in all of your money. If you currently have $10, then you must put in all of it to play. Now a coin is flipped. If it lands heads, you get back 10 times what you put in ($100). If not, then you lose it all. You can keep playing this game until you have no more money.

What does a perfectly rational expected value reasoner do?

Supposing that this reasoner’s sole goal is to maximize the quantity of money that they own, then the expected value for putting in the money is always greater than 0. If you put in $X, then you stand a 50% chance of getting $10X back and a 50% chance of losing $X. Thus, your expected value is 5X – X/2 = 9X/2.

This means that the expected value reasoner that wants to maximize their winnings would keep putting in their money until, eventually, they lose it all.

What’s wrong with this line of reasoning (if anything)? Does it serve as a reductio ad absurdum of expected value reasoning?

The Anthropic Dice Killer

Today we discuss anthropic reasoning.

The Problem

Imagine the following scenario:

A mad killer has locked you in a room. You are trapped and alone, with only your knowledge of your situation to help you out.

One piece of information that you have is that you are aware of the maniacal schemes of your captor. His plans began by capturing one random person. He then rolled a pair of dice to determine their fate. If the dice landed snake eyes (both 1), then the captive would be killed. If not, then they would be let free.

But if they are let free, the killer will search for new victims, and this time bring back ten new people and lock them alone in rooms. He will then determine their fate just as before, with a pair of dice. Snake eyes means they die, otherwise they will be let free and he will search for new victims.

His murder spree will continue until the first time he rolls snake eyes. Then he will kill the group that he currently has imprisoned and retire from the serial-killer life.

Now. You become aware of a risky way out of the room you are locked in and to freedom. The chances of surviving this escape route are only 50%. Your choices are thus either (1) to traverse the escape route with a 50% chance of survival or (2) to just wait for the killer to roll his dice, and hope that it doesn’t land snake eyes.

What should you do?

 

 

 

(Think about it before reading on)

 

 

 

A plausible-sounding answer

Your chance of dying if you stay and wait is just the chance that the dice lands snake eyes. The probability of snake eyes is just 1/36 (1/6 for each dice landing 1).

So your chance of death is only 1/36 (≈ 3%) if you wait, and it’s 50% if you try to run for it. Clearly, you are better off waiting!

But…

You guessed it, things aren’t that easy. You have extra information about your situation besides just how the dice works, and you should use it. In particular, the killing pattern of your captor turns out to be very useful information.

Ask the following question: Out of all of the people that have been captured or will be captured at some point by this madman, how many of them will end up dying? This is just the very last group, which, incidentally, is the largest group.

Consider: if the dice land snake eyes the first time they are rolled, then only one person is ever captured, and this person dies. So the fraction of those captured that die is 100%.

If they lands snake eyes the second time they are rolled, then 11 people total are captured, 10 of whom die. So the fraction of those captured that die is 10/11, or ≈ 91%.

If it’s the third time, then 111 people total are captured, 100 of whom die. Now the fraction is just over 90%.

In general, no matter how many times the dice rolls before landing snake eyes, it always ends up that over 90% of those captured end up being in the last round, and thus end up dying.

So! This looks like bad news for you… you’ve been captured, and over 90% of those that are captured always die. Thus, your chance of death is guaranteed to be greater than 90%.

The escape route with a 50% survival chance is looking nicer now, right?

Wtf is this kind of reasoning??

What we just did is called anthropic reasoning. Anthropic reasoning really just means updating on all of the information available to you, including indexical information (information about your existence, age, location, and so on). In this case, the initial argument neglected the very crucial information that you are one of the people that were captured by the killer. When updating on this information, we get an answer that is very very different from what we started with. And in this life-or-death scenario, this is an important difference!

You might still feel hesitant about the answer we got. After all, if you expect a 90% chance of death, this means that you expect a 90% chance for the dice to land snake eyes. But it’s not that you think the dice are biased or anything… Isn’t this just blatantly contradictory?

This is a convincing-sounding rebuttal, but it’s subtly wrong. The key point is that even though the dice are fair, there is a selection bias in the results you are seeing. This selection bias amounts to the fact that when the dice inevitably lands snake-eyes, there are more people around to see it. The fact that you are more likely than 1/36 to see snake-eyes is kind of like the fact that if you are given the ticket of a random concert-goer, you have a higher chance of ending seeing a really popular band than if you just looked at the current proportion of shows performed by really popular bands.

It’s kind of like the fact that in your life you will spend more time waiting in long lines than short lines, and that on average your friends have more friends than you. This all seems counterintuitive and wrong until you think closely about the selection biases involved.

