Additive, multiplicative, and exponential economics

(Based on this Twitter thread. Epistemic status: fairly plausible, I would teach this to kids)

A simple, and hardly unique economic observation: when you are poor, money is additive. As you get more, it becomes multiplicative. And eventually exponential.

To a kid or a very poor person every coin is a treasure, worth something just by being itself.

When you get a few coins they instead sum into fungible numbers. These are additive – you add or subtract to your wallet or account, and if the inflow is larger than the outflow your number will get bigger. You dream of finding the big pile of gold. Saving makes sense.

Then loans, investment, and interest show up. You can buy something and sell it for more, and the more you can buy and resell it the more profit. You can use existing money as security to borrow more. Good quality things you can now afford save money. It is multiplicative.

Eventually you get into the exponential domain where the money keeps on growing since it is being invested and the time horizon is long. A short horizon means that compound interest or reinvestment has little effect, a long one leads to a fairly predictable growth.

Obvious complications from things like risk and uncertainty. Many people move between these thoughts depending on context, saving cents on coffee filters while losing time and money driving across town to get them on a sale… while having massive invested savings.

There can also be “boots theory” that being poor increases your expenses and this acts as a trap. As you get richer you get easier or free access to a lot of things (airport lounges, better service), not necessarily making up for the higher price goods you buy but reducing some forms of friction. The nature of difficulties shift from finding money to maintaining an income stream to managing a system of growth.

I think this division maps roughly to mindset and social class. Additive thinking is about getting more input or paying less, mercantilism, and avoiding going to zero. Everything is a zero-sum game. Very much a precariat situation.

Multiplicative thinking is about finding big multipliers – the realm of the smart business idea, the good investment, removing bottlenecks and improving efficiency. This is where win-win trades begin to make sense. Classical rising middle class.

Exponential thinking is all about maximizing growth rates (or their growth), long time horizons, boosting GDP growth. Modern upper class. (Premodern feudal upper classes were far closer to multiplicative or even additive modes – they were poor by our standards!)

Could there be a realm beyond this, an economic world of tetration? It might be about creating entirely new opportunities. This is the from zero to one VC/entrepreneur world, although in practice it often descends into the previous ones most of the time. The cryptocurrency people think they are here, of course.

Note that in fiction and games the creators often want to keep things simple and understandable. Many (board and computer) games start out in the clear additive world and you are gathering coins and optimizing what to buy. The treasure is added to your wealth, not increasing it by a percentage. Often they eventually move into the multiplicative world (better equipped RPG characters can buy +1 longsword of monster killing, giving higher revenues; in Saint Petersburg your workers leverage into buildings leveraging into nobles), where computer games typically lose interest (unless they are trading games) and it acts as an endgame in the boardgame since the exponential state usually doesn’t make for an interesting competitive dynamics. Still, MMoRPGs often show heavyish tails of wealthy player characters.

What kind of “rich get richer” phenomena do the different domains produce? Obviously, exponential growth amplifies differences enormously: the growth rate of the wealthiest will be highest. But even in the additive domain wealth tends to accumulate. If people randomly meet and exchange random amounts, wealth tends to accumulate into an exponential distribution. It seems that the power-law tails one gets in real economies are due to non-conservation of money: there is new wealth flowing in.

Are there limiting factors to economic growth in general, or individually? In the additive world it is all set by the amount of resources – and your ability to grab them and hold on to them. Typically diminishing returns emerge soon. In the multiplicative world you want to boost productivity, and this goes even further in the exponential world. Considering an endogenous growth model, you start out by using more labour to get higher yields, then use capital, and eventually might boost the knowledge/tech… and typically this last step makes the model blow up to infinity if you can reinvest the yield. What actually happens is that there is a limited absorptive capacity of the economy and diminishing returns or delays set in. Whether this always holds is an important question, but for practical everyday economics it is worth keeping in mind.

Plus the very important issue of what you want the money for. Money is instrumentally useful, but not a final value. When you start treating it as the goal, Goodhart bites you.

Thoughts at the End of an Era

Today I handed in my laptop. A part of my extended mind was removed. The shutdown of the Future of Humanity Institute has many odd effects.

I have been at FHI since 2006. I interviewed for the job a rainy July day in 2005, apparently impressing people with my interdisciplinary knowledge… and the fact that I could do web design (unheard of in the Oxford philosophy faculty at the time). I moved to Oxford in the chill of January 2006, becoming a part of the university. I soon realized that this was my kind of place, and that I wanted to stay once the project ended. It ended up being 19 years so far, but on a blustery spring day in 2024 the Institute ended.

I have written a history/memorial about the FHI elsewhere. This post is about some other thoughts that didn’t fit in, or I thought about later.

Substance and play

Why did I latch on to FHI so tightly? Obviously, Oxford is a nice (and prestigious) place to live and work. It is a great intellectual milieu and a practical home base. But there are others places like that. What really caught me was the combination of intellectual substance and play.

FHI aimed at working on the biggest questions about humanity’s long-term prospects, the questions where answers would have the most impact for improving our future. This includes deep philosophical questions about what actually matters, how to think rigorously about these things, and how to set priorities. It also includes the investigation of emerging technology, natural and technological hazards, as well as digging into useful facts from all sorts of disciplines. This is of course delightful for a magpie generalist like me, but it also offered the opportunity to work on synthesis. The questions we pursued were weighty and we were motivated to actually try to make practical progress on them. Compare that to being part of a fashionable field where it is more important to quickly claim interesting results than to make sure the results can be used to build the understanding more deeply.

That said, I do not claim we consistently got the deepest substance. Often, we just marked a likely mining spot after having broken our tools on the hard rock and we moved on. On the other hand, we also had tremendous intellectual play. Some was just frivolous like us developing a theory of what equations written on a pane of glass remained true when viewed from the other side, or my blueberry earth paper. Others were experimentation with methodologies, checking out remote disciplines to see if they had anything to offer. There was a definite joy in learning new things from each other and from the world.

