The game theory of holidays

Dec 25, JDN 2457748

When this post goes live, it will be Christmas; so I felt I should make the topic somehow involve the subject of Christmas, or holidays in general.

I decided I would pull back for as much perspective as possible, and ask this question: Why do we have holidays in the first place?

All human cultures have holidays, but not the same ones. Cultures with a lot of mutual contact will tend to synchronize their holidays temporally, but still often preserve wildly different rituals on those same holidays. Yes, we celebrate “Christmas” in both the US and in Austria; but I think they are baffled by the Elf on the Shelf and I know that I find the Krampus bizarre and terrifying.

Most cultures from temperate climates have some sort of celebration around the winter solstice, probably because this is an ecologically important time for us. Our food production is about to get much, much lower, so we’d better make sure we have sufficient quantities stored. (In an era of globalization and processed food that lasts for months, this is less important, of course.) But they aren’t the same celebration, and they generally aren’t exactly on the solstice.

What is a holiday, anyway? We all get off work, we visit our families, and we go through a series of ritualized actions with some sort of symbolic cultural meaning. Why do we do this?

First, why not work all year round? Wouldn’t that be more efficient? Well, no, because human beings are subject to exhaustion. We need to rest at least sometimes.

Well, why not simply have each person rest whenever they need to? Well, how do we know they need to? Do we just take their word for it? People might exaggerate their need for rest in order to shirk their duties and free-ride on the work of others.

It would help if we could have pre-scheduled rest times, to remove individual discretion.

Should we have these at the same time for everyone, or at different times for each person?

Well, from the perspective of efficiency, different times for each person would probably make the most sense. We could trade off work in shifts that way, and ensure production keeps moving. So why don’t we do that?
Well, now we get to the game theory part. Do you want to be the only one who gets today off? Or do you want other people to get today off as well?

You probably want other people to be off work today as well, at least your family and friends so that you can spend time with them. In fact, this is probably more important to you than having any particular day off.

We can write this as a normal-form game. Suppose we have four days to choose from, 1 through 4, and two people, who can each decide which day to take off, or they can not take a day off at all. They each get a payoff of 1 if they take the same day off, 0 if they take different days off, and -1 if they don’t take a day off at all. This is our resulting payoff matrix:

1 2 3 4 None
1 1/1 0/0 0/0 0/0 0/-1
2 0/0 1/1 0/0 0/0 0/-1
3 0/0 0/0 1/1 0/0 0/-1
4 0/0 0/0 0/0 1/1 0/-1
None -1/0 -1/0 -1/0 -1/0 -1/-1


It’s pretty obvious that each person will take some day off. But which day? How do they decide that?
This is what we call a coordination game; there are many possible equilibria to choose from, and the payoffs are highest if people can somehow coordinate their behavior.

If they can actually coordinate directly, it’s simple; one person should just suggest a day, and since the other one is indifferent, they have no reason not to agree to that day. From that point forward, they have coordinated on a equilibrium (a Nash equilibrium, in point of fact).

But suppose they can’t talk to each other, or suppose there aren’t two people to coordinate but dozens, or hundreds—or even thousands, once you include all the interlocking social networks. How could they find a way to coordinate on the same day?

They need something more intuitive, some “obvious” choice that they can call upon that they hope everyone else will as well. Even if they can’t communicate, as long as they can observe whether their coordination has succeeded or failed they can try to set these “obvious” choices by successive trial and error.

The result is what we call a Schelling point; players converge on this equilibrium not because there’s actually anything better about it, but because it seems obvious and they expect everyone else to think it will also seem obvious.

This is what I think is happening with holidays. Yes, we make up stories to justify them, or sometimes even have genuine reasons for them (Independence Day actually makes sense being on July 4, for instance), but the ultimate reason why we have a holiday on one day rather than other is that we had to have it some time, and this was a way of breaking the deadlock and finally setting a date.

