A Conversation on “Morality Is Fractal”

Marcus: Hello everyone, welcome to the Econ Playground. Today I’m very happy to have some friends here to discuss a paper by Cameron Harwick called “Morality is Fractal.” Cameron is an associate professor at SUNY Brockport where he studies monetary theory, evolutionary economics, and complexity economics. And usually Cameron has some sort of anthropological flair to his research, I’ve noticed. I’ve also got two other friends with me. One who’s been on the channel before, one who hasn’t. The one who hasn’t is Kurtis Hingl. He’s a third year student at GMU studying the industrial organization of science. Kurtis and I have another paper that we’ve been working working on together for a while now and it’s very much related to Cameron’s paper. So I thought having him on board would be a lot of fun. And last but not least we’ve got Matt Kelly who’s an assistant professor of finance at Campbell University. He’s an expert in the uranium industry and he’s also proficient at Microsoft Excel, and he knows a lot about these kind of topics. So Cameron’s going to start giving us a little bit of an explanation of his paper with some visuals.

Thanks so much for having me on. This is a paper about something that all of you guys are also interested in, which is the incompleteness of formal rule sets. This is going to include social norms, moral rules, legal rules, contracts, any kind of thing like that. The basic argument of the paper is that the boundary of applicability of any kind of rule in any of these situations is going to be fractal. That is to say that any infinitesimal detail can potentially matter in a way that can’t be predicted in advance. For example if we think of the applicability of a rule like “you can move a a chess piece,” that only applies in the game of chess. But even rules with a really large domain of applicability, like Thou Shalt Not Kill, we recognize exceptions to this like self-defense.

Imagine that you’re on a jury and the question before you is, was this death wrongful, or was it self-defense. Does this count as murder or self-defense? It’s not going to be possible for you to enumerate in advance a list of necessary and sufficient conditions for that to have counted as self-defense. That could hinge on a signal as small as the size of a glove. “If the glove doesn’t fit, you must acquit.”

I’ll talk a little bit about the formal argument here, just because I know that uh this is relevant to some of the things that you guys are working on too. There’s basically two moving Parts in this formalization: the idea of signal space, and the idea of decision space, and the way those two interact.

Signal space I’m thinking of as all the observable characteristics of a situation, any kind of situation that I might be in. We’ll think of this as an infinite list of zeros and ones. For example, right now I’m sitting in front of a computer on Zoom. That takes a one. I’m talking to Marcus and Kurtis and Matt, that takes three ones. Marcus is not wearing a pink hat, so that takes a zero. Any kind of concrete situation I could be in is represented this way. And we’re going to project that to a 4×4 square in complex space. That might seem like a weird thing to do, but we’ll see the payoff to that shortly.

Once we project this signal space, any point in this 4×4 complex square is going to represent some kind of situation. Points that are farther apart are going to be more different situations. So from any given situation I can make some kind of decision. And different situations will have different sets of decisions that make sense for me. For example, if I’m standing in the bathroom about to take a shower, taking off my shirt is a sensible decision right now. Sitting on Zoom, taking off my shirt is probably not a sensible decision. So there’s a boundary of applicability of that decision. We’re going to think of decisions as a function, and that function is going to take you from a point in signal space – some situation – and move you to a new one.

We’re going to restrict the set of decisions to functions of the form z² + c. The reason we do that – we don’t have to do this; it could be any rational function – but z² + c is going to allow us to visualize our decision space. c is a complex point, and all of the functions we care about can be represented by that 4×4 square. So that’s going to be some infinite set of decisions. We could have different exponents; it doesn’t matter, we just get more axes of symmetry.

Let me show you how we are going to visualize this. On the right is signal space, the 4×4 square, and at a point in signal space –this blue point – this is some situation I might be in. On the left here is all the decisions that are valid from that point. And we’re going to think of a valid or sensible decision as being one that I can iterate. I can repeat it over and over and it makes sense for me. And if that converges either to a limit cycle or to a fixed point, then we’re going to say that that’s a viable decision. If that blows up to infinity, that shows that that is a decision that cannot be repeated; that’s probably not a wise decision.

The black area on the left here is all the decisions that are viable from this blue situation over here. And on the other hand this black area in signal space is all the situations that the decision in red can be applied at. So if you choose a different point in signal space, let’s move to a situation over here, and you get a different set of decisions that are viable. And the further out you go, the smaller the decision space is; the fewer decisions are viable there. And similarly the more central you are within this decision space, the larger the set of situations that you can apply that in. This would be a rule like “Thou shalt not kill.” You can apply that almost anywhere. But as I move out, especially toward the edge, now we get much weirder shapes. We get smaller shapes with these fractal boundaries.

Now what does this show us? On the right here is what we’re going to call a stability locus. These are the situations in which it makes sense to apply a decision over and over again. The further out we go with more and more esoteric decisions, that stability locus gets smaller and smaller and smaller. The boundary of that stability locus is what’s called the Julia set. And the Julia set is defined as a set of points where, in the neighborhood of that point, iteration generates chaotic behavior. That means if you are near the boundary then any movement in any direction – any given signal – can move you in or out of that stability locus and make it a good or a bad decision unpredictably.

The Julia set is fractal, which is to say it’s infinitely detailed. The length of that boundary is infinitely long, and humans can’t really do a whole lot with infinitely long things. So the next thing I talk about is resolution. And if we think of resolution as the maximum number of signals that I can pay attention to in any given situation – right now I’m paying attention to the fact that I’m on Zoom, the folks I’m talking with, but I’m not necessarily paying attention to the direction that Kurtis’s hair is combed. That’s not a thing that I need to pay attention to. So what I show is that the length of the description of the boundary of the Julia set increases in proportion to the resolution times the fractal dimension of the Julia set.