Anyway, I want to impress upon you that 90% really is the right answer, so I’ll throw some math at you. Let’s calculate in full detail what fraction of the group ends up surviving on average.

Screen Shot 2018-08-02 at 1.16.15 AM

By the way, the discrepancy between the baseline chance of death (1/36) and the anthropic chance of death (90%) can be made as large as you like by manipulating the starting problem. Suppose that instead of 1/36, the chance of the group dying was 1/100, and instead of the group multiplying by 10 in size each round, it grew by a factor of 100. Then the baseline chance of death would be 1%, and the anthropic probability would be 99%.

We can find the general formula for any such scenario:

Screen Shot 2018-08-02 at 4.54.30 AM.png

IF ANYBODY CAN SOLVE THIS, PLEASE TELL ME! I’ve been trying for too long now and would really like an analytic general solution. 🙂

There is a lot more to be said about this thought experiment, but I’ll leave it there for now. In the next post, I’ll present a slight variant on this thought experiment that appears to give us a way to get direct Bayesian evidence for different theories of consciousness! Stay tuned.

What do I find conceptually puzzling?

There are lots of things that I don’t know, like, say, what the birth rate in Sweden is or what the effect of poverty on IQ is. There are also lots of things that I find really confusing and hard to understand, like quantum field theory and monetary policy. There’s also a special category of things that I find conceptually puzzling. These things aren’t difficult to grasp because the facts about them are difficult to understand or require learning complicated jargon. Instead, they’re difficult to grasp because I suspect that I’m confused about the concepts in use.

This is a much deeper level of confusion. It can’t be adjudicated by just reading lots of facts about the subject matter. It requires philosophical reflection on the nature of these concepts, which can sometimes leave me totally confused about everything and grasping for the solid ground of mere factual ignorance.

As such, it feels like a big deal when something I’ve been conceptually puzzled about becomes clear. I want to compile a list for future reference of things that I’m currently conceptually puzzled about and things that I’ve become un-puzzled about. (This is not a complete list, but I believe it touches on the major themes.)

Things I’m conceptually puzzled about

What is the relationship between consciousness and physics?

I’ve written about this here.

Essentially, at this point every available viewpoint on consciousness seems wrong to me.

Eliminativism amounts to a denial of pretty much the only thing that we can be sure can’t be denied – that we are having conscious experiences. Physicalism entails the claim that facts about conscious experience can be derived from laws of physics, which is wrong as a matter of logic.

Dualism entails that the laws of physics by themselves cannot account for the behavior of the matter in our brains, which is wrong. And epiphenomenalism entails that our beliefs about our own conscious experience are almost certainly wrong, and are no better representations of our actual conscious experiences than random chance.

How do we make sense of decision theory if we deny libertarian free will?

Written about this here and here.

Decision theory is ultimately about finding the decision D that maximizes expected utility EU(D). But to do this calculation, we have to decide what the set of possible decisions we are searching is.

EU confusion

Make this set too large, and you end up getting fantastical and impossible results (like that the optimal decision is to snap your fingers and make the world into a utopia). Make it too small, and you end up getting underwhelming results (in the extreme case, you just get that the optimal decision is to do exactly what you are going to do, since this is the only thing you can do in a strictly deterministic world).

We want to find a nice middle ground between these two – a boundary where we can say “inside here the things that are actually possible for us to do, and outside are those that are not.” But any principled distinction between what’s in the set and what’s not must be based on some conception of some actions being “truly possible” to us, and others being truly impossible. I don’t know how to make this distinction in the absence of a robust conception of libertarian free will.

Are there objectively right choices of priors?

I’ve written about this here.

If you say no, then there are no objectively right answers to questions like “What should I believe given the evidence I have?” And if you say yes, then you have to deal with thought experiments like the cube problem, where any choice of priors looks arbitrary and unjustifiable.

(If you are going to be handed a cube, and all you know is that it has a volume less than 1 cm3, then setting maximum entropy priors over volumes gives different answers than setting maximum entropy priors over side areas or side lengths. This means that what qualifies as “maximally uncertain” depends on whether we frame our reasoning in terms of side length, areas, or cube volume. Other approaches besides MaxEnt have similar problems of concept dependence.)

How should we deal with infinities in decision theory?

I wrote about this here, here, here, and here.

The basic problem is that expected utility theory does great at delivering reasonable answers when the rewards are finite, but becomes wacky when the rewards become infinite. There are a huge amount of examples of this. For instance, in the St. Petersburg paradox, you are given the option to play a game with an infinite expected payout, suggesting that you should buy in to the game no matter how high the cost. You end up making obviously irrational choices, such as spending $1,000,000 on the hope that a fair coin will land heads 20 times in a row. Variants of this involve the inability of EU theory to distinguish between obviously better and worse bets that have infinite expected value.