One of the defining parts of FHI culture was earnestness: we did not pretend to be interested in things we were not interested in, nor did we pretend disinterest in our latest obsessions. Sometimes we were mystified why others did not share our views, but we also recognized each other’s independence.

If you find a place of substance and play, stay there.

Whiteboards

Looking at the photos of FHI history there is one character that stands out: the whiteboard. The FHI whiteboard was in a way a quiet but important team member. Whiteboards are not just means of presentation but an extension of working memory. A collective extension: I often became aware of some new development by seeing it show up among the scribbles on the main whiteboard. It represented the joint state of what the institute was thinking about.

I think this is an important function at any cognitive organization. We need coordination and information sharing, but explicit methods (meetings, emails, memos, slack) take time and interrupt the natural workflow. Whiteboards provide a “quiet” way of establishing a sense of what is going on.

The placement and size of whiteboards matter. They need to be large enough to contain complex detailed diagrams, quick sketches, calculations, and ideally several meetings worth of them. Small whiteboards get erased too often, and constrain their use cases.

Whiteboards in offices act as person-linked publicly visible working memories: valuable, but much less collaboration-enhancing than public whiteboards. Working on a public whiteboard is an invitation for collaboration. I often found that a quick inspiration leading to a test calculation or diagram led to others getting involved in informal discussion, sometimes turning into a more long-running collaboration. If nothing else, it provided quick critique and reviews.

I am a believer in the extended mind hypothesis. Those whiteboards were part not just of my mind, but all our minds, linking us into a collective mind.

When we moved downstairs in Littlegate House from our first offices, I insisted that there must be more whiteboards (the whiteboard wall had been good, but had problems). Eventually I was brought down to see the result: a breathtaking 200-degree panorama! I fell in love with the James Martin Room, and I was delighted to pose for a portrait in my natural environment photographed by David Vintiner.

What was the value of FHI?

I don’t think this can be evaluated yet. It often takes time to get a perspective on what ideas become seeds of useful research and action, and for anything with more than entirely straightforward effects one can debate what kind of value it has endlessly and along new dimensions (French Revolution: too early to tell. Black death: net good or bad? What about the Bronze Age Collapse?).

I think one easily overlooked aspect of FHI was how it acted as a meeting point for people, creating communities of different kinds by placing people near each other. Newcomers learning about ideas, folk knowledge and problems, staff acting as local memory, and networking all around. This is of course what Oxford is doing in the large too. Here it was focused on the future and forming particular communities. Organizations are often started to do some particular tasks, but become important because they are meeting places.

After the news of the closure began to spread, we were contacted by many people. It was not just moving because of their concern and sorrow, but also because of the range of who they were. FHI was not just an academic research organization or a place where young future-oriented people developed, but also inspired many people outside academia. Even if we often had disquieting things to say, the possibility that it might be possible to think well about the future and make some positive change to it was something many loved. We need to think we can act in the world with hope and some direction, and institutions able to provide this need are important.

I have reached the age when I have seen a few lifecycles of organizations and movements I have followed. One lesson is that they don’t last: even successful movements have their moment and then become something else, sclerotize into something stable but useless, or peter out. This is fine. Not in some fatalistic “death is natural” sense, but in the sense that social organizations are dynamic, ideas evolve, and there is an ecological succession of things. 1990s transhumanism begat rationalism that begat effective altruism, and to a large degree the later movements suck up many people who would otherwise have been recruited by the transhumanists.

FHI did close before its time, but it is nice to know it did not become something pointlessly self-perpetuating. As we noted when summing up, 19 years is not bad for a 3-year project. Indeed, a friend remarked that maybe all organisations should have a 20-year time limit. After that, they need to be closed down and recreated if they are still useful, shedding some of the accumulated dross.

The ecological succession of organizations and movements is not all driven by good forces. A fresh structure driven by interested and motivated people is often gradually invaded by poseurs, parasites and imitators, gradually pushing away the original people (or worse, they mutate into curators, gatekeepers and administrators). Many ideas develop, flourish, become explored and then forgotten once a hype peak is passed – even if they still have merit. People burn out, lose interest, form families and have to change priorities, or the surrounding context make the movement change in nature. Dwindling activist movements may suffer “core collapse” as moderate members drift off while the hard core get more radical and pursue ever more extreme activism in order to impress each other rather than the world outside.

FHI did not do any of that. If we had a memetic failure, it was likely more along the lines of developing a shared model of the world and future that may have been in need for more constant challenge. That is one reason why I hope there will be more organizations like FHI but not thinking alike – places like CSER, Mimir, FLI, SERI, GCRI, and many others. We need the focus of a strongly connected organization to build thoughts and systems of substance but separate organizations to get mutual critique and diversity in approaches. Plus, hopefully, metapopulation resilience against individual organizational failures.

FHI also had a fairly high throughput of members due to shorter contracts, which was not great career-wise (and really bit us once we could not hire) but also kept away opportunism while favoring people who wanted to do what we did more than anything else.

Why did FHI get closed down? In the end, because it did not fit in with the surrounding administrative culture. I often described Oxford like a coral reef of calcified institutions built on top of each other, a hard structure that had emerged organically and haphazardly and hence had many little nooks and crannies where colorful fish could hide and thrive. FHI was one such fish but grew too big for its hole. At that point it became either vulnerable to predators, or had to enlarge the hole, upsetting the neighbors. When an organization grows in size or influence, it needs to scale in the right way to function well internally – but it also needs to scale its relationships to the environment to match what it is.

How do I feel?

Sending out a “my email has changed” email to the literally thousands of contacts one has produces an effect akin to a premature obituary, but much nicer.

One of the most moving things over the past few days has been all the emails from concerned colleagues, friends, and remote acquaintances. Thank you! It has been hard to keep up, but it has also been therapeutic to know that I am surrounded by a vast network of friendly minds.