In fact, weekends are probably a more optimal solution to this coordination problem than holidays, because human beings need rest on a fairly regular basis, not just every few months. Holiday seasons now serve more as an opportunity to have long vacations that allow travel, rather than as a rest between work days. But even those we had to originally justify as a matter of religion: Jews would not work on Saturday, Christians would not work on Sunday, so together we will not work on Saturday or Sunday. The logic here is hardly impeccable (why not make it religion-specific, for example?), but it was enough to give us a Schelling point.

This makes me wonder about what it would take to create a new holiday. How could we actually get people to celebrate Darwin Day or Sagan Day on a large scale, for example? Darwin and Sagan are both a lot more worth celebrating than most of the people who get holidays—Columbus especially leaps to mind. But even among those of us who really love Darwin and Sagan, these are sort of half-hearted celebrations that never attain the same status as Easter, much less Thanksgiving or Christmas.

I’d also like to secularize—or at least ecumenicalize—the winter solstice celebration. Christianity shouldn’t have a monopoly on what is really something like a human universal, or at least a “humans who live in temperate climates” universal. It really isn’t Christmas anyway; most of what we do is celebrating Yule, compounded by a modern expression in mass consumption that is thoroughly borne of modern capitalism. We have no reason to think Jesus was actually born in December, much less on the 25th. But that’s around the time when lots of other celebrations were going on anyway, and it’s much easier to convince people that they should change the name of their holiday than that they should stop celebrating it and start celebrating something else—I think precisely because that still preserves the Schelling point.

Creating holidays has obviously been done before—indeed it is literally the only way holidays ever come into existence. But part of their structure seems to be that the more transparent the reasons for choosing that date and those rituals, the more empty and insincere the holiday seems. Once you admit that this is an arbitrary choice meant to converge an equilibrium, it stops seeming like a good choice anymore.

Now, if we could find dates and rituals that really had good reasons behind them, we could probably escape that; but I’m not entirely sure we can. We can use Darwin’s birthday—but why not the first edition publication of On the Origin of Species? And Darwin himself is really that important, but why Sagan Day and not Einstein Day or Niels Bohr Day… and so on? The winter solstice itself is a very powerful choice; its deep astronomical and ecological significance might actually make it a strong enough attractor to defeat all contenders. But what do we do on the winter solstice celebration? What rituals best capture the feelings we are trying to express, and how do we defend those rituals against criticism and competition?

In the long run, I think what usually happens is that people just sort of start doing something, and eventually enough people are doing it that it becomes a tradition. Maybe it always feels awkward and insincere at first. Maybe you have to be prepared for it to change into something radically different as the decades roll on.

This year the winter solstice is on December 21st. I think I’ll be lighting a candle and gazing into the night sky, reflecting on our place in the universe. Unless you’re reading this on Patreon, by the time this goes live, you’ll have missed it; but you can try later, or maybe next year.

In fifty years all the cool kids will be doing it, I’m sure.

Student debt crisis? What student debt crisis?

Dec 18, JDN 2457741
As of this writing, I have over $99,000 in student loans. This is a good thing. It means that I was able to pay for my four years of college, and two years of a master’s program, in order to be able to start this coming five years of a PhD. When I have concluded these eleven years of postgraduate education and incurred six times the world per-capita income in debt, what then will become of me? Will I be left to live on the streets, destitute and overwhelmed by debt?

No. I’ll have a PhD. The average lifetime income of individuals with PhDs in the United States is $3.4 million. Indeed, the median annual income for economists in the US is almost exactly what I currently owe in debt—so if I save well, I could very well pay it off in just a few years. With an advanced degree in economics like mine, or similarly high-paying fields such as physics, medicine, and law one can expect the higher end of that scale, $4 million or more; with a degree in a less-lucrative field such as art, literature, history, or philosophy, one would have to settle for “only” say $3 million. The average lifetime income in the US for someone without any college education is only $1.2 million. So even in literature or history, a PhD is worth about $2 million in future income.

On average, an additional year of college results in a gain in lifetime future earnings of about 15% to 20%. Even when you adjust for interest rates and temporal discounting, this is a rate of return that would make any stock trader envious.