And Pascal’s mugging is an even worse case. Roughly speaking, a person comes up to you and threatens you with infinite torture if you don’t submit to them and give them 20 dollars. Now, the probability that this threat is credible is surely tiny. But it is non-zero! (as long as you don’t think it is literally logically impossible for this threat to come true)

An infinite penalty times a finite probability is still an infinite expected penalty. So we stand to gain an infinite expected utility by just handing over the 20 dollars. This seems ridiculous, but I don’t know any reasonable formalization of decision theory that allows me to refute it.

Is causality fundamental?

Causality has been nicely formalized by Pearl’s probabilistic graphical models. This is a simple extension of probability theory, out of which naturally falls causality and counterfactuals.

One can use this framework to represent the states of fundamental particles and how they change over time and interact with one another. What I’m confused about is that in some ways of looking at it, the causal relations appear to be useful but un-fundamental constructs for the sake of easing calculations. In other ways of looking at it, causal relations are necessarily built into the structure of the world, and we can go out and empirically discover them. I don’t know which is right. (Sorry for the vagueness in this one – it’s confusing enough to me that I have trouble even precisely phrasing the dilemma).

How should we deal with the apparent dependence of inductive reasoning upon our choices of concepts?

I’ve written about this here. Beyond just the problem of concept-dependence in our choices of priors, there’s also the problem presented by the grue/bleen thought experiment.

This thought experiment proposes two new concepts: grue (= the set of things that are either green before 2100 or blue after 2100) and bleen (the inverse of grue). It then shows that if we reasoned in terms of grue and bleen, standard induction would have us concluding that all emeralds will suddenly turn blue after 2100. (We repeatedly observed them being grue before 2100, so we should conclude that they will be grue after 2100.)

In other words, choose the wrong concepts and induction breaks down. This is really disturbing – choices of concepts should be merely pragmatic matters! They shouldn’t function as fatal epistemic handicaps. And given that they appear to, we need to develop some criterion we can use to determine what concepts are good and what concepts are bad.

The trouble with this is that the only proposals I’ve seen for such a criterion reference the idea of concepts that “carve reality at its joints”; in other words, the world is composed of green and blue things, not grue and bleen things, so we should use the former rather than the latter. But this relies on the outcome of our inductive process to draw conclusions about the starting step on which this outcome depends!

I don’t know how to cash out “good choices of concepts” without ultimately reasoning circularly. I also don’t even know how to make sense of the idea of concepts being better or worse for more than merely pragmatic reasons.

How should we reason about self defeating beliefs?

The classic self-defeating belief is “This statement is a lie.” If you believe it, then you are compelled to disbelieve it, eliminating the need to believe it in the first place. Broadly speaking, self-defeating beliefs are those that undermine the justifications for belief in them.

Here’s an example that might actually apply in the real world: Black holes glow. The process of emission is known as Hawking radiation. In principle, any configuration of particles with a mass less than the black hole can be emitted from it. Larger configurations are less likely to be emitted, but even configurations such as a human brain have a non-zero probability of being emitted. Henceforth, we will call such configurations black hole brains.

Now, imagine discovering some cosmological evidence that the era in which life can naturally arise on planets circling stars is finite, and that after this era there will be an infinite stretch of time during which all that exists are black holes and their radiation. In such a universe, the expected number of black hole brains produced is infinite (a tiny finite probability multiplied by an infinite stretch of time), while the expected number of “ordinary” brains produced is finite (assuming a finite spatial extent as well).

What this means is that discovering this cosmological evidence should give you an extremely strong boost in credence that you are a black hole brain. (Simply because most brains in your exact situation are black hole brains.) But most black hole brains have completely unreliable beliefs about their environment! They are produced by a stochastic process which cares nothing for producing brains with reliable beliefs. So if you believe that you are a black hole brain, then you should suddenly doubt all of your experiences and beliefs. In particular, you have no reason to think that the cosmological evidence you received was veridical at all!

I don’t know how to deal with this. It seems perfectly possible to find evidence for a scenario that suggests that we are black hole brains (I’d say that we have already found such evidence, multiple times). But then it seems we have no way to rationally respond to this evidence! In fact, if we do a naive application of Bayes’ theorem here, we find that the probability of receiving any evidence in support of black hole brains to be 0!

So we have a few options. First, we could rule out any possible skeptical scenarios like black hole brains, as well as anything that could provide any amount of evidence for them (no matter how tiny). Or we could accept the possibility of such scenarios but face paralysis upon actually encountering evidence for them! Both of these seem clearly wrong, but I don’t know what else to do.

