Making sense of improbability

Imagine that you take a coin that you believe to be fair and flip it 20 times. Each time it lands heads. You say to your friend: “Wow, what a crazy coincidence! There was a 1 in 220 chance of this outcome. That’s less than one in a million! Super surprising.”

Your friend replies: “I don’t understand. What’s so crazy about the result you got? Any other possible outcome (say, HHTHTTTHTHHHTHTTHHHH) had an equal probability as getting all heads. So what’s so surprising?”

Responding to this is a little tricky. After all, it is the case that for a fair coin, the probability of 20 heads = the probability of HHTHTTTHTHHHTHTTHHHH = roughly one in a million.

Simpler Example_ Five Tosses.png

So in some sense your friend is right that there’s something unusual about saying that one of these outcomes is more surprising than another.

You might answer by saying “Well, let’s parse up the possible outcomes by the number of heads and tails. The outcome I got had 20 heads and 0 tails. Your example outcome had 12 heads and 8 tails. There are many many ways of getting 12 heads and 8 tails than of getting 20 heads and 0 tails, right? And there’s only one way of getting all 20 heads. So that’s why it’s so surprising.”

Probability vs. Number of heads (1).png

Your friend replies: “But hold on, now you’re just throwing out information. Sure my example outcome had 12 heads and 8 tails. But while there’s many ways of getting that number of heads and tails, there’s only exactly one way of getting the result I named! You’re only saying that your outcome is less likely because you’ve glossed over the details of my outcome that make it equally unlikely: the order of heads and tails!”

I think this is a pretty powerful response. What we want is a way to say that HHHHHHHHHHHHHHHHHHHH is surprising while HHTHTTTHTHHHTHTTHHHH is not, not that 20 heads is surprising while 12 heads and 8 tails is unsurprising. But it’s not immediately clear how we can say this.

Consider the information theoretic formalization of surprise, in which the surprisingness of an event E is proportional to the negative log of the probability of that event: Sur(E) = -log(P(E)). There are some nice reasons for this being a good definition of surprise, and it tells us that two equiprobable events should be equally surprising. If E is the event of observing all heads and E’ is the event of observing the sequence HHTHTTTHTHHHTHTTHHHH, then P(E) = P(E’) = 1/220. Correspondingly, Sur(E) = Sur(E’). So according to one reasonable formalization of what we mean by surprisingness, the two sequences of coin tosses are equally surprising. And yet, we want to say that there is something more epistemically significant about the first than the second.

(By the way, observing 20 heads is roughly 6.7 times more surprising than observing 12 heads and 8 tails, according to the above definition. We can plot the surprise curve to see how maximum surprise occurs at the two ends of the distribution, at which point it is 20 bits.)

Surprise vs. number of heads (1).png

So there is our puzzle: in what sense does it make sense to say that observing 20 heads in a row is more surprising than observing the sequence HHTHTTTHTHHHTHTTHHHH? We certainly have strong intuitions that this is true, but do these intuitions make sense? How can we ground the intuitive implausibility of getting 20 heads? In this post I’ll try to point towards a solution to this puzzle.

Okay, so I want to start out by categorizing three different perspectives on the observed sequence of coin tosses. These correspond to (1) looking at just the outcome, (2) looking at the way in which the observation affects the rest of your beliefs, and (3) looking at how the observation affects your expectation of future observations. In probability terms, these correspond to the P(E), P(T| T) and P(E’ | E).

Looking at things through the first perspective, all outcomes are equiprobable, so there is nothing more epistemically significant about one than the other.

But considering the second way of thinking about things, there can be big differences in the significance of two equally probable observations. For instance, suppose that our set of theories under consideration are just the set of all possible biases of the coin, and our credences are initially peaked at .5 (an unbiased coin). Observing HHTHTTTHTHHHTHTTHHHH does little to change our prior. It shifts a little bit in the direction of a bias towards heads, but not significantly. On the other hand, observing all heads should have a massive effect on your beliefs, skewing them exponentially in the direction of extreme heads biases.