I’ll share this one more time to show you the fractal dimension. What you can see is that for certain decisions this looks like a pretty easy shape. There’s not a whole lot going on in the edges. And if I do c=0, the Julia set is literally a circle. But as I go toward the edge, the boundary gets more and more complicated. Fractal dimension is a measure of the complexity of the boundary. And the further out we go, the more and more complex that boundary gets. And so descriptions of that boundary are going to increase in length faster the farther out we go. And that gives us a sense of the dynamics of institutions. For example, the length of the Code of Federal Regulations seems to rise at a reasonably steady pace for something like the last fifty years.

Now if you think of a regulator, for example the EPA, let’s think of this best-case scenario. The EPA has a substantive goal in mind like “care for the environment.” But the EPA cannot mandate that I care about the environment. The EPA can mandate that I do or don’t do particular things, or at least fine me for doing particular things. The mapping between that and good environmental outcomes is going to be a practical boundary. I’m going to have opportunities to signal formal compliance while substantively accomplishing my own goals in ways that the EPA might not want me to do. So this generates a kind of cat-and-mouse game in any kind of adversarial game, where I’m looking for these points in signal space where I’m accomplishing my substantive goals, but I’m still within the EPA’s finite-length rule set. And the EPA is going to have to increase its resolution of signals that it looks at over time to try to deal with that.

For another example, imagine you’re a loan officer or an investor. This draws on some previous work on blockchain finance that I’ve done with Jim Caton. You’re not going to be able to come up with a checklist of necessary and sufficient conditions for approving a loan, or for investing in a stock, based on observable signals. That’s going to be infinitely long eventually if you try. I also talk about the evolution of altruism, which depends on the ability of cooperators to recognize each other. How do I know whether you’re a cooperator, and therefore whether I should cooperate with you? Irrespective of whether or not you’re going to cooperate back with me, but are you the kind of person who would? There’s game-theoretic literature showing that this is an evolutionarily stable strategy, even though it’s not a Nash equilibrium, if and only if cooperators can assort with each other and recognize each other. And of course that leads to this arms-race dynamic where defectors want to get in under the radar, and cooperators need to get smarter to try to detect that, which drives the defectors to get smarter, and so on and so forth.

The last thing I’ll talk about here is two conceptions of learning. You might wonder how humans manage cooperation at all if there’s always going to be these edge cases where you can zoom into this little fractal hole that’s not covered by the finite-length rule. And if you’re looking at a given finite resolution, that’s always going to be true. But what humans seem to do is to not write down rules as necessary and sufficient conditions. Even very explicit legal norms are not fully explicit. There’s a lot of load-bearing fuzzy concepts like “negligence.” What are the necessary and sufficient conditions for something to be judged as negligent? I don’t know. That’s a matter of judgment. And judgment seems to be a situation where learning happens analogically. You perceive a situation as a whole, not as a set of necessary and sufficient conditions. In terms of the formalization, maybe you think of a distance measure in the signal space coordinate rather than a Hamming distance measure of your explicit signals. And that does seem to be basically how neural architecture works: learning and classification by holistic analogy of the whole situation, rather than by a list of necessary and sufficient conditions.

So one thing that might be worth exploring is, one of the previous papers I mentioned with Jim Caton is about the limits of doing things algorithmically. But how much of it could be done with, for example, neural networks, which seem to be doing something similar to what human brains are doing: taking a whole situation and then encoding it analogically using neural weights.

I’ll end there, and then I’m interested to hear your thoughts.

Discussion

Marcus: The first question I want to ask is just to make sure that I understand exactly how the different parts of the mathematics are then analogized to the actual decisions that people are making. It seems like what’s most important is the idea of whether or not a certain point in the decision space and the signal space cycle with one another. And for anyone that’s not familiar with the math, I’ll also link some videos from 3Blue1Brown, who has pretty good descriptions of this stuff. Kurtis shared those with me when I first sent him Cameron’s paper. So if there’s a cycle between the decision space and the signal space, then when I observe a signal I should expect a certain decision in response. Then that decision will create a signal that will create a decision that’ll create a signal that all balance themselves in some way such that I don’t need to know an infinite number of things, or an infinite number of ways that this process cycles, to get a general sense of what decisions are going to be made. So this signal actually generates predictability for me. Whereas if it’s a chaotic cycle then there’s possibly no algorithm, even with infinite time to compute, such that I could actually predict every single decision that’s going to be made, given that this rule set is followed in some sense.

So for the economist, we are sort of restricting this thought experiment to “if the rule was followed, this is what decisions people would be taking.” But as we know in econ, people have the choice to defect from rules all the time anyway. So what you’re saying is that if you make an analogy between signal space and rules – that is, when people observe this signal they will do this; they will follow this decision – rules are in a number of cases, specifically these boundary cases, insufficient to generate predictions about what other people are going to do.

There’s two different things going on there. There’s the question of whether a given decision will diverge after iteration – and that’s divergence, not necessarily chaotic behavior. The chaotic behavior arises at that boundary where if you change one signal – any signal – then it’s completely unpredictable whether it diverges or doesn’t diverge. But you’re right that converging to a fixed point or a limit cycle means that that decision is going to be predictable.

Matt: That last point was an important one, the difference between cycling and divergence. Maybe early on in the paper, for the members of your audience that aren’t as familiar with how the Mandelbrot set is defined mathematically as this iterative function, some time spent briefly covering that might be worthwhile. Just laying out, here’s literally the iterative function, here’s how the rule works, and maybe even like an example of starting from a starting point z and setting c somewhere in the plane can help illustrate that. Just kind of iterate it forward three or four times. I think that you describe that in the paper perfectly well; it’s just for illustrative purposes might be useful for some of your audience.