How should we reason about our own existence and indexical statements in general?

This is called anthropic reasoning. I haven’t written about it on this blog, but expect future posts on it.

A thought experiment: imagine a murderous psychopath who has decided to go on an unusual rampage. He will start by abducting one random person. He rolls a pair of dice, and kills the person if they land snake eyes (1, 1). If not, he lets them free and hunts down ten new people. Once again, he rolls his pair of die. If he gets snake eyes he kills all ten. Otherwise he frees them and kidnaps 100 new people. On and on until he eventually gets snake eyes, at which point his murder spree ends.

Now, you wake up and find that you have been abducted. You don’t know how many others have been abducted alongside you. The murderer is about to roll the dice. What is your chance of survival?

Your first thought might be that your chance of death is just the chance of both dice landing 1: 1/36. But think instead about the proportion of all people that are ever abducted by him that end up dying. This value ends up being roughly 90%! So once you condition upon the information that you have been captured, you end up being much more worried about your survival chance.

But at the same time, it seems really wrong to be watching the two dice tumble and internally thinking that there is a 90% chance that they land snake eyes. It’s as if you’re imagining that there’s some weird anthropic “force” pushing the dice towards snake eyes. There’s way more to say about this, but I’ll leave it for future posts.

Things I’ve become un-puzzled about

Newcomb’s problem – one box or two box?

To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.

– Nozick, 1969

I’ve spent months and months being hopelessly puzzled about Newcomb’s problem. I now am convinced that there’s an unambiguous right answer, which is to take the one box. I wrote up a dialogue here explaining the justification for this choice.

In a few words, you should one-box because one-boxing makes it nearly certain that the simulation of you run by the predictor also one-boxed, thus making it nearly certain that you will get 1 million dollars. The dependence between your action and the simulation is not an ordinary causal dependence, nor even a spurious correlation – it is a logical dependence arising from the shared input-output structure. It is the same type of dependence that exists in the clone prisoner dilemma, where you can defect or cooperate with an individual you are assured is identical to you in every single way. When you take into account this logical dependence (also called subjunctive dependence), the answer is unambiguous: one-boxing is the way to go.

Summing up:

Things I remain conceptually confused about:

  • Consciousness
  • Decision theory & free will
  • Objective priors
  • Infinities in decision theory
  • Fundamentality of causality
  • Dependence of induction on concept choice
  • Self-defeating beliefs
  • Anthropic reasoning

The Monty Hall non-paradox

I recently showed the famous Monty Hall problem to a friend. This friend solved the problem right away, and we realized quickly that the standard presentation of the problem is highly misleading.

Here’s the setup as it was originally described in the magazine column that made it famous:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

I encourage you to think through this problem for yourself and come to an answer. Will provide some blank space so that you don’t accidentally read ahead.

 

 

 

 

 

 

Now, the writer of the column was Marilyn vos Savant, famous for having an impossible IQ of 228 according to an interpretation of a test that violated “almost every rule imaginable concerning the meaning of IQs” (psychologist Alan Kaufman). In her response to the problem, she declared that switching gives you a 2/3 chance of winning the car, as opposed to a 1/3 chance for staying. She argued by analogy:

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

Notice that this answer contains a crucial detail that is not contained in the statement of the problem! Namely, the answer adds the stipulation that the host “knows what’s behind the doors and will always avoid the one with the prize.”

The original statement of the problem in no way implies this general statement about the host’s behavior. All you are justified to assume in an initial reading of the problem are the observational facts that (1) the host happened to open door No. 3, and (2) this door happened to contain a goat.

When nearly a thousand PhDs wrote in to the magazine explaining that her answer was wrong, she gave further arguments that failed to reference the crucial point; that her answer was only true given additional unstated assumptions.

My original answer is correct. But first, let me explain why your answer is wrong. The winning odds of 1/3 on the first choice can’t go up to 1/2 just because the host opens a losing door. To illustrate this, let’s say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you’ve chosen, we’ve learned nothing to allow us to revise the odds on the shell under your finger.

Notice that this argument is literally just a restatement of the original problem. If one didn’t buy the conclusion initially, restating it in terms of peas and shells is unlikely to do the trick!

This problem was made even more famous by this scene in the movie “21”, in which the protagonist demonstrates his brilliance by coming to the same conclusion as vos Savant. While the problem is stated slightly better in this scene, enough ambiguity still exists that the proper response should be that the problem is underspecified, or perhaps a set of different answers for different sets of auxiliary assumptions.