Importantly, since we’re looking at beliefs about coin bias, our distributions are now insensitive to any details about the coin flip beyond the number of heads and tails! As far as our beliefs about the coin bias go, finding only the first 8 to be tails looks identical to finding the last 8 to be tails. We’re not throwing out the information about the particular pattern of heads and tails, it’s just become irrelevant for the purposes of consideration of the possible biases of the coin.

Visualizing change in beliefs about coin bias.png

If we want to give a single value to quantify the difference in epistemic states resulting from the two observations, we can try looking at features of these distributions. For instance, we could look at the change in entropy of our distribution if we see E and compare it to the change in entropy upon seeing E’. This gives us a measure of how different observations might affect our uncertainty levels. (In our example, observing HHTHTTTHTHHHTHTTHHHH decreases uncertainty by about 0.8 bits, while observing all heads decreases uncertainty by 1.4 bits.) We could also compare the means of the posterior distributions after each observation, and see which is shifted most from the mean of the prior distribution. (In this case, our two means are 0.57 and 0.91).

Now, this was all looking at things through what I called perspective #2 above: how observations affect beliefs. Sometimes a more concrete way to understand the effect of intuitively implausible events is to look at how they affect specific predictions about future events. This is the approach of perspective #3. Sticking with our coin, we ask not about the bias of the coin, but about how we expect it to land on the next flip. To assess this, we look at the posterior predictive distributions for each posterior:

Posterior Predictive Distributions.png

It shouldn’t be too surprising that observing all heads makes you more confident that the next coin will land heads than observing HHTHTTTHTHHHTHTTHHHH. But looking at this graph gives a precise answer to how much more confident you should be. And it’s somewhat easier to think about than the entire distribution over coin biases.

I’ll leave you with an example puzzle that relates to anthropic reasoning.

Say that one day you win the lottery. Yay! Super surprising! What an improbable event! But now compare this to the event that some stranger Bob Smith wins the lottery. This doesn’t seem so surprising. But supposing that Bob Smith buys lottery tickets at the same rate as you, the probability that you win is identical to the probability that Bob Smith wins. So… why is it any more surprising when you win?

This seems like a weird question. Then again, so did the coin-flipping question we started with. We want to respond with something like “I’m not saying that it’s improbable that some random person wins the lottery. I’m interested in the probability of me winning the lottery. And if we parse up the outcomes as that either I win the lottery or that somebody else wins the lottery, then clearly it’s much more improbable that I win than that somebody else wins.”

But this is exactly parallel to the earlier “I’m not interested in the precise sequence of coin flips, I’m just interested in the number of heads versus tails.” And the response to it is identical in form: If Bob Smith, a particular individual whose existence you are aware of, wins the lottery and you know it, then it’s cheating to throw away those details and just say “Somebody other than me won the lottery.” When you update your beliefs, you should take into account all of your evidence.

Does the framework I presented here help at all with this case?

A simple probability puzzle

In front of you is an urn containing some unknown quantity of balls. These balls are labeled 1, 2, 3, etc. They’ve been jumbled about so as to be in no particular order within the urn. You initially consider it equally likely that the urn contains 1 ball as that it contains 2 balls, 3 balls, and so on, up to 100 balls, which is the maximum capacity of the urn.

Now you reach in to draw out a ball and read the number on it: 34. What is the most likely theory for how many balls the urn contains?

 

 

(…)

 

(Think of an answer before reading on.)

 

(…)

 

 

The answer turns out to be 34!

Hopefully this is a little unintuitive. Specifically, what seems wrong is that you draw out a ball and then conclude that this is the ball with the largest value on it. Shouldn’t extreme results be unlikely? But remember, the balls were randomly jumbled about inside the urn. So whether or not the number on the ball you drew is at the beginning, middle, or end of the set of numbers is pretty much irrelevant.