So the logistic map is an idea that I think complexity scientists are familiar with. And the logistic map is one cross-section of the Mandelbrot set in the real domain. Laying that out for people might be useful also because the logistic map is a useful illustration for showing how a fairly simple iterative growth function can explode into these chaotic ways. And showing how, hey, if we extend that idea into the complex plane, you can see how the same story is emerging. You just have a different geometric representation of this similar idea. So those who are familiar with the logistic map, and who the Mandelbrot set might be new to and maybe have heard that it’s related to the Mandelbrot set, it might give them something to glom on to and recognize the generality of what you’re saying and how it extends to multiple domains.

Like this thing you say about rules by analogy toward the end is also really evocative and interesting. Some citations or connecting to the literature in that part of the paper might be helpful. My impression is that Hofstadter is big on that idea. So connecting that might give you a quotation, because he’s rather eloquent most of the time. Might be a useful way to make clear to your audience what it is that’s so important about that idea and nailing down how necessary and sufficient conditions aren’t enough, or why reasoning by analogy is different from a known endpoint.

I think you say it at some point, but the fact that signal space really does nest other more simplistic non-fractal “signal spaces” or rules – like saying that you want a signal space where there’s a well-defined boundary, and it’s like a circle or something – yeah we got that, that’s covered. You set this meta-parameter to zero and you get all that, and then you get a lot more than that. And also conveying those bits of information with theorems would be useful. The nesting of that as a special case as a theorem would be a nice self-contained way to put the matter. Something that I think will be easily readable to most economists, but also make clear to everybody else that the homework has been done.

Another sort of theorem to state here – I’m pretty sure I’ve seen it stated in a complex analysis course that I took, but I’m forgetting what the name of the theorem is – but this idea that for any level of fine-grainedness – for any nth iteration of increasing the fidelity of the picture of the set of the boundary, you have this probability function that you define, the p(z) and d and ρ, you could maybe state it. You could say what you’ve said in English and translate it into a theorem just by saying for any n<∞ – for any finite n – there exists a z such that p(z)<1. That might be a concise way to put what you’re saying, where this ability to always go an iteration further and see a greater complexity to the boundary creates a kind of uncertainty. That if you wanted to predict whether a point is or isn’t in the boundary after m more iterations or something, that there’s uncertainty there.

I’ll stop prognosticating comments here and ask a proper question. So these black regions in the interactive – I think I know how you’ve said it but I’ll ask it just to make it clear for everybody in the audience: this divergence. So if we pick a point and we iterate forward, and where you iterate to – or the output I suppose – if it diverges, does that mean there’s a rule that we’ve identified that says that that action or that signal indicates yes or no, or good or bad? Suppose I wanted to put it this way: there’s a signal and I want to know is some sort of project profitable or not, or will they pay back a loan, let’s use the example that you bring up. So if it diverges, does that mean that they won’t pay the loan, or if it converges does that mean that they do pay the loan? Or is it that if it diverges we don’t know whether it pays the loan or not?

Marcus: Yeah, because in standard contract incompleteness literature, the way they talk about these things is people have this reservation expectation. Given that the contract is incomplete some specific event will happen which we might not like very much. So actual law and econ people that have meditated on this problem say, if the contract is incomplete, a judge should just not enforce the contract, or just set the price to zero or something like that, such that people would actually want to avoid that chaotic situation, so that they don’t basically treat the court system as a common pool resource that’s going to write the rest of their contract for them.

Thank you for all those suggestions. And anything that I should be connecting to that I have not connected, definitely tell me. On the meaning of divergence – in the loan case divergence means that you don’t get paid back. Now there are situations where divergence is not apparent for some number of iterations. It might not be clear after 10, 20, even 30, 40, 50 iterations yet whether this is going to eventually diverge or not. I might interpret this as a situation where you’re testing the boundaries of what you can get away with. Imagine two kids in the back seat of a car and little brother’s going “I’m not touching you, I’m not touching you.” How close can he get before big sister turns around and slugs you? And there’s some threshold where if you do that a little bit and then back off it’s going to be fine, but if you iterate that—if you keep doing that—then you’re going to get slugged. And so an evolutionarily stable decision is going to be one that I can keep applying and she’s not going to slug me.

Matt: I get the impression that some of these signals are uninterpretable. Or – where does the undecidability come in? What you’ve described is like, here are the regions where the binary answer to “will this diverge” is a zero or one. Are there points where we don’t know? And is that a matter of how many times you go through this costly iteration to gather more information about what outcome a signal will apply? There’s yes, there’s no, and then there’s not decidable, right? So where does that come in?

The undecidability comes in at the boundary of a stability locus, or of a rule in signal space. This would be the proper Julia set, which is actually the border. This would be a situation where at the border it’s unclear whether moving in any given direction is going to move me in or out or whether I am already in or out. For example, think of yourself as an investor and you’re trying to decide whether to buy a stock. This is going to be the mapping from observable signals to expectations: what might you look at to figure out whether a stock is a good investment? There are easy bets. But then efficient market hypothesis says easy bets get arbitraged away. So the only way you’re going to make money is with actual difficult bets, and by looking at higher resolution than other investors are looking. So you maybe go to board meetings, you look at the earnings reports, you comb through the balance sheets, you follow the executive team on Twitter and see what they’re up to. So the higher resolution you can incorporate into your decision, the more you’re going to be able to form a better expectation about whether that stock is indeed going to be profitable or not.