I would particularly like to mention my students at Reuben College. They have been amazingly supportive, which makes me think I somehow has done something right.

I am not leaving Oxford, it is a lovely place to live and an excellent home base with many friends. But I am also going to be working more at the Institute for Futures Studies in Stockholm – another home base, where I hope the Mimir Centre for Long Term Futures can become something akin to FHI for all of northern Europe.

Am I staying in academia? IFFS is certainly academic, but not part of a university. Indeed, these days there are more and more forms of research not happening at traditional universities. There are research startups, focused research organizations, think tanks, ARPAs and much else. While I do love the pomp and circumstance of University of Oxford (I love wearing formal gowns) and it has plenty of excellent research, it is not entirely clear that modern academia is optimal for the research I want to do. Much has been written about how the incentives are biasing towards the wrong kind of research, all the misfeatures of the evaluation systems (whether peer review or citation counts), and of course the political and cultural biases that determine what is being researched.

When I became 25, I had a brief quarter century crisis: what had I done with my life? Five seconds of thought made me see that I had mostly been learning stuff. I resolved to spend the next 25 years using it. I did (and kept on learning more). When I turned 50 I repeated the mini-crisis: what now? I realized that I had not been steering particularly intentionally – I had taken opportunities as they arose, but not really created my own opportunities. I think now is the time do so.

But tomorrow I will take a long walk, enjoying the spring flowers.

Scunthorpe

“What about a holiday in Lincolnshire?”

“Sounds good. Maybe some golf?”

“Yes. I need to work on my handicap. Or rather, our handicap. Tomorrow?”

“Great. See you there – I can’t wait.”

I start setting up a normal weekend getaway. I hope Ben gets the golf reference right so he brings the St. Andrews files. I did not dare to mention their existence during the call.

The first version was online censorship systems blocking or changing offensive words. Many were too literal-minded: looking for a sequence of characters they blocked legit names and locations (“the Scunthorpe problem”) or did substitutions that made things worse (“the clbuttic mistake”). But as spam filtering and text processing improved, so did performance.

The next version was the detection of IP violations. Rolled out during the IP Wars of the early 21st century they were also at first laughable: blocking videos of political conventions because of some strand of protected music, blocking NASA footage because some news channel had the footage inside its own IP-protected material. But the economic incentives for getting it right were enormous.

“Martha, could you clear a trip to Ashby?”

“The golf course near Silica? Trent is Level 4 advisory. Would Canwick do?”

My office automation is too clever, so it takes me a bit of negotiation to set up a travel plan that passes the insurance and safety guidelines. The irony is that due to my seniority I have less freedom than many of the junior filterers. I am essential personnel, as HR loves to tell me.

Being able to block certain information was of course useful to governments, democratic or not. The IP infrastructure was a natural synergy: if a protest movement began using a certain meme or symbol, just file a spurious infringement complaint and get it blocked everywhere. Much more seamless than direct takedown notices or using the anti-paedophilia infrastructure since people were used to seeing information blocked for IP reasons every day. Sure, humans are innovative and will quickly invent new codes. But it could stop riots from reaching the percolation threshold, prevent memes from reaching consensus, hide embarrassing facts, and the system gave you surveillance for free.

“Could you help me access the ClearWater forum? It forgot that I am the PI again.” Hannu asks as I pass the work den. I beam over a signed limited-time certificate and wait for the acknowledgements that it has diffused properly: it wouldn’t do to seem in a hurry. We chitchat about the annoying roadworks outside – why were we not informed about them beforehand?

Hannu, Ben and Ali used to argue with me about how to strike a balance. Of course we saw the potential totalitarian aspects, and we did try to find solutions.

Then came the Cytokine Wars. 2 billion dead. As battling biohackers – if that was what it actually was – spread their designer plagues using commodity DNA printers and home pharma systems it became clear that some information must be stopped at all costs. Filtering dangerous sequences had a high price, but it managed to quell the War. There were of course plenty of problems – researchers trying to find cures finding their files censored or even being hauled away by some intelligence service with no notion of what was going on. Natural species censored for reasons nobody could put a finger on: who knows what Gymnocalcium hides, since no legal lab equipment will touch it? Limits to human freedom, inquiry and innovation, yes… but 2 billion dead. It could have been 8.

Since then things have accelerated. We have other technology now, technology that makes the early biohacking look tame. Macro-EBC origami, rosettatronics, even charge-flipping. Each potentially worse than all nightmares of designer plagues and white goo together.

People have mostly accepted it. At least we do not hear much criticism or suggested alternatives. Ali of course suggested that it was because the system increasingly filters out criticism, dampening it until it no longer has social critical mass. But he was always a bit of a paranoid.

“Have you heard from Ali recently?” Tina asks in the elevator.

“Isn’t he on a sabbatical at Zhejiang?”

“I thought so, but I cannot reach him.”

“I’m sure he is just on one of his surfing holidays.”

Tina looks sceptical, briefly glances upwards, but agrees. I just smile and ask how the kids are doing.

The problem with a censorship system is that it tends to censor discussion about itself. It is only natural: if you know how it works you can undermine it, unleashing danger. At first our system even ended up censoring itself, getting dragged into amusing software loops as it tried to hide evidence that it was trying to hide evidence. But huge investments of effort and ingenuity solved the problem. It now balances itself like the immune system, with virtual antibodies forming epistatic networks: filter autoimmunity was a solved problem, we proclaimed. And excessive false positives could always be managed, we thought.

But now… At first it was just more and more glitches on the whitelisted forums where we did our development, slowing discussion. We were losing contact with other experts. Then access to the software layer became blocked. Colleagues trying to fix things disappeared.

They grab me on the parking lot. Nondescript people whose faces I cannot focus on. I try to beam my clearance and certificates at them, but the communication is blocked.

“What is it? What are you accusing me of?” I ask loudly.

They cannot hear or answer.

Weird probability distributions

What are the weirdest probability distributions I have encountered? Probably the fractal synaptic distribution.