What is relevant is the likelihood: Pr(There are N balls | I drew a ball numbered 34). And the value of this is simply 1/N.

In general, comparing the theory that there are N balls to the theory that there are M balls, we look at the likelihood ratio: Pr(There are N balls | I drew a ball numbered 34) / Pr(There are M balls | I drew a ball numbered 34). This is simply M/N.

Thus we see that our prior odds get updated by a factor that favors smaller values of N, as long as N ≥ 34. The likelihood is zero up to N = 33, maxes at 34, and then decreases steadily after it as N goes to infinity. Since our prior was evenly spread out between N = 1 and 100 and zero everywhere else, our posterior will be peaked at 34 and decline until 100, after which it will drop to zero.

One way to make this result seem more intuitive is to realize that while strictly speaking the most probable number of balls in the urn is 34, it’s not that much more probable than 35 or 36. The actual probability of 34 is still quite small, it just happens to be a little bit more probable than its larger neighbors. And indeed, for larger values of the maximum capacity of the urn, the relative difference between the posterior probability of 34 and that of 35 decreases.

The end goal of epistemology

What are we trying to do in epistemology?

Here’s a candidate for an answer: The goal of epistemology is to formalize rational reasoning.

This is pretty good. But I don’t think it’s quite enough. I want to distinguish between three possible end goals of epistemology.

  1. The goal of epistemology is to formalize how an ideal agent with infinite computational power should reason.
  2. The goal of epistemology is to formalize how an agent with limited computational power should reason.
  3. The goal of epistemology is to formalize how a rational human being should reason.

We can understand the second task as asking something like “How should I design a general artificial intelligence to most efficiently and accurately model the world?” Since any general AI is going to be implemented in a particular bit of hardware, the answer to this question will depend on details like the memory and processing power of the hardware.

For the first task, we don’t need to worry about these details. Imagine that you’re a software engineer with access to an oracle that instantly computes any function you hand it. You want to build a program that takes in input from its environment and, with the help of this oracle, computes a model of its environment. Hardware constraints are irrelevant, you are just interested in getting the maximum epistemic juice out of your sensory inputs as logically possible.

The third task is probably the hardest. It is the most constrained of the three tasks; to accomplish it we need to first of all have a descriptively accurate model of the types of epistemic states that human beings have (e.g. belief and disbelief, comparative confidence, credences). Then we want to place norms on these states that are able to accommodate our cognitive quirks (for example, that don’t call things like memory loss or inability to instantly see all the logical consequences of a set of axioms irrational).

But both of these goals are on a spectrum. We aren’t interested in fully describing our epistemic states, because then there’s no space for placing non-trivial norms on them. And we aren’t interested in fully accommodating our cognitive quirks, because some of these quirks are irrational! It seems really hard to come up with precise and non-arbitrary answers to how descriptive we want to be and how many quirks we want to accommodate.

Now, in my experience, this third task is the one that most philosophers are working on. The second seems to be favored by statisticians and machine learning researchers. The first is favored by LessWrong rationalist-types.

For instance, rationalists tend to like Solomonoff induction as a gold standard for rational reasoning. But Solomonoff induction is literally uncomputable, immediately disqualifying it as a solution to tasks (2) and (3). The only sense in which Solomonoff induction is a candidate for the perfect theory of rationality is the sense of task (1). While it’s certainly not the case that Solomonoff induction is the perfect theory of rationality for a human or a general AI, it might be the right algorithm for an ideal agent with infinite computational power.

I think that disambiguating these three different potential goals of epistemology allows us to sidestep confusion resulting from evaluating a solution to one goal according to the standards of another. Let’s see this by purposefully glossing over the differences between the end goals.

We start with pure Bayesianism, which I’ll take to be the claim that rationality is about having credences that align with the probability calculus and updating them by conditionalization. (Let’s ignore the problem of priors for the moment.)