Fitting the law of diminishing returns, the rates of return on education in poor countries are even larger, often mind-bogglingly huge; the increase in lifetime income from a year of college education in Botswana was estimated at 38%. This implies that someone who graduates from college in Botswana earns four times as much money as someone who only finished high school.

We who pay $100,000 to receive an additional $2 to $3 million can hardly be called unfortunate.

Indeed, we are mind-bogglingly fortunate; we have been given an opportunity to better ourselves and the society we live in that is all but unprecedented in human history granted only to a privileged few even today. Right now, only about half of adults in the most educated countries in the world (Canada, Russia, Israel, Japan, Luxembourg, South Korea, and the United States) ever go to college. Only 30% of Americans ever earn a bachelor’s degree, and as recently as 1975 that figure was only 20%. Worldwide, the majority of people never graduate from high school. The average length of schooling in developing countries today is six yearsthat is, sixth grade—and this is an enormous improvement from the two years of average schooling found in developing countries in 1950.

If we look a bit further back in history, the improvements in education are even more staggering. In the United States in 1910, only 13.5% of people graduated high school, and only 2.7% completed a bachelor’s degree. There was no student debt crisis then, to be sure—because there were no college students.

Indeed, I have been underestimating the benefits of education thus far, because education is both a public and private good. The figures I’ve just given have been only the private financial return on education—the additional income received by an individual because they went to college. But there is also a non-financial return, such as the benefits of working in a more appealing or exciting career and the benefits of learning for its own sake. The reason so many people do go into history and literature instead of economics and physics very likely has to do with valuing these other aspects of education as highly as or even more highly than financial income, and it is entirely rational for people to do so. (An interesting survey question I’ve alas never seen asked: “How much money would we have to give you right now to convince you to quit working in philosophy for the rest of your life?”)

Yet even more important is the public return on education, the increased productivity and prosperity of our society as a result of greater education—and these returns are enormous. For every $1 spent on education in the US, the economy grows by an estimated $1.50. Public returns on college education worldwide are on the order of 10%-20% per year of education. This is over and above the 15-20% return already being made by the individuals going to school. This means that raising the average level of education in a country by just one year raises that country’s income by between 25% and 40%.

Indeed, perhaps the simplest way to understand the enormous social benefits of education is to note the strong correlation between education level and income level. This graph comes from the UN Human Development Report Data Explorer; it plots the HDI education index (which ranges from 0, least educated, to 1, most educated) and the per-capita GDP at purchasing power parity (on a log scale, so that each increment corresponds to a proportional increase in GDP); as you can see, educated countries tend to be rich countries, and vice-versa.


Of course, income drives education just as education drives income. But more detailed econometric studies generally (though not without some controversy) show the same basic result: The more educated a country’s people become, the richer that country becomes.

And indeed, the United States is a spectacularly rich country. The figure of “$1 trillion in college debt” sounds alarming (and has been used to such effect in many a news article, ranging from the New York Daily News, Slate, and POLITICO to USA Today and CNN all the way to Bloomberg, MarketWatch, and Business Insider, and even getting support from the Consumer Financial Protection Bureau and The Federal Reserve Bank of New York).

But the United States has a total GDP of over $18.6 trillion, and total net wealth somewhere around $84 trillion. Is it really so alarming that our nation’s most important investment would result in debt of less than two percent of our total nation’s wealth? Democracy Now asks who is getting rich off of $1.3 trillion in student debt? All of us—the students especially.

In fact, the probability of defaulting on student loans is inversely proportional to the amount of loans a student has. Students with over $100,000 in student debt default only 18% of the time, while students with less than $5,000 in student debt default 34% of the time. This should be shocking to those who think that we have a crisis of too much student debt; if student debt were an excess burden that is imposed upon us for little gain, default rates should rise as borrowing amounts increase, as we observe, for example, with credit cards: there is a positive correlation between carrying higher balances and being more likely to default. (This also raises doubts about the argument that higher debt loads should carry higher interest rates—why, if the default rate doesn’t go up?) But it makes perfect sense if you realize that college is an investment—indeed, almost certainly both the most profitable and the most socially responsible investment most people will ever have the opportunity to make. More debt means you had access to more credit to make a larger investment—and therefore your payoff was greater and you were more likely to be able to repay the debt.

