The potential of an advertising tax

Jan 7, JDN 2458126

Advertising is everywhere in our society. You may see some on this very page (though if I hit my next Patreon target I’m going to pay to get rid of those). Ad-blockers can help when you’re on the Web, and premium channels like HBO will save you from ads when watching TV, but what are you supposed to do about ads on billboards as you drive down the highway, ads on buses as you walk down the street, ads on the walls of the subway train?

And Banksy isn’t entirely wrong; this stuff can be quite damaging. Based on decades of research, the American Psychological Association has issued official statements condemning the use of advertising to children for its harmful psychological effects. Medical research has shown that advertisements for food can cause overeating—and thus, the correlated rise of advertising and obesity may be no coincidence.

Worst of all, political advertising distorts our view of the world. Though we may not be able to blame advertising per se for Trump; most of his publicity was gained for free by irresponsible media coverage.

And yet, advertising is almost pure rent-seeking. It costs resources, but it doesn’t produce anything. In most cases it doesn’t even raise awareness about something or find new customers. The primary goal of most advertising is to get you to choose that brand instead of a different brand. A secondary goal (especially for food ads) is to increase your overall consumption of that good, but since the means employed typically involve psychological manipulation, this increase in consumption is probably harmful to social welfare.

A general principle of economics that has almost universal consensus is the Pigou Principle: If you want less of something, you should put a tax on it. So, what would happen if we put a tax on advertising?

The amazing thing is that in this case, we would probably not actually reduce advertising spending, but we would reduce advertising, which is what we actually care about. Moreover, we would be able to raise an enormous amount of revenue with zero social cost. Like the other big Pigovian tax (the carbon tax), this a rare example of a tax that will give you a huge amount of revenue while actually yielding a benefit to society.

This is far from obvious, so I think it is worth explaining where it comes from.

The key point is that advertising doesn’t typically increase the overall size of the market (though in some cases it does; I’ll get back to that in a moment). Rather that a conventional production function like we would have for most types of expenditure, advertising is better modeled by what is called a contest function (something that our own Stergios Skaperdas at UCI is actually a world-class expert in). In a production function, inputs increase the total amount of output. But in a contest function, inputs only redistribute output from one place to another. Contest functions thus provide a good model of rent-seeking, which is what most advertising is.

Suppose there’s a total market M for some good, where M is the total profits that can be gained from capturing that entire market.
Then, to keep it simple, let’s suppose there are only two major firms in the market, a duopoly like Coke and Pepsi or Boeing and Airbus.

Let’s say Coke decides to spend an amount x on advertising, and Pepsi decides to spend an amount y.

For now, let’s assume that total beverage consumption won’t change; so the total profits to be had from the market are always M.

What advertising does is it changes the share of that market which each firm will get. Specifically, let’s use the simplest model, where the share of the market is equal to the share of advertising spending.

Then the net profit for Coke is the following:

The share they get, x/(x+y), times the size of the whole market, M, minus the advertising spending x.

max M*x/(x+y) – x

We can maximize this with the usual first-order condition:

y/(x+y)^2 M – 1 = 0

(x+y)^2 = My

Since the game is symmetric, in a Nash equilibrium, Pepsi will use the same reasoning:

(x+y)^2 = Mx

Thus we have:

x = y

(2x)^2 = Mx

x = M/4

In this very simple model, each firm will spend one-fourth of the market’s value, and the total advertising spending will be equal to half the size of the market. Then, each company’s net income will be equal to its advertising spending. This is a pretty good estimate for Coca-Cola in real life, which spends about $3.3 billion on advertising and receives about $2.8 billion in net income each year.

What would happen if we introduce a tax? Let’s say we introduce a proportional tax r on all advertising spending. That is, for every dollar you spend on advertising, you must pay the government $r in tax. The really remarkable thing is that companies who advertise shouldn’t care what we make the tax; the only ones who will care are the advertising companies themselves.

If Coke pays x, the actual amount of advertising they receive is x – r x = x(1-r).

Likewise, Pepsi’s actual advertising received is y(1-r).

But notice that the share of total advertising spending is completely unchanged!