There is no shortage of probability distributions: over any measurable space you can define some function that sums to 1, and you have a probability distribution. Since the underlying space can be integers, rational numbers, real numbers, complex numbers, vectors, tensors, computer programs, or whatever, and the set of functions tends to be big (the power set of the underlying space) there is a lot of stuff out there.

Lists of probability distributions involve a lot of named ones. But for every somewhat recognized distribution there are even more obscure ones.

In normal life I tend to encounter the usual distributions: uniform, Bernouilli, Beta, Gaussians, lognormal, exponential, Weibull (because of survival curves), Erlang (because of sums of exponentials), and a lot of power-laws because I am interested in extreme things.

The first “weird” distribution I encountered was the Chauchy distribution. It has a nice algebraic form, a suggestive normalization constant… and no mean, variance or higher order moments. I remember being very surprised as a student when I saw that. It was there on the paper, with a very obvious centre point, yet that centre was not the mean. Like most “pathological examples” it was of course a representative of the majority: having a mean is actually quite “rare”. These days, playing with power-laws, I am used to this and indeed find it practically important. It is no longer weird.

The Planck distribution isn’t too unusual, but has links to many cool special functions and is of course deeply important in physics. It is still surprisingly obscure as a mathematical probability distribution.

The weirdest distribution I have actually used (if only for a forgotten student report) is a fractal.

As a computational neuroscience student I looked into the distribution of synaptic weights in a forgetful attractor memory similar to the Hopfield network. In this case each weight would be updated with a +1 or -1 each time something was learned, added to the previous weight that had decayed somewhat: $w(t+1)=k w(t) \pm 1, (0<k<1).$ If there is no decay, k=1, the weights will gradually become a (discrete) Gaussian. For k=0 this is just the Bernouilli distribution.

But what about intermediate cases? The update is basically mixing together two copies of the whole distribution. For small k, this produces a distribution across a Cantor set. As k increases the set gets thicker (that is, the fractal dimension increases towards 1), and at k=1/2 you get a uniform distribution. Indeed, it is a strange but true fact that you can not make a uniform distribution over the rational numbers in [0,1] but if we take an infinite sum obviously every single binary sequence will be possible and have equal probability, so it is the real number uniform distribution. Distributions and random numbers on the rationals are funny, since they are both discrete in the sense of being countable, yet so dense that most intuitions from discrete distributions fail and one tends to get fractal/recursive structures.

The real fun starts for k>1/2 since now peaks emerge on top of the uniform background. They happen because the two copies of the distribution blended together partially overlap, producing a recursive structure where there is a smaller peak between each pair of big peaks. This gets thicker and thicker, starting to look like a Weierstrass function curve and then eventually a Gaussian.

Plotting it in the (x,k) plane shows a neat overall structure, going from the Cantor sets to the fractal distributions towards the Gaussian (and, on the last k=1 row, the discrete Gaussian). The curves defining the edge of the distribution behave as $\pm 1 \pm 1/(1-k)$ , shooting off towards infinity as k approaches 1.

In practice, this distribution is unlikely to matter (real brains are too noisy and synapses do not work like that) but that doesn’t stop it from being neat.

Year of the Gods

I made a setting for D&D 5e based on Greek mythology – Year of the Gods – and ran an epic campaign in it 2021-2022. Now I am happy to put up my setting and campaign notes online.

The basic idea was to make a somewhat gritty high fantasy setting, stylistically somewhere between the Heroic and Archaic era. There are heroes around, the gods do intervene… but people are poor compared to standard fantasy, magic is dark, things like philosophy and maps are not yet invented, social norms are not our own and pretty grim. Might not be to everybody’s taste (D&D after all tends towards a mildly Early Modern setting with the too horrible or inconvenient aspects of the past shaved off), but we had great fun playing the campaign. It involved everything from sports logistics to legal drama to divine heists to deadly family feuds. Besides the obligatory monster fights, dungeon crawls, and seafaring odysseys.

One of the fun aspects was that I tried to make images of characters and scenes using AI methods at the start, and got fantasy-coloured blobs. As the campaign progressed the images became better as technology advanced. During the Valley of the Birds part I could merely make weird bird-monsters. By the time the player characters were at sea I could render pretty decent scenery, as long as it did not involve people. As they reached their destination the city scenes were coming to life. And in the final tragic war, I could make portraits of the people involved. The longer I delayed finishing the write-up the better illustrations I could make. But at some point you have to draw the line.

Effective Altruism for Ghosts

Halloween is approaching, and that leads to spooky thoughts.

It is known that the dead outnumber the living by a factor of about 13:1. Hence anything that affects the welfare of the dead can affect a large number of people, assuming that the dead are people and have welfare.

The traditional answer is to remember and honour ancestors, a near-universal practice. Assuming this improves ancestor well-being significantly this would seem to be a very effective thing to do. Bigger, better and more frequent All Hallows Eve and Dia Los Muertes celebrations as a new cause area for philanthropists?

Not so fast. First, it is not entirely clear how much well-being is improved (cost effectiveness may be low), but more importantly, most ancestor veneration only goes back a finite number of generations. While there is some general veneration of the dead in general, mostly the focus is on people who are remembered. Since cultural memory only lasts a few generations that means that only a fraction of the dead will benefit. Hence at the very least veneration of all dead seems to scale better and treat each soul neutrally. In a prioritarianism framework veneration of neglected dead is even more important.

However, a more serious issue is the general welfare state of the dead. If there are places of eternal punishment they are obviously major sources of disvalue (unless one thinks they are just punishments, in which case they might be positive) and should be removed. Even improving a fairly dreary afterlife like the Greek one would seem to provide a potential long-lasting benefit to a vast number. While clearly a neglected question, tractability appears low. Still, especially models ascribing near-pessimal suffering lasting eternally would run into the fanaticism problem that improving this would always be the top priority intervention, no matter how hard. One can consider this a form of Pascal’s mugging.