Yes, job prospects were bad for college graduates right after the Great Recession—because it was right after the Great Recession, and job prospects were bad for everyone. Indeed, the unemployment rate for people with college degrees was substantially lower than for those without college degrees, all the way through the Second Depression. The New York Times has a nice little gadget where you can estimate the unemployment rate for college graduates; my hint for you is that I just said it’s lower, and I still guessed too high. There was variation across fields, of course; unsurprisingly computer science majors did extremely well and humanities majors did rather poorly. Underemployment was a big problem, but again, clearly because of the recession, not because going to college was a mistake. In fact, unemployment for college graduates (about 9%) has always been so much lower than unemployment for high school dropouts that the maximum unemployment rate for young college graduates is less than the minimum unemployment rate for young high school graduates (10%) over the entire period since the year 2000. Young high school dropouts have fared even worse; their minimum unemployment rate since 2000 was 18%, while their maximum was a terrifying Great Depression-level of 32%. Education isn’t just a good investment—it’s an astonishingly good investment.

There are a lot of things worth panicking about, now that Trump has been elected President. But student debt isn’t one of them. This is a very smart investment, made with a reasonable portion of our nation’s wealth. If you have student debt like I do, make sure you have enough—or otherwise you might not be able to pay it back.

What good are macroeconomic models? How could they be better?

Dec 11, JDN 2457734

One thing that I don’t think most people know, but which immediately obvious to any student of economics at the college level or above, is that there is a veritable cornucopia of different macroeconomic models. There are growth models (the Solow model, the Harrod-Domar model, the Ramsey model), monetary policy models (IS-LM, aggregate demand-aggregate supply), trade models (the Mundell-Fleming model, the Heckscher-Ohlin model), large-scale computational models (dynamic stochastic general equilibrium, agent-based computational economics), and I could go on.

This immediately raises the question: What are all these models for? What good are they?

A cynical view might be that they aren’t useful at all, that this is all false mathematical precision which makes economics persuasive without making it accurate or useful. And with such a proliferation of models and contradictory conclusions, I can see why such a view would be tempting.

But many of these models are useful, at least in certain circumstances. They aren’t completely arbitrary. Indeed, one of the litmus tests of the last decade has been how well the models held up against the events of the Great Recession and following Second Depression. The Keynesian and cognitive/behavioral models did rather well, albeit with significant gaps and flaws. The Monetarist, Real Business Cycle, and most other neoclassical models failed miserably, as did Austrian and Marxist notions so fluid and ill-defined that I’m not sure they deserve to even be called “models”. So there is at least some empirical basis for deciding what assumptions we should be willing to use in our models. Yet even if we restrict ourselves to Keynesian and cognitive/behavioral models, there are still a great many to choose from, which often yield inconsistent results.

So let’s compare with a science that is uncontroversially successful: Physics. How do mathematical models in physics compare with mathematical models in economics?

Well, there are still a lot of models, first of all. There’s the Bohr model, the Schrodinger equation, the Dirac equation, Newtonian mechanics, Lagrangian mechanics, Bohmian mechanics, Maxwell’s equations, Faraday’s law, Coulomb’s law, the Einstein field equations, the Minkowsky metric, the Schwarzschild metric, the Rindler metric, Feynman-Wheeler theory, the Navier-Stokes equations, and so on. So a cornucopia of models is not inherently a bad thing.

Yet, there is something about physics models that makes them more reliable than economics models.

Partly it is that the systems physicists study are literally two dozen orders of magnitude or more smaller and simpler than the systems economists study. Their task is inherently easier than ours.

But it’s not just that; their models aren’t just simpler—actually they often aren’t. The Navier-Stokes equations are a lot more complicated than the Solow model. They’re also clearly a lot more accurate.