(x(1-r))/(x(1-r) + y(1-r)) = x/(x+y)

Since the payoff for Coke only depends on how much Coke spends and what market share they get, it is also unchanged. Since the same is true for Pepsi, nothing will change in how the two companies behave. They will spend the same amount on advertising, and they will receive the same amount of net income when all is said and done.

The total quantity of advertising will be reduced, from x+y to (x+y)(1-r). That means fewer billboards, fewer posters in subway stations, fewer TV commercials. That will hurt advertising companies, but benefit everyone else.

How much revenue will we get for the government? r x + r y = r(x+y).

Since the goal is to substantially reduce advertising output, and it won’t distort other industries in any way, we should set this tax quite high. A reasonable value for r would be 50%. We might even want to consider something as high as 90%; but for now let’s look at what 50% would do.

Total advertising spending in the US is over $200 billion per year. Since an advertising tax would not change total advertising spending, we can expect that a tax rate of 50% would simply capture 50% of this spending as revenue, which is to say $100 billion per year. That would be enough to pay for the entire Federal education budget, or the foreign aid and environment budgets combined.
Another great aspect of how an advertising tax is actually better than a carbon tax is that countries will want to compete to have the highest advertising taxes. If say Canada imposes a carbon tax but the US doesn’t, industries will move production to the US where it is cheaper, which hurts Canada. Yet the total amount of pollution will remain about the same, and Canada will be just as affected by climate change as they would have been anyway. So we need to coordinate across countries so that the carbon taxes are all the same (or at least close), to prevent industries from moving around; and each country has an incentive to cheat by imposing a lower carbon tax.

But advertising taxes aren’t like that. If Canada imposes an advertising tax and the US doesn’t, companies won’t shift production to the US; they will shift advertising to the US. And having your country suddenly flooded with advertisements is bad. That provides a strong incentive for you to impose your own equal or even higher advertising tax to stem the tide. And pretty soon, everyone will have imposed an advertising tax at the same rate.

Of course, in all the above I’ve assumed a pure contest function, meaning that advertisements are completely unproductive. What if they are at least a little bit productive? Then we wouldn’t want to set the tax too high, but the basic conclusions would be unchanged.

Suppose, for instance, that the advertising spending adds half its value to the value of the market. This is a pretty high estimate of the benefits of advertising.

Under this assumption, in place of M we have M+(x+y)/2. Everything else is unchanged.

We can maximize as before:

max (M+(x+y)/2)*x/(x+y) – x

The math is a bit trickier, but we can still solve by a first-order condition, which simplifies to:

(x+y)^2 = 2My

By the same symmetry reasoning as before:

(2x)^2 = 2Mx

x = M/2

Now, total advertising spending would equal the size of the market without advertising, and net income for each firm after advertising would be:

2M(1/2) – M/2 = M/2

That is, advertising spending would equal net income, as before. (A surprisingly robust result!)

What if we imposed a tax? Now the algebra gets even nastier:

max (M+(x+y)(1-r)/2)*x/(x+y) – x

But the ultimate outcome is still quite similar:

(1+r)(x+y)^2 = 2My

(1+r)(2x)^2 = 2Mx

x = M/2*1/(1+r)

Advertising spending will be reduced by a factor of 1/(1+r). Even if r is 50%, that still means we’ll have 2/3 of the advertising spending we had before.

Total tax revenue will then be M*r/(1+r), which for r of 50% would be M/3.

Total advertising will be M(1-r)/(1+r), which would be M/3. So we managed to reduce advertising by 2/3, while reducing advertising spending by only 1/3. Then we would receive half of that spending as revenue. Thus, instead of getting $100 billion per year, we would get $67 billion, which is still just about enough to pay for food stamps.

What’s the downside of this tax? Unlike most taxes, there really isn’t one. Yes, it would hurt advertising companies, which I suppose counts as a downside. But that was mostly waste anyway; anyone employed in advertising would be better employed almost anywhere else. Millions of minds are being wasted coming up with better ways to sell Viagra instead of better treatments for cancer. Any unemployment introduced by an advertising tax would be temporary and easily rectified by monetary policy, and most of it would hit highly educated white-collar professionals who have high incomes to begin with and can more easily find jobs when displaced.

The real question is why we aren’t doing this already. And that, I suppose, has to come down to politics.