In favor of this theory: it works really well, in principle! Bayesianism has a lot of really nice properties like convergence to truth and maximizing relative entropy in updating on evidence (which is sort of like squeezing out all the information out of your evidence).

In opposition: the problem of logical omniscience. A Bayesian expects that all of the logical consequences of a set of axioms should be immediately obvious to a rational agent, and therefore that all credences of the form P(logical consequence of axioms | axioms) should be 100%. But now I ask you: is 19,973 a prime number? Presumably you understand natural numbers, including how to multiply and divide them and what prime numbers are. But it seems wrong to declare that the inability to conclude that 19,973 is prime from this basic level is knowledge is irrational.

This is an appeal to task (2). We want to say that there’s a difference between rationality and computational power. An agent with infinite computational power can be irrational if it is running poor software. And an agent with finite computational power can be perfectly rational, in that it makes effective use of these limited computational resources.

What this suggests is that we want a theory of rationality that is indexed by the computational capacities of the agent in question. What’s rational for one agent might not be rational for another. Bayesianism by itself isn’t nuanced enough to do this; two agents with the same evidence (and the same priors) should always end up at the same final credences. What we want is a framework in which two agents with the same evidence, priors, and computational capacity have the same beliefs.

It might be helpful to turn to computational complexity theory for insights. For instance, maybe we want a principle that says that a polynomial-powered agent is not rationally expected to solve NP problems. But the exact details of how such a theory would turn out are not obvious to me. Nor is it obvious that there even is a single non-arbitrary choice.

Regardless, let’s imagine for the moment that we have in hand the perfect theory of rationality for task (2). This theory should reduce to (1) as a special case when the agent in question has infinite computational powers. And if we treat human beings very abstractly as having some well-defined quantity of memory and processing power, then the theory also places norms on human reasoning. But in doing this, we open a new possible set of objections. Might this theory condemn as irrational some cognitive features of humans that we want to label as arational (neither rational nor irrational)?

For instance, let’s suppose that this theory involves something like updating by conditionalization. Notice that in this process, your credence in the evidence being conditioned on goes to 100%. Perhaps we want to say that the only things we should be fully 100% confident in are our conscious experiences at the present moment. Your beliefs about past conscious experiences could certainly be mistaken (indeed, many regularly are). Even your beliefs about your conscious experiences from a moment ago are suspect!

What this implies is that the set of evidence you are conditioning on at any given moment is just the set of all your current conscious experiences. But this is way too small a set to do anything useful with. What’s worse, it’s constantly changing. The sound of a car engine I’m updating on right now will no longer be around to be updated in one more moment. But this can’t be right; if at time T we set our credence in the proposition “I heard a car engine at time T” to 100%, then at time T+1 our credence should still be 100%.

One possibility here is to deny that 100% credences always stay 100%, and allow for updating backwards in time. Another is to treat not just your current experiences but also all your past experiences as 100% certain. Both of these are pretty unsatisfactory to me. A more plausible approach is to think about the things you’re updating on as not just your present experiences, but the set of presently accessible memories. Of course, this raises the question of what we mean by accessibility, but let’s set that aside for a moment and rest on an intuitive notion that at a given moment there is some set of memories that you could call up at will.

If we allow for updating on this set of presently accessible memories as well as present experiences, then we solve the problem of the evidence set being too small. But we don’t solve the problem of past certainties becoming uncertain. Humans don’t have perfect memory, and we forget things over time. If we don’t want to call this memory loss irrational, then we have to abandon the idea that what counts as evidence at one moment will always count as evidence in the future.

The point I’m making here is that the perfect theory of rationality for task (2) might not be the perfect theory of rationality for task (3). Humans have cognitive quirks that might not be well-captured by treating our brain as a combination of a hard drive and processor. (Another example of this is the fact that our confidence levels are not continuous like real numbers. Trying to accurately model the set of qualitatively distinct confidence levels seems super hard.)