The feature that models in physics seem to have that models in economics do not is something we might call nesting, or maybe consistency. Models in physics don’t come out of nowhere; you can’t just make up your own new model based on whatever assumptions you like and then start using it—which you very much can do in economics. Models in physics are required to fit consistently with one another, and usually inside one another, in the following sense:

The Dirac equation strictly generalizes the Schrodinger equation, which strictly generalizes the Bohr model. Bohmian mechanics is consistent with quantum mechanics, which strictly generalizes Lagrangian mechanics, which generalizes Newtonian mechanics. The Einstein field equations are consistent with Maxwell’s equations and strictly generalize the Minkowsky, Schwarzschild, and Rindler metrics. Maxwell’s equations strictly generalize Faraday’s law and Coulomb’s law.
In other words, there are a small number of canonical models—the Dirac equation, Maxwell’s equations and the Einstein field equation, essentially—inside which all other models are nested. The simpler models like Coulomb’s law and Newtonian mechanics are not contradictory with these canonical models; they are contained within them, subject to certain constraints (such as macroscopic systems far below the speed of light).

This is something I wish more people understood (I blame Kuhn for confusing everyone about what paradigm shifts really entail); Einstein did not overturn Newton’s laws, he extended them to domains where they previously had failed to apply.

This is why it is sensible to say that certain theories in physics are true; they are the canonical models that underlie all known phenomena. Other models can be useful, but not because we are relativists about truth or anything like that; Newtonian physics is a very good approximation of the Einstein field equations at the scale of many phenomena we care about, and is also much more mathematically tractable. If we ever find ourselves in situations where Newton’s equations no longer apply—near a black hole, traveling near the speed of light—then we know we can fall back on the more complex canonical model; but when the simpler model works, there’s no reason not to use it.

There are still very serious gaps in the knowledge of physics; in particular, there is a fundamental gulf between quantum mechanics and the Einstein field equations that has been unresolved for decades. A solution to this “quantum gravity problem” would be essentially a guaranteed Nobel Prize. So even a canonical model can be flawed, and can be extended or improved upon; the result is then a new canonical model which we now regard as our best approximation to truth.

Yet the contrast with economics is still quite clear. We don’t have one or two or even ten canonical models to refer back to. We can’t say that the Solow model is an approximation of some greater canonical model that works for these purposes—because we don’t have that greater canonical model. We can’t say that agent-based computational economics is approximately right, because we have nothing to approximate it to.

I went into economics thinking that neoclassical economics needed a new paradigm. I have now realized something much more alarming: Neoclassical economics doesn’t really have a paradigm. Or if it does, it’s a very informal paradigm, one that is expressed by the arbitrary judgments of journal editors, not one that can be written down as a series of equations. We assume perfect rationality, except when we don’t. We assume constant returns to scale, except when that doesn’t work. We assume perfect competition, except when that doesn’t get the results we wanted. The agents in our models are infinite identical psychopaths, and they are exactly as rational as needed for the conclusion I want.

This is quite likely why there is so much disagreement within economics. When you can permute the parameters however you like with no regard to a canonical model, you can more or less draw whatever conclusion you want, especially if you aren’t tightly bound to empirical evidence. I know a great many economists who are sure that raising minimum wage results in large disemployment effects, because the models they believe in say that it must, even though the empirical evidence has been quite clear that these effects are small if they are present at all. If we had a canonical model of employment that we could calibrate to the empirical evidence, that couldn’t happen anymore; there would be a coefficient I could point to that would refute their argument. But when every new paper comes with a new model, there’s no way to do that; one set of assumptions is as good as another.

Indeed, as I mentioned in an earlier post, a remarkable number of economists seem to embrace this relativism. “There is no true model.” they say; “We do what is useful.” Recently I encountered a book by the eminent economist Deirdre McCloskey which, though I confess I haven’t read it in its entirety, appears to be trying to argue that economics is just a meaningless language game that doesn’t have or need to have any connection with actual reality. (If any of you have read it and think I’m misunderstanding it, please explain. As it is I haven’t bought it for a reason any economist should respect: I am disinclined to incentivize such writing.)