“DSGE or GTFO”: Macroeconomics took a wrong turn somewhere

Dec 31, JDN 2458119

The state of macro is good,” wrote Oliver Blanchard—in August 2008. This is rather like the turkey who is so pleased with how the farmer has been feeding him lately, the day before Thanksgiving.

It’s not easy to say exactly where macroeconomics went wrong, but I think Paul Romer is right when he makes the analogy between DSGE (dynamic stochastic general equilbrium) models and string theory. They are mathematically complex and difficult to understand, and people can make their careers by being the only ones who grasp them; therefore they must be right! Nevermind if they have no empirical support whatsoever.

To be fair, DSGE models are at least a little better than string theory; they can at least be fit to real-world data, which is better than string theory can say. But being fit to data and actually predicting data are fundamentally different things, and DSGE models typically forecast no better than far simpler models without their bold assumptions. You don’t need to assume all this stuff about a “representative agent” maximizing a well-defined utility function, or an Euler equation (that doesn’t even fit the data), or this ever-proliferating list of “random shocks” that end up taking up all the degrees of freedom your model was supposed to explain. Just regressing the variables on a few years of previous values of each other (a “vector autoregression” or VAR) generally gives you an equally-good forecast. The fact that these models can be made to fit the data well if you add enough degrees of freedom doesn’t actually make them good models. As Von Neumann warned us, with enough free parameters, you can fit an elephant.

But really what bothers me is not the DSGE but the GTFO (“get the [expletive] out”); it’s not that DSGE models are used, but that it’s almost impossible to get published as a macroeconomic theorist using anything else. Defenders of DSGE typically don’t even argue anymore that it is good; they argue that there are no credible alternatives. They characterize their opponents as “dilettantes” who aren’t opposing DSGE because we disagree with it; no, it must be because we don’t understand it. (Also, regarding that post, I’d just like to note that I now officially satisfy the Athreya Axiom of Absolute Arrogance: I have passed my qualifying exams in a top-50 economics PhD program. Yet my enmity toward DSGE has, if anything, only intensified.)

Of course, that argument only makes sense if you haven’t been actively suppressing all attempts to formulate an alternative, which is precisely what DSGE macroeconomists have been doing for the last two or three decades. And yet despite this suppression, there are alternatives emerging, particularly from the empirical side. There are now empirical approaches to macroeconomics that don’t use DSGE models. Regression discontinuity methods and other “natural experiment” designs—not to mention actual experiments—are quickly rising in popularity as economists realize that these methods allow us to actually empirically test our models instead of just adding more and more mathematical complexity to them.

But there still seems to be a lingering attitude that there is no other way to do macro theory. This is very frustrating for me personally, because deep down I think what I would like to do as a career is macro theory: By temperament I have always viewed the world through a very abstract, theoretical lens, and the issues I care most about—particularly inequality, development, and unemployment—are all fundamentally “macro” issues. I left physics when I realized I would be expected to do string theory. I don’t want to leave economics now that I’m expected to do DSGE. But I also definitely don’t want to do DSGE.

Fortunately with economics I have a backup plan: I can always be an “applied micreconomist” (rather the opposite of a theoretical macroeconomist I suppose), directly attached to the data in the form of empirical analyses or even direct, randomized controlled experiments. And there certainly is plenty of work to be done along the lines of Akerlof and Roth and Shiller and Kahneman and Thaler in cognitive and behavioral economics, which is also generally considered applied micro. I was never going to be an experimental physicist, but I can be an experimental economist. And I do get to use at least some theory: In particular, there’s an awful lot of game theory in experimental economics these days. Some of the most exciting stuff is actually in showing how human beings don’t behave the way classical game theory predicts (particularly in the Ultimatum Game and the Prisoner’s Dilemma), and trying to extend game theory into something that would fit our actual behavior. Cognitive science suggests that the result is going to end up looking quite different from game theory as we know it, and with my cognitive science background I may be particularly well-positioned to lead that charge.

Still, I don’t think I’ll be entirely satisfied if I can’t somehow bring my career back around to macroeconomic issues, and particularly the great elephant in the room of all economics, which is inequality. Underlying everything from Marxism to Trumpism, from the surging rents in Silicon Valley and the crushing poverty of Burkina Faso, to the Great Recession itself, is inequality. It is, in my view, the central question of economics: Who gets what, and why?