The wiki page on this ‘paradox’ describes it as a veridical paradox, “because the correct choice (that one should switch doors) is so counterintuitive it can seem absurd, but is nevertheless demonstrably true.”

Later on the page, we see the following:

In her book The Power of Logical Thinking, vos Savant (1996, p. 15) quotes cognitive psychologist Massimo Piattelli-Palmarini as saying that “no other statistical puzzle comes so close to fooling all the people all the time,” and “even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer.”

There’s something to be said about adequacy reasoning here; when thousands of PhDs and some of the most brilliant mathematicians in the world are making the same point, perhaps we are too quick to write it off as “Wow, look at the strength of this cognitive bias! Thank goodness I’m bright enough to see past it.”

In fact, the source of all of the confusion is fairly easy to understand, and I can demonstrate it in a few lines.

Solution to the problem as presented

Initially, all three doors are equally likely to contain the car.
So Pr(1) = Pr(2) = Pr(3) = ⅓

We are interested in how these probabilities update upon the observation that 3 does not contain the car.
Pr(1 | ~3) = Pr(1)・Pr(~3 | 1) / Pr(~3)
= (⅓ ・1) / ⅔ = ½

By the same argument,
Pr(2 | ~3) = ½

Voila. There’s the simple solution to the problem as it is presented, with no additional presumptions about the host’s behavior. Accepting this argument requires only accepting three premises:

(1) Initially all doors are equally likely to be hiding the car.

(2) Bayes’ rule.

(3) There is only one car.

(3) implies that Pr(the car is not behind a door | the car is behind a different door) = 100%, which we use when we replace Pr(~3 | 1) with 1.

The answer we get is perfectly obvious; in the end all you know is that the car is either in door 1 or door 2, and that you picked door 1 initially. Since which door you initially picked has nothing to do with which door the car was behind, and the host’s decision gives you no information favoring door 1 over door 2, the probabilities should be evenly split between the two.

It is also the answer that all the PhDs gave.

Now, why does taking into account the host’s decision process change things? Simply because the host’s decision is now contingent on your decision, as well as the actual location of the car. Given that you initially opened door 1, the host is guaranteed to not open door 1 for you, and is also guaranteed to not open up a door hiding the car.

Solution with specified host behavior

Initially, all three doors are equally likely to contain the car.
So Pr(1) = Pr(2) = Pr(3) = ⅓

We update these probabilities upon the observation that 3 does not contain the car, using the likelihood formulation of Bayes’ rule.

Pr(1 | open 3) / Pr(2 | open 3)
= Pr(1) / Pr(2)・Pr(open 3 | 1) / Pr(open 3 | 2)
= ⅓ / ⅓・½ / 1 = ½

So Pr(1 | open 3) = ⅓ and Pr(2 | open 3) = ⅔

Pr(open 3 | 2) = 1, because the host has no choice of which door to open if you have selected door 1 and the car is behind door 2.

Pr(open 3 | 1) = ½, because the host has a choice of either opening 2 or 3.

In fact, it’s worth pointing out that this requires another behavioral assumption about the host that is nowhere stated in the original post, or Savant’s solution. This is that if there is a choice about which of two doors to open, the host will pick randomly.

This assumption is again not obviously correct from the outset; perhaps the host chooses the larger of the two door numbers in such cases, or the one closer to themselves, or the one or the smaller number with 25% probability. There are an infinity of possible strategies the host could be using, and this particular strategy must be explicitly stipulated to get the answer that Wiki proclaims to be correct.

It’s also worth pointing out that once these additional assumptions are made explicit, the ⅓ answer is fairly obvious and not much of a paradox. If you know that the host is guaranteed to choose a door with a goat behind it, and not one with a car, then of course their decision about which door to open gives you information. It gives you information because it would have been less likely in the world where the car was under door 1 than in the world where the car was under door 2.

In terms of causal diagrams, the second formulation of the Monty Hall problem makes your initial choice of door and the location of the car dependent upon one another. There is a path of causal dependency that goes forwards from your decision to the host’s decision, which is conditioned upon, and then backward from the host’s decision to which door the car is behind.

Any unintuitiveness in this version of the Monty Hall problem is ultimately due to the unintuitiveness of the effects of conditioning on a common effect of two variables.

Monty Hall Causal

In summary, there is no paradox behind the Monty Hall problem, because there is no single Monty Hall problem. There are two different problems, each containing different assumptions, and each with different answers. The answers to each problem are fairly clear after a little thought, and the only appearance of a paradox comes from apparent disagreements between individuals that are actually just talking about different problems. There is no great surprise when ambiguous wording turns out multiple plausible solutions, it’s just surprising that so many people see something deeper than mere ambiguity here.