Creating such a canonical model would no doubt be extremely difficult. Indeed, it is a task that would require the combined efforts of hundreds of researchers and could take generations to achieve. The true equations that underlie the economy could be totally intractable even for our best computers. But quantum mechanics wasn’t built in a day, either. The key challenge here lies in convincing economists that this is something worth doing—that if we really want to be taken seriously as scientists we need to start acting like them. Scientists believe in truth, and they are trying to find it out. While not immune to tribalism or ideology or other human limitations, they resist them as fiercely as possible, always turning back to the evidence above all else. And in their combined strivings, they attempt to build a grand edifice, a universal theory to stand the test of time—a canonical model.

Experimentally testing categorical prospect theory

Dec 4, JDN 2457727

In last week’s post I presented a new theory of probability judgments, which doesn’t rely upon people performing complicated math even subconsciously. Instead, I hypothesize that people try to assign categories to their subjective probabilities, and throw away all the information that wasn’t used to assign that category.

The way to most clearly distinguish this from cumulative prospect theory is to show discontinuity. Kahneman’s smooth, continuous function places fairly strong bounds on just how much a shift from 0% to 0.000001% can really affect your behavior. In particular, if you want to explain the fact that people do seem to behave differently around 10% compared to 1% probabilities, you can’t allow the slope of the smooth function to get much higher than 10 at any point, even near 0 and 1. (It does depend on the precise form of the function, but the more complicated you make it, the more free parameters you add to the model. In the most parsimonious form, which is a cubic polynomial, the maximum slope is actually much smaller than this—only 2.)

If that’s the case, then switching from 0.% to 0.0001% should have no more effect in reality than a switch from 0% to 0.00001% would to a rational expected utility optimizer. But in fact I think I can set up scenarios where it would have a larger effect than a switch from 0.001% to 0.01%.

Indeed, these games are already quite profitable for the majority of US states, and they are called lotteries.

Rationally, it should make very little difference to you whether your odds of winning the Powerball are 0 (you bought no ticket) or 0.000000001% (you bought a ticket), even when the prize is $100 million. This is because your utility of $100 million is nowhere near 100 million times as large as your marginal utility of $1. A good guess would be that your lifetime income is about $2 million, your utility is logarithmic, the units of utility are hectoQALY, and the baseline level is about 100,000.

I apologize for the extremely large number of decimals, but I had to do that in order to show any difference at all. I have bolded where the decimals first deviate from the baseline.

Your utility if you don’t have a ticket is ln(20) = 2.9957322736 hQALY.

Your utility if you have a ticket is (1-10^-9) ln(20) + 10^-9 ln(1020) = 2.9957322775 hQALY.

You gain a whopping 40 microQALY over your whole lifetime. I highly doubt you could even perceive such a difference.

And yet, people are willing to pay nontrivial sums for the chance to play such lotteries. Powerball tickets sell for about $2 each, and some people buy tickets every week. If you do that and live to be 80, you will spend some $8,000 on lottery tickets during your lifetime, which results in this expected utility: (1-4*10^-6) ln(20-0.08) + 4*10^-6 ln(1020) = 2.9917399955 hQALY.
You have now sacrificed 0.004 hectoQALY, which is to say 0.4 QALY—that’s months of happiness you’ve given up to play this stupid pointless game.

Which shouldn’t be surprising, as (with 99.9996% probability) you have given up four months of your lifetime income with nothing to show for it. Lifetime income of $2 million / lifespan of 80 years = $25,000 per year; $8,000 / $25,000 = 0.32. You’ve actually sacrificed slightly more than this, which comes from your risk aversion.

Why would anyone do such a thing? Because while the difference between 0 and 10^-9 may be trivial, the difference between “impossible” and “almost impossible” feels enormous. “You can’t win if you don’t play!” they say, but they might as well say “You can’t win if you do play either.” Indeed, the probability of winning without playing isn’t zero; you could find a winning ticket lying on the ground, or win due to an error that is then upheld in court, or be given the winnings bequeathed by a dying family member or gifted by an anonymous donor. These are of course vanishingly unlikely—but so was winning in the first place. You’re talking about the difference between 10^-9 and 10^-12, which in proportional terms sounds like a lot—but in absolute terms is nothing. If you drive to a drug store every week to buy a ticket, you are more likely to die in a car accident on the way to the drug store than you are to win the lottery.