That is a fundamentally macro question, but you can’t even talk about that issue in DSGE as we know it; a “representative agent” inherently smooths over all inequality in the economy as though total GDP were all that mattered. A fundamentally new approach to macroeconomics is needed. Hopefully I can be part of that, but from my current position I don’t feel much empowered to fight this status quo. Maybe I need to spend at least a few more years doing something else, making a name for myself, and then I’ll be able to come back to this fight with a stronger position.

In the meantime, I guess there’s plenty of work to be done on cognitive biases and deviations from game theory.

Why New Year’s resolutions fail

Jan 1, JDN 2457755

Last week’s post was on Christmas, so by construction this week’s post will be on New Year’s Day.

It is a tradition in many cultures, especially in the US and Europe, to start every new year with a New Year’s resolution, a promise to ourselves to change our behavior in some positive way.

Yet, over 80% of these resolutions fail. Why is this?

If we are honest, most of us would agree that there is something about our own behavior that could stand to be improved. So why do we so rarely succeed in actually making such improvements?

One possibility, which I’m guessing most neoclassical economists would favor, is to say that we don’t actually want to. We may pretend that we do in order to appease others, but ultimately our rational optimization has already chosen that we won’t actually bear the cost to make the improvement.

I think this is actually quite rare. I’ve seen too many people with resolutions they didn’t share with anyone, for example, to think that it’s all about social pressure. And I’ve seen far too many people try very hard to achieve their resolutions, day after day, and yet still fail.

Sometimes we make resolutions that are not entirely within our control, such as “get a better job” or “find a girlfriend” (last year I made a resolution to publish a work of commercial fiction or a peer-reviewed article—and alas, failed at that task, unless I somehow manage it in the next few days). Such resolutions may actually be unwise to make in the first place, as it can feel like breaking a promise to yourself when you’ve actually done all you possibly could.

So let’s set those aside and talk only about things we should be in control over, like “lose weight” or “save more money”. Even these kinds of resolutions typically fail; why? What is this “weakness of will”? How is it possible to really want something that you are in full control over, and yet still fail to accomplish it?

Well, first of all, I should be clear what I mean by “in full control over”. In some sense you’re not in full control, which is exactly the problem. Your conscious mind is not actually an absolute tyrant over your entire body; you’re more like an elected president who has to deal with a legislature in order to enact policy.

You do have a great deal of power over your own behavior, and you can learn to improve this control (much as real executive power in presidential democracies has expanded over the last century!); but there are fundamental limits to just how well you can actually consciously will your body to do anything, limits imposed by billions of years of evolution that established most of the traits of your body and nervous system millions of generations before there even was such a thing as rational conscious reasoning.

One thing that makes a surprisingly large difference lies in whether your goals are reduced to specific, actionable objectives. “Lose weight” is almost guaranteed to fail. “Lose 30 pounds” is still unlikely to succeed. “Work out for 2 hours per week,” on the other hand, might have a chance. “Save money” is never going to make it, but “move to a smaller apartment and set aside $200 per month” just might.

I think the government metaphor is helpful here; if you President of the United States and you want something done, do you state some vague, broad goal like “Improve the economy”? No, you make a specific, actionable demand that allows you to enforce compliance, like “increase infrastructure spending by 24% over the next 5 years”. Even then it is possible to fail if you can’t push it through the legislature (in the metaphor, the “legislature” is your habits, instincts and other subconscious processes), but you’re much more likely to succeed if you have a detailed plan.

Another technique that helps is to visualize the benefits of succeeding and the costs of failing, and keep these in your mind. This counteracts the tendency for the costs of succeeding and the benefits of giving up to be more salient—losing 30 pounds sounds nice in theory, but that treadmill is so much work right now!

This salience effect has a lot to do with the fact that human beings are terrible at dealing with the future.

Rationally, we are supposed to use exponential discounting; each successive moment is supposed to be worth less to us than the previous by a fixed proportion, say 5% per year. This is actually a mathematical theorem; if you don’t discount this way, your decisions will be systematically irrational.

And yet… we don’t discount that way. Some behavioral economists argue that we use hyperbolic discounting, in which instead of discounting time by a fixed proportion, we use a different formula that drops off too quickly early on and not quickly enough later on.