The undecidability comes in in that at any given finite resolution, you’re drawing an approximation around that stability locus around the Julia set. And that’s going to fail to capture some points that are indeed in or out. You might make a decision where you misinterpret some remark. I think about Alan Greenspan’s “irrational exuberance” remark and everybody says “oh my gosh he’s telling us something, sell sell sell,” and then he’s like “I didn’t mean anything by that.” Or maybe he’s dropping hints and trying to tell people things, but they don’t believe him, and they fail to act on something he is telling them. That would be situations both in and out of the actual stability locus. So that mismatch between the finite approximation and the actual stability locus – where you think it’s going to diverge but it doesn’t, or vice versa – there’s always some point close enough to the actual Julia set where any finite resolution is going to fail to arrive at that decision correctly.

Kurtis: Thanks for this excellent paper. I was thinking about analogy stuff, but first I want to look at this paper in a different framing. Firstly the setup is great. So you have some way of formalizing – now it’s funny because you’re doing some complexity stuff, but it really is a simple model, and then you have all these applications. I love examples like that: like how can we explain as much as possible with a simple model? What I’m interested in is, how can we add to these applications, how can we dissect these further? You talk about adversarial games. I’m wondering if there’s some kind of way we can formalize something like moral gerrymandering. Obviously there’s people on both sides of the border who would can profit if the decision goes their way. Even in the introduction you talked about like the glove fitting in our conversation here, but there’s probably tons of cases that you can say “what if we had this extra piece of evidence, how would that flip the sign?” And then how much focus each party will have on including this additional signal in this set that we’re agreeing to. So that’s where I was interested in: can we say anything else about this adversarial nature where different parties are trying to increment that resolution by one signal? That marginal ρ. And maybe I’m misinterpreting how you’re using this, but that’s how I was thinking about it: there’s parties on either side of the border who are trying to include that signal. And then also the difference in different spheres. We talked about finance and investing a little bit. It seems like where there’s big gains to be had, like in finance, we would go back and forth quicker. We’d say “that one’s more important, and then that one’s more important.” But there’s other spheres where it takes a long time: EMH is still true, but the timescale is just a lot longer.

I really like this idea of moral gerrymandering. One of the things I talked about in the paper is deriving a costly signaling theory, in the sense of my ability to verify the correctness of certain signals being given off by you versus your ability to at some cost emit signals that maybe mislead me. For example, Enron falsifying their balance sheet versus my ability to say there’s something fishy going on here. But moral gerrymandering is a little different from that, and I like that because what you have here is a competition over what signals we pay attention to. This almost gets us to a behavioral story where we have limited attention, or maybe saliency bias. That also lets you think of the adversarial nature of analogical learning. One thing I’ve been thinking about recently – not with this paper but separately – is the ways that people can adversarially exploit analogies. For example if you remember a year or two ago, there was a big infrastructure bill. And then the sharks come in and smell the blood in the water, and they say “everything that I’m on about is infrastructure,” drawing this analogy where it may or may not exist: “Child care is infrastructure, paid leave is infrastructure, healthcare is infrastructure.” Because you get money if you can draw that analogy.

Matt: Or, “my company is an ESG company”

Exactly. What counts as ESG or not now that we have these rules for who gets money? So the vulnerability to that sort of analogizing is another one of these adversarial games. If you are overly willing to update on these kinds of analogies, then the sharks come in – I should say piranhas because they’re little nibbles – the piranhas come and nibble you away and you don’t survive. You get exploited into oblivion. Whereas if you’re more skeptical you can say “no, child care is great but it’s not infrastructure.” Then you’re going to be more resistant to being exploited; you’re going to be safer within that stability locus as opposed to the situation where you let everybody push you off into infinity and you blow up your budget constraints. So I really like that connection to moral gerrymandering.

Marcus: Some of what Kurtis and I have been working on would actually have great visual analogies to the stuff that you’re doing. The way that we’ve been treating it so far is like “we’re working in some system with some rule, and we hit the border and things go chaotic.” But people anticipate things like that. People can anticipate whether or not the decision rule that we’re setting is going to converge or diverge or have chaotic results. We want it to be as predictable as possible. And when we’re designing this rule we do understand this space to some degree. So in the standard costly signaling literature – at least in econ mostly based on Iannacone’s paper on it – is thinking about trying to get people to not be free riders on a club good. But one of the other things it does that’s kind of underrated is it forces people to act in certain relatively stable domains, or to decide and meditate on rules which are only going to have certain signals, such that we’ve stabilized a large portion of the signal space, such that we’re only moving very small distances from one another, such that it’s much less likely that we’re going to hit the fractal boundaries of our morality. If you literally think of the Mandelbrot set as like a place that we live and we want to have kids play in the center so that they learn the decision rules slowly over time. As a kid you’re maybe for the first time going through the iterative process and everything feels new and you’re learning it. But once you’re an adult, hanging out in the center of the set is old hat. You’re not learning anything. But the heroes or the explorers of our society are the people that venture closer and closer to the edge of the set. And maybe these people need to be trained through other methods. They can’t just be told to follow the decision rule, because where they’re going to go the old tools are not going to work anymore. So to connect it to the costly signaling, one of the things that costly signaling does is to make it more costly for people to explore close to the edge of the set such that everything else in our society continues to work. Basically minimize the costs of having to introduce new analogies, or minimize the costs of losing coordination because someone is operating at the edge of morality.

Now what you say in the abstract of the paper is that there is then a tradeoff here between predictability and certain gains from trade. And that’s essentially what the explorer or the social deviant or something is saying: “look at all these gains from trade that by your morality you’re leaving on the table.” And essentially you’re giving a response to them – at least this is how the argument would go – it’s not clear exactly which side dominates. It’s something like “the reason that we’re forgoing those gains from trade is so that we can maintain predictability in the rest of our actions. You don’t realize what other things are dependent on it.” So it’s a very much a Burkean point in that you can want to change things, and there’s nothing wrong in principle with wanting to improve society of course. But sometimes you simply don’t realize what an individual change will do, or you don’t realize what changing one rule will do by changing the rest of the process.