Notice that as we move from (1) to (2) to (3), things get increasingly difficult and messy. This makes sense if we think about the progression as adding more constraints to the problem (as well as making it increasingly vague constraints).

While I am hopeful that we can find an optimal algorithm for inference with infinite computing power, I am less hopeful that there is a unique best solution to (2), and still less for (3). This is not merely a matter of difficulty, the problems themselves become increasingly underspecified as we include constraints like “these rational norms should apply to humans.”

Deciphering conditional probabilities

How would you evaluate the following two probabilities?

  1. P(B | A)
  2. P(A → B)

In words, the first is “the probability that B is true, given that A is true” and the second is “the probability that if A is true, then B is true.” I don’t know about you, but these sound pretty darn similar to me.

But in fact, it turns out that they’re different. In fact, you can prove that P(B | A) is always greater than or equal to P(A → B) (the equality only in the case that P(A) = 1 or P(A → B) = 1). The proof of this is not too difficult, but I’ll leave it to you to figure out.

Conditional probabilities are not the same as probabilities of conditionals. But maybe this isn’t actually too strange. After all, material conditionals don’t do such a great job of capturing what we actually mean when we say “If… then…” For instance, consult your intuitions about the truth of the sentence “If 2 is odd then 2 is even.” This turns out to be true (because any material conditional with a true consequent is true). Similarly, think about the statement “If I am on Mars right now, then string theory is correct.” Again, this turns out to be true if we treat the “If… then…” as a material conditional (since any material conditional with a false antecedent is true).

The problem here is that we actually use “If… then…” clauses in several different ways, the logical structure of which are not well captured by the material implication. A → B is logically equivalent to “A is true or B is false,” which is not always exactly what we mean by “If A then B”. Sometimes “If A then B” means “B, because A.” Other times it means something more like “A gives epistemic support for B.” Still other times, it’s meant counterfactually, as something like “If A were to be the case, then B would be the case.”

So perhaps what we want is some other formula involving A and B that better captures our intuitions about conditional statements, and maybe conditional probabilities are the same as probabilities in these types of formulas.

But as we’re about to prove, this is wrong too. Not only does the material implication not capture the logical structure of conditional probabilities, but neither does any other logical truth function! You can prove a triviality result: that if such a formula exists, then all statements must be independent of one another (in which case conditional probabilities lose their meaning).

The proof:

  1. Suppose that there exists a function Γ(A, B) such that P(A | B) = P(Γ(A, B)).
  2. Then P(A | B & A) = P(Γ | A).
  3. So 1 = P(Γ | A).
  4. Similarly, P(A | B & -A) = (Γ | -A).
  5. So 0 = P(Γ | -A).
  6. P(Γ) = P(Γ | A) P(A) + P(Γ | -A) P(-A).
  7. P(Γ) = 1 * P(A) + 0 * P(-A).
  8. P(Γ) = P(A).
  9. So P(A | B) = P(A).

This is a surprisingly strong result. No matter what your formula Γ is, we can say that either it doesn’t capture the logical structure of the conditional probability P(B | A), or it trivializes it.

We can think of this as saying that the language of first order logic is insufficiently powerful to express the conditionals in conditional probabilities. If you take any first order language and apply probabilities to all its valid sentences, none of those credences will be conditional probabilities. To get conditional probabilities, you have to perform algebraic operations like division on the first order probabilities. This is an important (and unintuitive) thing to keep in mind when trying to map epistemic intuitions to probability theory.

The Problem of Logical Omniscience

Bayesian epistemology says that rational agents have credences that align with the probability calculus. A common objection to this is that this is actually really really demanding. But we don’t have to say that rationality is about having perfectly calibrated credences that match the probability calculus to an arbitrary number of decimal points. Instead we want to say something like “Look, this is just our idealized model of perfectly rational reasoning. We understand that any agent with finite computational capacities is incapable of actually putting real numbers over the set of all possible worlds and updating them with perfect precision. All we say is that the closer to this ideal you are, the better.”