But I am increasingly convinced that human beings don’t actually use discounting at all. We have a series of rough-and-ready heuristics for making future judgments, which can sort of act like discounting, but require far less computation than actually calculating a proper discount rate. (Recent empirical evidence seems to be tilting this direction.)

In any case, whatever we do is clearly not a proper rational discount rate. And this means that our behavior can be time-inconsistent; a choice that seems rational at one time can not seem rational at a later time. When we’re planning out our year and saying we will hit the treadmill more, it seems like a good idea; but when we actually get to the gym and feel our legs ache as we start running, we begin to regret our decision.

The challenge, really, is determining which “version” of us is correct! A priori, we don’t actually know whether the view of our distant self contemplating the future or the view of our current self making the choice in the moment is the right one. Actually, when I frame it this way, it almost seems like the self that’s closer to the choice should have better information—and yet typically we think the exact opposite, that it is our past self making plans that really knows what’s best for us.

So where does that come from? Why do we think, at least in most cases, that the “me” which makes a plan a year in advance is the smart one, and the “me” that actually decides in the moment is untrustworthy.

Kahneman has a good explanation for this, in his model of System 1 and System 2. System 1 is simple and fast, but often gets the wrong answer. System 2 usually gets the right answer, but it is complex and slow. When we are making plans, we have a lot of time to think, and we can afford to expend the extra effort to engage the full power of System 2. But when we are living in the moment, choosing what to do right now, we don’t have that luxury of time, and we are forced to fall back on System 1. System 1 is easier—but it’s also much more likely to be wrong.

How, then, do we resolve this conflict? Commitment. (Perhaps that’s why it’s called a New Year’s resolution!)

We make promises to ourselves, commitments that we will feel bad about not following through.

If we rationally discounted, this would be a baffling thing to do; we’re just imposing costs on ourselves for no reason. But because we don’t discount rationally, commitments allow us to change the calculation for our future selves.

This brings me to one last strategy to use when making your resolutions: Include punishment.

“I will work out at least 2 hours per week, and if I don’t, I’m not allowed to watch TV all weekend.” Now that is a resolution you are actually likely to keep.

To see why, consider the decision problem for your System 2 self today versus your System 1 self throughout the year.

Your System 2 self has done the cost-benefit analysis and ruled that working out 2 hours per week is worthwhile for its health benefits.

If you left it at that, your System 1 self would each day find an excuse to procrastinate the workouts, because at least from where they’re sitting, working out for 2 hours looks a lot more painful than the marginal loss in health from missing just this one week. And of course this will keep happening, week after week—and then 52 go by and you’ve had few if any workouts.

But by adding the punishment of “no TV”, you have imposed an additional cost on your System 1 self, something that they care about. Suddenly the calculation changes; it’s not just 2 hours of workout weighed against vague long-run health benefits, but 2 hours of workout weighed against no TV all weekend. That punishment is surely too much to bear; so you’d best do the workout after all.

Do it right, and you will rarely if ever have to impose the punishment. But don’t make it too large, or then it will seem unreasonable and you won’t want to enforce it if you ever actually need to. Your System 1 self will then know this, and treat the punishment as nonexistent. (Formally the equilibrium is not subgame perfect; I am gravely concerned that our nuclear deterrence policy suffers from precisely this flaw.) “If I don’t work out, I’ll kill myself” is a recipe for depression, not healthy exercise habits.

But if you set clear, actionable objectives and sufficient but reasonable punishments, there’s at least a good chance you will actually be in the minority of people who actually succeed in keeping their New Year’s resolution.

And if not, there’s always next year.

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.

The Tragedy of the Commons

JDN 2457387

In a previous post I talked about one of the most fundamental—perhaps the most fundamental—problem in game theory, the Prisoner’s Dilemma, and how neoclassical economic theory totally fails to explain actual human behavior when faced with this problem in both experiments and the real world.

As a brief review, the essence of the game is that both players can either cooperate or defect; if they both cooperate, the outcome is best overall; but it is always in each player’s interest to defect. So a neoclassically “rational” player would always defect—resulting in a bad outcome for everyone. But real human beings typically cooperate, and thus do better. The “paradox” of the Prisoner’s Dilemma is that being “rational” results in making less money at the end.