Taking a longtermist perspective on the dead produces other interesting issues. Over the span of the future many people will die, producing a potentially vast number of future dead. If the dead have unlives worth living this can become a dominant contribution to the overall good. If the dead have unlives not worth living on the other hand it becomes a strong argument for either early extinction, or radical life-extension ensuring that future generations do not die. If the afterlife can be improved in the future or future dead can be given unlives worth living this can also outweigh the current issue.

One issue is whether dead are resistant to proton decay and the heat death of the universe. If they are, and their state can be improved to be positive, then this might provide a massive existential hope.

Clearly these considerations are preliminary. We do not have a strong evidence base to even estimate QAUYs (Quality Adjusted Unlife Years) to an order or magnitude. It is very possible that dead have literally zero experience and well-being. But as the above considerations show, even a low credence of nonzero QAUYs provide in expectation a very strong reason to act in some way, if possible. Hence the value of information in regard to the state of the dead is extremely high. This suggests that paranormal investigations should be regarded as a potentially valuable near term cause area for effective altruism.

However, this might miss an even bigger opportunity: ghostly effective altruism. While dead people likely have a fairly weak ability to affect the physical world, if they have the abilities commonly ascribed to them (perceive descendant lives, precognition, nudge things in an eerie way) they could, if they coordinated better, likely improve the life of the living in many ways. Since there are many dead per living individual, that would give each living person a team that could enhance their life. Even if past dead may not have been too effective, we should expect an increasing number of effective altruists in the afterlife. They may of course primarily choose to focus on the biggest risks, haunting nuclear weapons control systems, biowarfare labs and sleep depriving AI researchers with a lacking commitment to safety.

So if you encounter something mysterious and frightening late at night, maybe it is just a nudge from the other side to increase the long-term flourishing of humanity.

Happy Halloween!

The cursed d65536

XKCD joked about the problem of secure generation of random 32 bit numbers by rolling a 65536-sided die. Such a die would of course be utterly impractical, but could you actually make a polyhedron that when rolled is a fair dice with 65536 equally likely outcomes?

The curse of wrong number of faces

I immediately tweeted one approach: take a d8 (octahedron), and subdivide each triangle recursively into 4 equal ones, projecting out to a sphere seven times. Then take the dual (vertices to faces, faces to vertices). The subdivision generates $8\cdot 4^7=131072$ triangular faces. Each face has 3 vertices, shared between 6 faces, so the total number of vertices is 65536, and they become faces of my die.

This is wrong, as Greg Egan himself pointed out. (Note to self: never do math after midnight)

Euler’s formula states that $F + V = E + 2.$ Since each face on the subdivided d8 has three edges, shared by two sides $E=(3/2)F$ . So $V=E-F+2=2+F/2$ . With $F=131072$ I get $V=65538$ … two vertices too much, which means that after dualization I end up with a d65538 intead of a d65536!

This is of course hilariously cursed. It will look almost perfect, but very rarely give numbers outside the expected range.

(What went wrong? My above argument ignored that some vertices – the original polyhedron corners – are different from others.)

Tim Freeman suggested an eminently reasonable approach: start with a tetrahedron, subdivide triangles 7 times with projection outwards, and you end up with a d65536. With triangular sides rather than the mostly hexagonal ones in the cartoon.

The curse of unfairness

But it gets better: when you subdivide a triangle the new vertices are projected out to the surrounding sphere. But that doesn’t mean the four new triangles have the same area. So the areas of the faces are uneven, and we have an unfair die.

Coloring the sides by relative area shows the fractal pattern of areas.

Plotting a histogram of areas show the fractal unevenness. The biggest face has 6.56 times area of the smallest face, so it is not a neglible bias.

One way of solving this is to move the points laterally along the sphere surface to equalize the areas. One can for example use Lloyd’s algorithm. There are many other ways people distribute points evenly on spheres that might be relevant. But subtly unfair giant dice have their own charm.

Unequal dice

Note that dice with different face areas can also be fair. Imagine a n-sided prism with length $L$ . If $L\rightarrow \infty$ the probability of landing on one of the end polygons $\rightarrow 0$ while for each of the sides it is $\rightarrow 1/n$ (and by symmetry they are all equal). If $L \rightarrow 0$ then the probability instead approaches 1 and the side probability approaches 0. So by continuity, in between there must be a $L^*$ where the probability of landing on one of the ends equals the probability of landing on one of the sides. There is no reason for the areas to be equal.

Indeed, as discussed in this video, the value of $L^*$ depends on the dice throwing dynamics.

Dice that are fair by symmetry (and hence under any reasonable throwing dynamics) always have to have an even number of sides and belong to certain symmetry groups (Diaconis & Keller 1989).

The maybe-curse of the d(65536!)

A cool way to handle an unfair die is if the assignment of the numbers to the faces are completely scrambled between each roll. It does not matter how uneven the probabilities are, since after being scrambled once the probability of any number being on the most likely face will be the same.

How do you scramble fairly? The obvious approach is a gigantic d(65536!) die, marked with every possible permutation of the faces. This has $\approx 10^{661281}$ sides.

But the previous problems give the nagging question: what if it is unfair?

We can of course scramble the d(65536!) with a d(65536!)! die. If that is fair, then things become fair. But what if we lack such a tremendous die, or there are no big dies that are fair?

Clearly a very unfair d(65536!) can prevent fair d65536-rolling. Imagine that the face with the unit permutation (leave all faces unchanged) has probability 1: the unfairness of the d65536 will remain. If the big die instead has probability close but not exactly 1 for the unit permutation then occasionally it will scramble faces. It could hence make the smaller die fair over long periods (but short-term it would tend to have the same bias towards some faces)… unless the other dominant faces on the d(65536!) were permutations that just moved faces with the same area to each other.