Of course, these are not experimental conditions. So I need to devise a similar game, with smaller stakes but still large enough for people’s brains to care about the “almost impossible” category; maybe thousands? It’s not uncommon for an economics experiment to cost thousands, it’s just usually paid out to many people instead of randomly to one person or nobody. Conducting the experiment in an underdeveloped country like India would also effectively amplify the amounts paid, but at the fixed cost of transporting the research team to India.

But I think in general terms the experiment could look something like this. You are given $20 for participating in the experiment (we treat it as already given to you, to maximize your loss aversion and endowment effect and thereby give us more bang for our buck). You then have a chance to play a game, where you pay $X to get a P probability of $Y*X, and we vary these numbers.

The actual participants wouldn’t see the variables, just the numbers and possibly the rules: “You can pay $2 for a 1% chance of winning $200. You can also play multiple times if you wish.” “You can pay $10 for a 5% chance of winning $250. You can only play once or not at all.”

So I think the first step is to find some dilemmas, cases where people feel ambivalent, and different people differ in their choices. That’s a good role for a pilot study.

Then we take these dilemmas and start varying their probabilities slightly.

In particular, we try to vary them at the edge of where people have mental categories. If subjective probability is continuous, a slight change in actual probability should never result in a large change in behavior, and furthermore the effect of a change shouldn’t vary too much depending on where the change starts.

But if subjective probability is categorical, these categories should have edges. Then, when I present you with two dilemmas that are on opposite sides of one of the edges, your behavior should radically shift; while if I change it in a different way, I can make a large change without changing the result.

Based solely on my own intuition, I guessed that the categories roughly follow this pattern:

Impossible: 0%

Almost impossible: 0.1%

Very unlikely: 1%

Unlikely: 10%

Fairly unlikely: 20%

Roughly even odds: 50%

Fairly likely: 80%

Likely: 90%

Very likely: 99%

Almost certain: 99.9%

Certain: 100%

So for example, if I switch from 0%% to 0.01%, it should have a very large effect, because I’ve moved you out of your “impossible” category (indeed, I think the “impossible” category is almost completely sharp; literally anything above zero seems to be enough for most people, even 10^-9 or 10^-10). But if I move from 1% to 2%, it should have a small effect, because I’m still well within the “very unlikely” category. Yet the latter change is literally one hundred times larger than the former. It is possible to define continuous functions that would behave this way to an arbitrary level of approximation—but they get a lot less parsimonious very fast.

Now, immediately I run into a problem, because I’m not even sure those are my categories, much less that they are everyone else’s. If I knew precisely which categories to look for, I could tell whether or not I had found it. But the process of both finding the categories and determining if their edges are truly sharp is much more complicated, and requires a lot more statistical degrees of freedom to get beyond the noise.

One thing I’m considering is assigning these values as a prior, and then conducting a series of experiments which would adjust that prior. In effect I would be using optimal Bayesian probability reasoning to show that human beings do not use optimal Bayesian probability reasoning. Still, I think that actually pinning down the categories would require a large number of participants or a long series of experiments (in frequentist statistics this distinction is vital; in Bayesian statistics it is basically irrelevant—one of the simplest reasons to be Bayesian is that it no longer bothers you whether someone did 2 experiments of 100 people or 1 experiment of 200 people, provided they were the same experiment of course). And of course there’s always the possibility that my theory is totally off-base, and I find nothing; a dissertation replicating cumulative prospect theory is a lot less exciting (and, sadly, less publishable) than one refuting it.

Still, I think something like this is worth exploring. I highly doubt that people are doing very much math when they make most probabilistic judgments, and using categories would provide a very good way for people to make judgments usefully with no math at all.