I think that’s a great metaphor, thinking of institutions as a way of pushing you away from the boundary. So the kid in the back seat’s trying to push the limits of where he can get close to his sister, but before he gets even close to there, as a parent I turn around and say “hey stay on your own side.” So we never even get close to the point where she hits him.

Marcus: And the kid perceives that as unfair because the low-resolution rule “don’t touch your sibling” is the way that the parent has communicated morality to the kid. The kid has to know that he’s breaking the spirit of the rule, but he does feel that there’s some sort of unfairness because the parent seems to have added a new rule, or made an arbitrary decision, or falsely applied a rule. Whereas we know that the parent is actually reaching for something higher, more important, like not annoying the sister, or teaching the child to not distress other people. That seems to be the tradeoff in not using the decision rule at all. So when moral gerrymandering is occurring such that someone is exploiting the way that this rule is limited, the way that you correct it is by breaking the rule itself in a sense, or transcending it, or introducing a new rule. So there’s this other level of unpredictability there. And the way Kurtis and I have started to think about it is, what you’ve assumed with the decision rule is consistency. The decisions are happening according to a consistent pattern, like an eight-year-old that’s done addition could go through every step of the iterative process. That’s not what’s complex about this complex math. And that is consistent the whole time. So in order to correct the problems that this consistency leads you into, you have to be inconsistent. And inconsistency leads to other sorts of unpredictability than the problem with fractals.

Matt: You mind if I tack one question onto there? I think that discussion can be summarized by asking, is this a market failure? Is this inconsistency a global inconsistency? You make this distinction between the spirit of the law and the letter of the law. So is the inconsistency in the letter of the law needed to make the spirit consistent?

I think that’s exactly right. Let me circle back in the long way around to this market failure question. The letter of the law, the spirit of the law, that’s something key here. And it does entail inconsistency, in at least any finite rule set. The original inspiration for this paper was looking at the generative process for these fractals, and you have this process where you generate a shape and you have some rule for what counts and what doesn’t. But then you get to the border and you see “oh wait a minute you need to make an exception here. You have to stick a little piece on or carve a little piece out.” And then you get to the border of that and you still got to carve a little piece out or stick a little piece on, and the border just never resolves. And so anything is going to necessarily have those inconsistencies at the border, because that’s what the generative process – at least for an adversarial game – entails.

Which also raises a moral philosophy question. There’s some moral philosophy papers that ask the question like, what is the thing – the real thing, the spirit of the law – underneath our finite-length moral rules? Is there a spirit of the law? Should we be moral realists? And what I want to argue here is that that thing underneath is actually evolutionary stability. That a moral rule should at the very least – and this is not a sufficient condition, but it’s a necessary condition – a moral rule should at least provide for its own continuation. Now there’s a debate recently about effective altruism and the Singer-type approach to morality. For Singer the underlying thing, the spirit of the law, is utilitarianism. We want to do the most good for the most people. And my argument against that would be: is that an evolutionarily stable strategy? If it’s not, then a moral rule of always doing the most good for the most people is not going to be a rule that provides for its own persistence. Being an effective altruist today is going to impair the existence of effective altruists in the future. And so what’s necessary is some sense of what is a sustainable strategy. Effective altruism is going to diverge, blow off to infinity. Can we find something maybe close to effective altruism – as close as you like – that doesn’t do that? What are the minimal changes to that moral rule that we have to make to make sure that it provides for its own persistence and is not exploitable off to infinity?

Matt: Another thing for the audience: calling it a market failure helps ground all this stuff into a context that may be useful. You also mentioned parents telling kids in the back seat has an analog to a central planner. Is there some sort of rationale for intervention of some kind?

The market failure angle is interesting and you’re right I didn’t end up coming back to that because this would arise in these kinds of adverse selection asymmetric information cases which are typically thought of as market failures. And maybe we can continue to think of them as market failures, although I’m not sure that’s a justification for intervention. Because if anything there is going to be more scope for judgment with private action than with a regulation, which has to be at least somewhat explicit, it being ex post and well-defined legally. And this is something that has typically annoyed me about public goods discourse too. People think about public goods as a market failure therefore the government steps in and provides them and problem solved. Well no, because the government is itself a giant public good. So if you can assume a government, then great, but that’s not a general solution to a public goods game. And similarly I’m not sure that intervention would be a general solution to these sorts of adverse selection problems because you’re going to get adversarial exploitation of regulation as well – regulatory capture, malicious compliance, and what not.

Matt: Do you think it’s fair to say that what you’re laying out here is neutral to those questions? Because yeah it applies to players as well as the referee is one way to put it. Or is there maybe sort of a larger meta-point about the complexity of the world being too much for a central planner to handle? Or is all this kind of suggestive of that but not quite – because the logistic map and the Mandelbrot set are deterministic if you know it, but nonetheless is probabilistic if you don’t know. I mean do you think that this is neutral to those questions, or does it have something of import to the role of government more broadly?

I want to think of this as neutral with respect to these questions. One question that has been asked about this paper is “what about the self-similarity aspect of fractals?” And I have to this point not really had a good answer to that question. It’s just not really important.

Marcus: What’s the question? I don’t understand. What does self-similarity mean?