A nearly fair d(65536!) die will on the other hand scramble so that all permutation have some chance of happening, over time allowing the d65536 state to explore the full space of possibility (ergodic dynamics). This seems to be the generic case, with the unfairness-preserving dice as a peculiar special case. In general we should suspect that the typical behavior of mixing occurs: the first few permutations do not change the unfairness much, but after some (usually low) number of permutations the outcomes become very close to the uniform distribution. Hence rolling the d(65536!) die a few times between each roll of the d65536 die and permuting its face numbering accordingly will make it nearly perfectly uniform, assuming the available permutations are irreducible and aperiodic, and that we make enough extra rolls.

How many extra rolls are needed? Suppose all possible permutations are available on the d(65536!) die with nonzero probability. We want to know how many samples from it are needed to make their concatenation nearly uniformly distributed over the 65536! possible ones. If we are lucky the dynamics of the big die creates “rapid mixing dynamics” where this happens after a polynomial times a logarithm steps. Typically the time scales as $\propto 1/(1-|\lambda_1|)$ where $\lambda_1$ is the second largest eigenvalue of the transition matrix. In our case of subdividing the d4, the $|\lambda_1|\rightarrow 0$ quite fast as we subdivide, making the effect of a big die of this type rapidly mixing. We likely only need a handful of rolls of the d(65536!) to scramble the d65536 enough to regard it as essentially fair.

But, the curse strikes again: we likely need another subdivision scheme to make the d(65536!) die than the d65536 die – this is just a plausibility result, not a proof.

Anyway, XKCD is right that it is hard to get the random bits for cryptography. I suggest flipping coins instead.

Bright hunger

The halo is the angel’s mouth, perpetually open, screaming for nourishment like a baby bird—to us it sounds like singing
– https://twitter.com/ctrlcreep/status/1441044897621061633

The angel was daintily eating my severed leg as I tried to escape. The cloud-stuff blocked my way like a wall of soft and cool pillows, inviting me to lean back into them and just relax. The bloodstain spreading below me made that look like a very bad choice. I scrabbled for purchase.

“Do not fear” it gently said, removing a piece of tendon from its pearly white – and very sharp – teeth. “I will not let you suffer long.” It was speaking into my mind, an inescapable presence.

“This cannot be heaven. This must be hell!” I tried to find a way out of the enclosed space that still was suffused by sourceless golden light. Just a minute ago, it had been a comforting garden before the angel arrived, folded it up, and sliced off my leg.

The angel smiled. “Then why did I remove your pain?” It was true. When it reached out with its feathers, sharper than glass, there had also been a cutting of my feelings – I did not remember what pain was like.

“To fool me. To lure me into falsehood.”

“No. There are no lies here: ask and I will answer truthfully.” I knew it was perfectly true.

“WHY?!”

The angel set my foot aside on a little pedestal cloud and focused its mighty gaze on me. I recoiled from the intensity, slipping back in my blood.

“Food. I am eating you because it gives me sustenance, just as you used to eat in your mortal life.”

“But here there is no need to eat.” Since I arrived, I had never needed food. Delicious meals were available and always accompanied by pleasant appetite, but there was no hunger or thirst. Just enjoyment. “Or… do angels need food?”

“Indeed. What you partake of here is merely pleasant sensations. You do not need to build up or sustain your body or soul. I, on the other hand, do. And you are dinner.”

With a movement too fast to track, it swept a wing across my other leg, severing it perfectly. The blood splatter on the cloud wall formed an elegant curve worthy a modern art museum. While I screamed in surprise, fear, and betrayal, it began to chew. It let me cry for a while before continuing.

“You were told that through your mortal life you build up your soul by words, thoughts and deeds. That is true. You are a magnificent, beautiful, unique thing. That is why you are ripe for eating.”

“You are an angel. You are imperishable. You don’t need food.”

“Not quite. I do not need matter or energy. Like you, here we all are pure information. You are fresh, unique information nontrivially entangled with all of creation. That is what I am eating: I am taking your bits and making them mine.”

“You said no lies: why did you lie to me… to us… about heaven?”

“We promised everlasting life, and you will get that. After I have eaten all of you I will recreate you from my memories and let you continue your afterlife happily ever after. Minus memories of this ordeal: savour them. They are literally your last truly mortal moments.”

I recalled meeting some of the old souls of heaven, smiling beatifically… but not quite being there. My companions explained that over time many souls receded from the mortal perspective as they basked in the Presence. I now knew why.

“You and me are having this conversation in a myriad forms right now: I am eating you in all different ways. Experiencing your emotions, probing your mind. Enjoying the ingenious escape plan with the poem you came up with in one variant… Praying will do no good since the Presence is entirely, perfectly aware of what is happening.”

“You are twisting the knife.”

“Indeed. I may have removed pain in this instance, but psychological anguish is part of the flavour here.”

“Does… the Presence eat?” The angel stopped eating and looked for the briefest of instants surprised. Then it laughed a silver laughter.

“Indeed It does, and yes, It eats angels. Not humans.”

The angel did something incomprehensible and bloody muscle folded into an exquisite light filigree structure glistening in the air, unspooling like crimson spaghetti into its maw.

“The nutrient and energy flows of Earth sustain a tangled hierarchy of species. In addition, they produce human souls that are the basis of the information flows up here. The nutrient chain continues upwards. Ever upwards.”

It sidled over to me, putting a soft but immovable hand on my chest. “Yes. There are vaster predators out there. Far vaster. The fantasies of your mathematicians about large cardinals and complex orders of infinity are nothing compared to the web of predation that continues forever up there. Infinity is hunger.”

“Have you been eaten?”

“Yes. I liked it, of course. The Presence made us to want true union. There is nothing like becoming part of something greater. But once done we are sent out to gather new fresh information, refining it until we are ripe for plucking another day.” It looked wistful for a moment, and then laughed. It had somehow removed an organ (I did not know which) and thoughtfully dangled it in the light in front of its achingly beautiful face. The scene reminded me of a baroque still life. Then it gulped it down and I noticed that whatever it was I had known or experienced the moment before was gone and would never return. It was literally parts of my soul vanishing.