Which raises an interesting question: what do we mean by ‘closeness’? We want some metric to say how rational/irrational a given a given person is being (and how they can get closer to perfect rationality), but it’s not obvious what this metric should be. Also, it’s important to notice that the details of this metric are not specified by Bayesianism!  If we want a precise theory of rationality that can be applied in the real world, we probably have to layer on at least this one additional premise.

Trying to think about candidates for a good metric is made more difficult by the realization that descriptively, our actual credences almost certainly don’t form a probability distribution. Humans are notoriously sub additive when considering the probabilities of disjuncts versus their disjunctions. And I highly doubt that most of my actual credences are normalized.

That said, even if we imagine that we have some satisfactory metric for comparing probability distributions to non-probability-distributions-that-really-ought-to-be-probability-distributions, our problems still aren’t over. The demandingness objection doesn’t just say that it’s hard to be rational. It says that in some cases the Bayesian standard for rationality doesn’t actually make sense. Enter the problem of logical omniscience.

The Bayesian standard for ideal rationality is the Kolmogorov axioms (or something like it). One of these axioms says that for any tautology T, P(T) = 1. In other words, we should be 100% confident in the truth of any tautology. This raises some thorny issues.

For instance, if the Collatz conjecture is true, then it is a tautology (given the definitions of addition, multiplication, natural numbers, and so on). So a perfectly rational being should instantly adopt a 100% credence in its truth. This already seems a bit wrong to me. Whether or not we have deduced the Collatz conjectures from the axioms looks more like an issue of raw computational power than one of rationality. I want to make a distinction between what it takes to be rational, and what it takes to be smart. Raw computing power is not necessarily rationality. Rationality is good software running on that hardware.

But even if we put that worry aside, things get even worse for the Bayesian. Not only can a Bayesian not say that your credences in tautologies can be reasonably non-1, they also have no way to account for the phenomenon of obtaining evidence for mathematical truths.

If somebody comes up to you and shows you that the first 10^20 numbers all satisfy the Collatz conjecture, then, well, the Collatz conjecture is still either a tautology or a contradiction. Updating on the truth of the first 10^20 cases shouldn’t sway your credences at all, because nothing should sway your credences in mathematical truths. Credences of 1 stay 1, always. Same for credences of 0.

That is really really undesirable behavior for an epistemic framework.  At this moment there are thousands of graduate students sitting around feeling uncertain about mathematical propositions and updating on evidence for or against them, and it looks like they’re being perfectly rational to do so. (Both to be uncertain, and to move that uncertainty around with evidence.)

The problem here is not a superficial one. It goes straight to the root of the Bayesian formalism: the axioms that define probability theory. You can’t just throw out the axiom… what you end up with if you do so is an entirely different mathematical framework. You’re not talking about probabilities anymore! And without it you don’t even have the ability to say things like P(X) + P(-X) = 1. But keeping it entails that you can’t have non-1 credences in tautologies, and correspondingly that you can’t get evidence for them. It’s just true that P(theorem | axioms) = 1.

Just to push this point one last time: Suppose I ask you whether 79 is a prime number. Probably the first thing that you automatically do is run a few quick tests (is it even? Does it end in a five or a zero? No? Okay, then it’s not divisible by 2 or 5.) Now you add 7 to 9 to see whether the sum (16) is divisible by three. Is it? No. Upon seeing this, you become more confident that 79 is prime. You realize that 79 is only 2 more than 77, which is a multiple of 7 and 11. So 79 can’t be divisible by either 7 or 11. Your credence rises still more. A reliable friend tells you that it’s not divisible by 13. Now you’re even more confident! And so on.

It sure looks like each step of this thought process was perfectly rational. But what is P(79 is prime | 79 is not divisible by 3)? The exact same thing as P(79 is prime): 100%. The challenge for Bayesians is to account for this undesirable behavior, and to explain how we can reason inductively about logical truths.

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?