The Julia set and the Mandelbrot set are self-similar. You can see here on the Julia set that it looks like it’s constructed of a piece, and then you rotate the piece and then you make it smaller and then you stick that on the end. And as far as you zoom in, it looks similar to how it looks at a larger scale. And the Mandelbrot similarly: you’re taking this cardioid shape, and you’re sticking it on top of each other on the end and all around, smaller pieces again, and so smaller parts look like bigger parts. I haven’t done anything with that in the paper itself, but to Matt’s question, I have a sense – and I have not rigorously proved this, and I don’t even know if you can rigorously make this analogy – but my sense is that you could get something like self-similarity out of the fact that this is an nth-order public goods problem. You have the normal public goods problem where it’s hard to prevent free riders: easy peasy, you get an enforcer to make everybody not free ride and then you solve the problem. But then how do you make the enforcer do this? And now you have a self-similar problem just stepped up a level. And then you have constraints on the enforcer – who enforces those? And no matter how many levels you go you have this similar problem. So I get the sense that this is – I don’t want to say orthogonal or neutral with respect to market versus government – but that it applies to both of them at an n+1 level.

Matt: Looking at the whole picture of the set, you can see these similarities. But when you zoom in – and if anybody at home wants to try smoking some pot and looking at the fractal zoom – you see these patterns just keep showing up, and then you zoom a bit further and an identical pattern will show up, and you’ll see these shapes that are almost identical or near-identical mutations of the same general form. And you could just keep zooming and zooming and zooming and you have these recurring patterns. I mentioned the zoom-in part because that zoom phenomenon of self-similarity happens at the boundary, which is exactly where the interesting things happen in this model.

Kurtis: I think’s the nth-level public goods game is a great example for zooming in. But there’s the flip side of that, which is the evolutionary niche of investments, and finding that alpha. So again having some rule that says we’ll look at these five signals, and these five signals yield a pretty good return, but then somebody else can always come along and say “well look at those five signals plus this other one” and then we’re gaining and then again we can zoom way in. And instead of having the problem of enforcer we’re now having the problem of extractor or something like this. So I think the self-similarity actually is an interesting part of your story, namely that we can have predictions about the kinds of actions people take around these boundaries. Well no one’s going to come along and jump right up to the n+600 level enforcer problem, and we start with these five signals in the investment problem – no one’s going to come along and say immediately we’re going to look at these 100 signals. We’re always going to have this iteration of one more, one more, one more on both sides of the problem. So I think that’s an interesting prediction. We could look at some of these cases on some of these rule sets and say, where did they actually increase the granularity of the rule? And it’s always right on that next little margin. And then the next one ends up just being on that next little margin.

Yeah I like that. You don’t just happen to find yourself in the 50th lobe of the Mandelbrot or the far corner of the Julia set. You get there iteratively by exploring new signals.

Kurtis: One more thing I wanted to ask about: you were talking about the effective altruists. I think you’re right to say that they aren’t evolutionarily stable. But there’s a counterargument that says, if we only care about evolutionarily stable rule sets or moral intuitions, then we’re leaving all these gains on the table, or in their case ways we could be helping humanity. So do we have a meta-rule for establishing how close to the boundary you want to get? That’s impossible because the boundary is infinitely granular. But then you might have some kind of meta-rule of how far from the center, from z₀ – you might want to get. And maybe that’s what Marcus was talking about, reducing your domain to cases where we can coordinate, or reducing our signal space of things we can talk about. So while I agree that I don’t think EA is evolutionarily stable, there is a case to say, no it’s not, but because there are gains to be had on the table, we should grab them when we can because in the long run we’re all dead, or because you guys aren’t doing it, or because whatever we should still take this opportunity. Not because we think that we can be as granular as possible, but because we do think we are within one radius of the center.

Marcus: I just also read this past week the Roy Rapoport article that you cite in the paper, because you had recommended it to me before. And then right after that I read the book The Sacred and the Profane by Mircea Eliade. And if anyone is familiar with Jordan Peterson, people think of Peterson as a Jungian thinker. And Jung has definitely influenced him, but I think Mircea Eliade and his writings have influenced his way of thought much more directly. And it is exactly what Kurtis said, because the way that Eliade relates to this stuff is, he talks about religious symbolism as creating a domain that one sees as sacred and ordered and stable, versus the profane, which is not things that are evil, but things that are not yet incorporated into our sacred order. And so almost by definition then something that is part of the set of non-ordered things doesn’t have a clear definition for when it begins and when it doesn’t begin. And so in ancient ritual the way that you would define order is not by whether things are of this class or of that class, but rather by whether things are closer or further away from our totemic center or further or closer away to the ritual that we practice. So the center of the Mandelbrot set – the z₀ point – is like our totem pole in the middle of our village. Or he says later it’s like the cross in Christianity. Iit’s a great book because he’s able to go all the way back into what we know of more primitive religions all the way up to essentially modern Christianity. And he talks about for example the Spanish going into the new world and presenting the cross. They thought of themselves as bringing order or sacredness to a fundamentally profane domain. So that’s actually an awesome way to think about it. Closeness to the center is a better way of defining order than not-in-the-chaos. Think about the beach for example: the land is order, the ocean is chaos. But there’s waves. You never stand at the edge of land unless you live somewhere with like massive cliff faces. So if you’re in the cliffs of Dover or something like that you can feel like, yeah I’m literally right at the edge of order. If I take one more step or take an infinitesimal step I will fall over the edge. But if you’re at a beach, you have more of a sense of the chaotic part of the world is constantly lapping up against the edge of order.