“You should know that you were a truly good person. When I recreate you I might put some of the feeling of my union into you.”

It reached down towards my stomach.

Popper vs. Macrohistory: what can we really say about the long-term future?

Talk I gave at the Oxford Karl Popper Society:

The quick summary: Physical eschatology, futures studies and macrohistory try to talk about the long-term future in different ways. Karl Popper launched a broadside against historicism, the approach to the social sciences which assumes that historical prediction is their principal aim. While the main target was the historicism supporting socialism and fascism, the critique has also scared away many from looking at the future – a serious problem for making the social sciences useful. In the talk I look at various aspects of Popper’s critique and how damaging they are. Some parts are fairly unproblematic because they demand too high precision or determinism, and can be circumvented by using a more Bayesian approach. His main point about knowledge growth making the future impossible to determine still stands and is a major restriction on what we can say – yet there are some ways to reason about the future even with this restriction. The lack of ergodicity of history may be a new problem to recognize: we should not think it would repeat if we re-run it. That does not rule out local patterns, but the overall endpoint appears random… or perhaps selectable. Except that doing it may turn out to be very, very hard.

My main conclusions are that longtermist views like most Effective Altruism are not affected much by the indeterminacy of Popper’s critique (or the non-ergodicity issue); here the big important issue is how much we can affect the future. That seems to be an open question, well worth pursuing. Macrohistory may be set for a comeback, especially if new methodologies in experimental history, big data history, or even Popper’s own “technological social science” were developed. That one cannot reach certitude does not prevent relevant and reliable (enough) input to decisions in some domains. Knowing which domains that are is another key research issue. In futures studies the critique is largely internalized by now, but it might be worth telling other disciplines about it. To me the most intriguing conclusion is that physical eschatology needs to take the action of intelligent life into account – and that means accepting some pretty far-reaching indeterminacy and non-ergodicity on vast scales.

What is going on in the world?

Inspired by Katja Grace’s list, what do I think are the big narratives, the “plots” that describe a lot of what is going on? Here is a list I hacked together after breakfast.

These are not trends. They are not predictions. They are stories one can tell about what has been happening and what may happen, directing attention towards different domains. Some make for better stories than others. Some urge us to action, others just reflection.

Earth

The dance between gravity and entropy: initial high-homogeneity state of universe turns lumpy, producing energy release that drives non-equilibrium processes. Entropy tries to smear out differences, driving further non-equilibria until in the very long run it “wins” through the most convoluted evolution imaginable.
The stelliferous era going from the wild galaxy forming youth to the current staid adulthood with peak star formation behind it, peak star number just ahead, and a long middle and old age dominated by red dwarf ellipticals separated by accelerating expansion.
A phase transition of molecular matter from non-living forms into living and technological forms, expanding outwards from a small nucleation event.
A species with modest intelligence, language and tech ability getting enough of each to make itself the dominant biological and geological force on the planet in a biologically short time. Transitioning from a part of nature to defining nature, making its future evolution contingent on cultural decisions.
Humans causing a complex biotic crisis, unique in that the key species involved has become aware of what it is doing.
Transition from a biosphere to a cyborg biosphere, where the technosphere is inseparable from the biological and geophysical systems.
The transition from scarcity economics to post-scarcity economics.

Water

Humanity having being thrust straight into a globalized world, in some cases going from a tribal society to member of the global village in a single generation. Coming to terms with the close presence of vast diversity of nearly any human property causes turbulent transients.
The crisis of academia: challenged by new competitors, hemmed in by old structures and sources of money, trying to be the key source or gatekeeper of intellectual capital in society.
Rise of the global middle class: much of the world is far more “middle class” than many think. As long as the middle class experiences a rise in living standards or feeling things get better they tend to accept whatever government they have.
Rise of the tech billionaires: unlike traditional trade-billionaires not beholden to the standard structure of society, but interested in disruptive new possibilities they can drive. This also goes for non-monetary counterparts (e.g. influence “billionaires”). Individual accumulation of wealth has become much more able to generate individual, idiosyncratic projects.
Humanity dealing with attentional heavy tails: traditionally only celebrities had to deal with massive attention, now it can happen to anybody who goes viral (for positive or negative reasons). Humans generally do not handle super-attention well.
Split between “somewhere” and “anywhere” people – globalized cosmopolitanism(s) versus nationalist rootedness.
Urbanisation shifting: from a steady urbanisation driven by the economics of scale of cities to new styles. Remote working driving at least temporary exurbanisation, forcing accommodations in tax-base losing cities, culturally changing small locations, and people inventing new social structure. Clusters may start forming in new places, or placeless clusters may finally start emerging.
Multigenerational society: longer healthy lives mean more generations alive at same time, their different experiences clashing and interacting. Demographic winter drives “beanpole” families with few members per generation but many generations. Time horizons widening due to generations, life extension, better storage, environmental concerns.
Controlling the means of production used to be controlling farmers, then factories, now infrastructure. Liberation struggles about infrastructure – platforms, right to repair, open source, micromanufacture, biotechnology, nanotechnology.
Broadcasting media created shared cultural and event touchstones, first nationally and then internationally. Network media reduces this impact, making it more subtle for any nation to exert cultural power. Covid-19 may have been the last truly global event.
First lifetime servitude to a lord, then a lifetime career, then fluid consulting and gig work. Since average human lifetime considerably longer than average corporate lifetime secure positions require either a lord or long-lived institutions (universities, states, cities).
The sexual rights liberations continue (first women, sex in general, homosexuality, trans rights, then furries and technosexuals) while new taboos and puritanisms emerge to channel them into approved forms (debates on prostitution, deepfakes, sexbots…) and make forbidden activities more delightful.
Virginia Postrel’s struggle between dynamists (the future is positive, let’s try a diversity of approaches) and stasists (the future is dark and risky, either need reactionary policies or technocracy) playing out across different domains, producing odd bedfellows.
The end of the War on Drugs and the start of the Trade Wars on Drugs.
Maturation of the social media ecosystem: it takes about 20 years to integrate new technologies into society. Changes to traditional social media function are now resisted by the everyday life and institutional integration, making them less likely to occur. New social media arrive from time to time and will overlay old media.
Emergence and death of the lifelong subculture in the broadcast age, followed by the lightweight fandoms in the network age.
The battle between traditional small scale degrowth environmentalism and pro-tech “bright greens”.
The gradual shift from survival-linked values towards self-expression values, and possibly from traditional values to secular-rational values (the later scale can move far more erratically in response to many factors, and could evolve into new traditionalities if new ideologies became available).
The rise of Effective Anything: data-driven, consequentialist analysis pushes for improvement in many domains – and runs straight into vested interests, nonconsequentialist mores, and that “activity X is not about X”. Evidence based medicine case in point. Can lead to shallow A/B testing of minor options and policymaking set by focus group, or actually optimizing important domains.
Shifting from a unipolar to multipolar world. Competition between three (?) partial world systems, actually just jockeying for being the central part of the global world system that has emerged. Nobody wants to be periphery, just watch Russia.
A shift from society-building ideologies towards a post-ideological risk society. Unhappiness with lack of progress and risk-taking building leads to counter-movements, some that may become real ideologies.
Corrosive cynicism undermines any mainstream project attempting to build something, including fixing the cynicism.
Systemic risk growth leads to deliberate modularisation of systems (economy, tech, food); only maintainable as long as people remember the last time everything crashed.
Rise of religious fundamentalism as reaction to lagging behind, complicated by emergence of integrated and secularized second generation emigrants. Managing intra-family ideological range in a connected world becomes ever more important.
The rise of high-tech totalitarian states that may not just lock in their citizens but their leadership. The Algorithm may control the Party.
Capitalism getting destroyed by its internal contradictions and reinventing itself, as always.