The religious history of this question is really interesting. For example in Catholicism, you have a pretty clear sacred-profane distinction, and there are people who are more holy than other people. So if you want to be the most holy you become a priest and forgo a bunch of worldly pleasures. Whereas the Protestants come along and say there’s actually no distinction; everything is, or at least can be, holy. A merchant businessman can be just as holy as the priest, and therefore the priest doesn’t have to forgo worldly pleasures because those can be sanctified too. But also you can order everything toward that – as you say – totemic center. So that raises the question about how can everything be so ordered. And I wonder if that’s the same question here or if that’s kind of orthogonal. My sense is that it would not be possible to singly order everything; that you would need some kind of distinction between sacred and profane. On the other hand one doesn’t want to necessarily make too much of that because there’s obvious failure modes on both sides. But I’ll throw that out there and see what you think.

Marcus: On your characterization of Catholicism versus Protestantism: in Catholicism there is certainly a sense that everything can be made sacred, but not all at once. So the purpose of something like the ascetic practice of the priest, or keeping the church building holy, is very related to what we’ve been talking about in that eventually everyone can be brought into the church, but we don’t do it all at once, because then we would be picking a decision rule which brings us immediately outside of the sacred, and then we’re flooded. The best analogy is something like a flood, as opposed to the opposite, which is to treat everything as ordered. And then you end up with a desert in which nothing grows. What you want is something in between like a garden.

Another point that I wanted to bring up earlier which is related: I’ve been reading Charles Taylor’s A Secular Age. And one of the readings that he has of the history of Western religion, and also philosophy, is that there’s this transition from when people are building ordered systems, understanding that the edges of their systems – they wouldn’t understand this formally, obviously – but they’re fractal-like, in that we don’t have a clear boundary for when the ordered part of the world and the chaotic part of the world begins and ends. And then we move to an understanding of the entire world as part of a single order, or God’s order, or eventually the fixed order of nature. So the way that your paper is framed is something like, you might have these decision rules which are standing in for morality. But given that morality is a fractal, these are going to fail in certain ways, and so much the worse for morality. And the way that a lot of people see it is actually the opposite, where they go “well the world must have a morality which is not going to explode, so it must be this one that we’ve received or that we’ve already committed to.” So once you pick this decision rule, but it turns out that this edge case appears such that now the rule is ambiguous we don’t know what we do – we just eliminate the edge case, or we cover it up, or we destroy it. And that’s essentially the spirit of something like totalitarianism, which is: we have these rules, and anyone that dares – not even just anyone that dares defect from these rules, but anyone that dares make these rules incoherent must be eliminated. So all of the people that don’t fit a certain kind of mold, all the people that don’t genuflect in front of whatever the sacred ideology at the center of that set of rules is – is to be immediately concentrated to a certain domain so that other people don’t interact with them, and those edge cases don’t actually appear. So when I was working on Kurtis’s paper, I read a little bit of Arendt’s Origins of Totalitarianism, and this is one of the things that she continuously refers to: that what totalitarianism requires people to do is train them to believe in inconsistencies as if they’re at the center of the ideology. Everyone’s familiar with 1984 where Winston is required to believe that 2+2=5 – that is that the decision rule system is always generating true statements even if they are at some level arbitrary or at some level referring to something else. He has to commit to it so that all of those edge cases are eliminated or irrelevant.

I think that’s exactly the right way to think about totalitarianism: the attempt to bring everything into consistency with a central proposition. Whether that’s a political thing or whether that’s a moral proposition – like I would call Singerism a totalitarian moral system in the same sense as if you tried to make that a political center. And what’s interesting is the temptation of universalism and totalitarianism – maybe those are even the same thing. Why is that such a tempting thing to do, given that it has not always been a temptation, and given that the failure modes are so apparent? Why are we convinced that we just haven’t found the right principle yet?

Marcus: And closer to home for some of us coming from libertarian-ish circles, I’ve really realized that that’s what turned me off ultimately to someone like Rothbard: he’s is a libertarian totalitarian. The non-aggression principle is just his principle that’s supposed to have no fractal edges to it whatsoever. And so for the crowd that commits hard to Rothbard, when there are these edge cases which we’re not familiar with whether or not they fit the rules – I might be mischaracterizing this a little bit, but we push them out of society, or we only want people in the libertarian society whose actions with respect to the non-aggression principle are non-arbitrary or easily predictable. And then everything else is free. So it feels very much an ideology about freedom, but at the core since it has the totalitarian attitude, it ends up requiring certain things that seem like they’re almost the exact opposite of freedom, or very exclusionary at minimum.

It was interesting to me how – as a sociological question – the most hardline Rothbardians have been the ones that swung hardest for Trump out of anybody in the libertarian community. That was always puzzling, but I think you’ve landed on something important here: that arises from trying to make things consistent which are not, and cannot necessarily be made so.

Matt: Would you say that Mises’s action axiom is similar on that front? It’s a very logical axiomatic approach to science, but it is somewhat different in kind from Rothbard’s non-aggression principle in that the latter is an inference to say “given where I’ve started from, the non-aggression principle seems to be the only universally applicable rule that I could have.” I’m in libertarianism, and I’ve always sensed we seem to really prize logical consistency. And I’m okay with that, but I’m definitely not okay with this sort of commitment to conclusions that the evidence just does not bear out. and it’s easy to come up with exceptions to this rule of course.

Let’s take this in a Popper versus Kuhn direction in philosophy of science. Going back to the question about the temptation of the totalizing system, I think for even for myself historically, the sense has been that you have to have some sort of formal logical structure to have any thought at all. And to some extent that seems to be true, and Mises seems to be trying to set up a scaffolding for how we understand certain phenomena conceptually. But Rothbard’s trying to do a moral system. So the failure modes are a little more dire there than in a scientific system. But I see what you mean that even for a scientific system, or a theoretical system, the same issues would apply. What do you do with observations that don’t slot in easily into Mises’s taxonomy? You could do carveouts at the boundary, you could ignore them, you could assimilate them, but it’s not necessarily deterministic what to do with those kinds of observations.

Matt: This is why the edgelord is such a fascinating and aptly named character.

Literally at the edges of moral space.

Matt: I’m finding myself most fascinated by the latent asset pricing implications. You bring up the loan example, which is a very good motivation, but really that’s very close to what modern asset pricing does. You’ve got some signal, some risk factor, which can sometimes be the same thing as a return from a tradable portfolio, or it could just be some exogenous signal like GDP or employment or total consumption. And from that you can describe the statistical characteristics of returns among a whole big cross-section of assets by saying some are very exposed to this signal, meaning some rise and fall with this signal a lot and others do not, and some have a high level of idiosyncratic noise that’s not related to the signal, and others are earning something in excess like an alpha no matter what the signals are doing. This architecture is quite general. So thinking of these signals as something like “does or does not converge” based on where this signal arises, it could be quite applicable. So maybe it’s too big to go into further here given the confines of this discussion, but I do just want to get that out there that the possibilities are quite open there.

I have a couple of wild comments that are frankly kind of vague for me, but I want to vomit them out there. You mentioned this probability function p and the arguments are z, d, and ρ – z being the signal, d being this decision rule that tells you where it’s going to iterate to next, and ρ the resolution being the how fine-grained of an iteration we’re going to have. So in quantum mechanics, instead of a probability function being a primitive, there’s this thing called a probability amplitude. The probability amplitude is basically the value of this wave function that is much like a probability function, but how you use it to arrive at probability is you take this amplitude, which is a complex number – which is why I think this is relevant, is that the probability amplitude is a complex valued function. You got this complex valued variable being your signal, you can take a function of that, and you’ve got another complex number, and that can be a probability amplitude. And what you do is you take the modulus – that is, the magnitude or the absolute value – and you square that and that’s going to be equal to the probability. That’s at least the interpretation a lot of people use. And I’m mentioning that set of tools to say that there’s a rich set of results that pop out of it. Namely there’s a property known as non-commutativity: something is not commutative if there’s some operation A on B – it commutes if the output is the same as B on A, and it’s not commutative if instead you get some different value. And where this arises as important is that if you observe some signal A and then observe another signal B, the conclusion that you could draw from that is different from if you were to observe B and then A. So I’m just going to gesture to that and leave it there to say that there’s this rich set of similar other stuff that I think takes us down a very interesting rabbit hole. It remains right now kind of a speculative area of mathematics – this overlap and similarity between Gödelian incompleteness type results, which is which you guys are all talking about. It’s talked about in computer science. And I’ll share a paper with you Cameron – it’s called “From Heisenberg to Gödel via Chaitin” or something like that. It basically lays out how there are some necessary implications that have to do with Heisenberg’s uncertainty principle and Gödelian incompleteness. That may be a fruitful sort of path to go down.

Another thing is that there’s this thing called stereographic projection where a point in the complex plane can be mapped to a point on the surface of a three-sphere known as a Riemann sphere, a unit sphere. People use stereographic projection to do things like cartography – making maps and so forth – creating a two-dimensional representation of some geographic formation on a globe. And what’s neat about it is that you can take this signal that you’re thinking inhabits a two-dimensional space, and there’s a one-to-one mapping between it and a sphere, which allows you to contain more information or have a signal present more information. And that might be useful to people who want to put this idea into some kind of applied area. And again just gesturing toward the potential similarities between this and quantum mechanics and quantum computing is that a qubit – which is much like a bit in computing, a one or zero, this bit way of expressing information – the qubit is represented as a point on the Riemann sphere. So results in quantum computing may sort of have interesting parallels or connections to what you’re doing.

And so without going further into that I’ll just say why I think that’s so interesting from sort of a libertarian economic perspective: there’s this analogy – and maybe I’m being freewheeling with an analogy like you mentioned earlier – between the market and a method of computational calculation for determining which projects are viable, what their economic meaning is and what their economic value will be, whether one should or should not undertake a project. This is of course talked about a lot by Mises and Hayek and others in the calculation debate. And without rehashing any of that I’ll just say that information is really weird. Information has some unexpectedly strange properties that make mapping a signal to a decision a much weirder kind of a process than the standard classical computation analogy at first suggests. So perhaps the market is more like a quantum computer. Perhaps it can do operations that classical computers are slow or bad at very well. And again these are just kind of gestures and suggestions, but those tools and those ideas may be useful ways to sort of extend these Hayekian insights. And I see what you’re doing as a very important and necessary step on that travel.

A lot of interesting things to think about there. I know you’ve written about quantum wave functions. Could you send me something like that to give me a sense of the relevance of it? On the asset pricing question, this is one reason why I’m kind of sympathetic toward rational expectations as a project. And Roger Koppl who does similar things yelled at me for this one time because he’s not sympathetic toward Lucas and rational expectations. But I think the core insight of rational expectations is doing a kind of diagonalization argument to recognize the fact that the mapping from signals to expectations is not necessarily mechanical or deterministic, and that this is the same sort of incompleteness problem. Now rational expectations doesn’t cover all aspects of that problem, in the sense that it doesn’t really have room for disagreement, or – one of the other things that Roger’s been working on – is frame relativity. Rational expectations doesn’t capture that. But it does capture the non-algorithmic quality of mapping from signals to expectations.

The non-commutativity point is interesting because when I presented this at Markets and Society, somebody said “have you done anything with quaternions?” And I was like I don’t know, but those are non-commutative. So this might actually make me go and see if those are relevant, because that question of the order of the signals appearing would be an interesting way of operationalizing that.