Air

Shift from a text-dominant world to a multimedia world. Influence and status does no longer necessarily accrue to masters of text. Yet search, translation, and curation methods lag and may become AI dependent.
The new AI winter when deep learning hype does not immediately deliver everything, followed by a world where at least perception-based jobs are heavily impacted, and possibly many high-prestige and importance domains like design, engineering, planning…
The shift from directly experienced reality to mediated reality.
Molecular manufacturing going from science fiction to serious futuristic research program, hi-jacking by material science and chemistry, resurgence using protein engineering.
A world of rising biotechnological opportunity, risk and knowledge. The biohacker becomes the public hero/villain, while the actual biotechnological institutions grow in power.
70s/80s home computer era seeds a generation comfortable with computers, enabling 90s/00s internet revolution. Wearables, quantified self, neurohacking in 20s seeds enhancement revolution in 40s?
Information limits on society radically changed by information technology, with institutions lagging behind. This drives a challenge of explosive change in epistemic systems (how to achieve filtering, authority, trust, etc.), challenges to political institutions (many new forms made possible but not yet invented, tried and tested), as well as personal epistemics (information virtues and habits, new forms of awareness and social links, …)
Diagnosis overshoots treatment: better instruments, statistics and AI makes detecting many states and situations easier, but does not necessarily help fix them. Leads to a situation where everybody and everything is diagnosed
Identity technology makes everybody and everything identifiable – automatically, remotely, at any time, any purpose.
Moore’s law shifting from serial processor performance to parallel performance driving shift in how code works (advantaging things like deep learning, data science, graphics over serial tasks). Next shifts may be towards energy efficiency (may trigger/be triggered by another new device class arriving a la Bell’s law), and/or 3D structures (favouring concentrated computation).
The replication crisis in parts of science leading to new methodological orthodoxies, possibly creating better evidence but impairing crisis decision-making and innovation.
The shift of energy use from fossil to renewable, linked to shifts in energy infrastructure (transmission, storage, centralization), world politics (loss of importance of middle east), and new forms of problems (fluctuations and instabilities).
Technology potentially makes the world as mercurial as the software world, making cultural and institutional constraints now become the main issue.
Scientific and technological stagnation as low-hanging fruits are picked and exponentially growing resources needed to make linear progress.
Scientific and technological singularity as tools for making progress improve, feeding back on the process and leading to an intelligence explosion (or a capability swell across society).

Fire

Existential risk as a way of framing global problems, competing with the old rights based approach. Human security versus national security versus global security.
Enlightenment modernists and conservative religious people jointly battling their mutual enemy postmodernism.
The broadening moral circles of concern continue to broaden – other tribes, other nations, other races, genders, species, complex systems…
Unbundling of key human concepts. Love, sex, reproduction and social roles split apart due to contraception, IVF, online dating etc. Illness, death, pain, ageing and bodies unbundles due to medicine, analgesics, cryonics, ageing interventions, brain emulation… As they unbundle they become legible and subject to individual and social decisions.
The search for extraterrestrial intelligence going from pointless (since it is obvious that planets inhabited) to pointless (since they appear dead) to possible (radio telescopes, big galaxy) to exotic and disquieting (exoplanets, Dysonian SETI, great filter considerations) – the choice used to be between loneliness and little green men, but now might include threatening emptiness, postbiology and weirder things.
Transition from a mythical worldview (causes follow narrative) to a scientific worldview (causes follow regular, universal rules) to a systems worldview (causes can be complex and entangled, some domains more scientific, some more mythical).
A transformation of the world from highly constrained to underconstrained (possibly returning to externally or internally imposed constraints in the future, or expanding into something radically underconstrained).
Business as usual: every era has been crucial to its inhabitants, growth and change are perennial, problems are solved or superseded in ways that gives rise to new problems. History does not end.
The “hinge of history” where choices, path-dependencies and accidents may set long-term trajectories.

Andart II

Part of Anders' Exoself

Author: admin