Obviously, this is not actually a good definition of rational behavior. Being short-sighted and ignoring the impact of your behavior on others doesn’t actually produce good outcomes for anybody, including yourself.

But the Prisoner’s Dilemma only has two players. If we expand to a larger number of players, the expanded game is called a Tragedy of the Commons.

When we do this, something quite surprising happens: As you add more people, their behavior starts converging toward the neoclassical solution, in which everyone defects and we get a bad outcome for everyone.

Indeed, people in general become less cooperative, less courageous, and more apathetic the more of them you put together. K was quite apt when he said, “A person is smart; people are dumb, panicky, dangerous animals and you know it.” There are ways to counteract this effect, as I’ll get to in a moment—but there is a strong effect that needs to be counteracted.

We see this most vividly in the bystander effect. If someone is walking down the street and sees someone fall and injure themselves, there is about a 70% chance that they will go try to help the person who fell—humans are altruistic. But if there are a dozen people walking down the street who all witness the same event, there is only a 40% chance that any of them will help—humans are irrational.

The primary reason appears to be diffusion of responsibility. When we are alone, we are the only one could help, so we feel responsible for helping. But when there are others around, we assume that someone else could take care of it for us, so if it isn’t done that’s not our fault.

There also appears to be a conformity effect: We want to conform our behavior to social norms (as I said, to a first approximation, all human behavior is social norms). The mere fact that there are other people who could have helped but didn’t suggests the presence of an implicit social norm that we aren’t supposed to help this person for some reason. It never occurs to most people to ask why such a norm would exist or whether it’s a good one—it simply never occurs to most people to ask those questions about any social norms. In this case, by hesitating to act, people actually end up creating the very norm they think they are obeying.

This can lead to what’s called an Abilene Paradox, in which people simultaneously try to follow what they think everyone else wants and also try to second-guess what everyone else wants based on what they do, and therefore end up doing something that none of them actually wanted. I think a lot of the weird things humans do can actually be attributed to some form of the Abilene Paradox. (“Why are we sacrificing this goat?” “I don’t know, I thought you wanted to!”)

Autistic people are not as good at following social norms (though some psychologists believe this is simply because our social norms are optimized for the neurotypical population). My suspicion is that autistic people are therefore less likely to suffer from the bystander effect, and more likely to intervene to help someone even if they are surrounded by passive onlookers. (Unfortunately I wasn’t able to find any good empirical data on that—it appears no one has ever thought to check before.) I’m quite certain that autistic people are less likely to suffer from the Abilene Paradox—if they don’t want to do something, they’ll tell you so (which sometimes gets them in trouble).

Because of these psychological effects that blunt our rationality, in large groups human beings often do end up behaving in a way that appears selfish and short-sighted.

Nowhere is this more apparent than in ecology. Recycling, becoming vegetarian, driving less, buying more energy-efficient appliances, insulating buildings better, installing solar panels—none of these things are particularly difficult or expensive to do, especially when weighed against the tens of millions of people who will die if climate change continues unabated. Every recyclable can we throw in the trash is a silent vote for a global holocaust.

But as it no doubt immediately occurred to you to respond: No single one of us is responsible for all that. There’s no way I myself could possibly save enough carbon emissions to significantly reduce climate change—indeed, probably not even enough to save a single human life (though maybe). This is certainly true; the error lies in thinking that this somehow absolves us of the responsibility to do our share.

I think part of what makes the Tragedy of the Commons so different from the Prisoner’s Dilemma, at least psychologically, is that the latter has an identifiable victimwe know we are specifically hurting that person more than we are helping ourselves. We may even know their name (and if we don’t, we’re more likely to defect—simply being on the Internet makes people more aggressive because they don’t interact face-to-face). In the Tragedy of the Commons, it is often the case that we don’t know who any of our victims are; moreover, it’s quite likely that we harm each one less than we benefit ourselves—even though we harm everyone overall more.

Suppose that driving a gas-guzzling car gives me 1 milliQALY of happiness, but takes away an average of 1 nanoQALY from everyone else in the world. A nanoQALY is tiny! Negligible, even, right? One billionth of a year, a mere 30 milliseconds! Literally less than the blink of an eye. But take away 30 milliseconds from everyone on Earth and you have taken away 7 years of human life overall. Do that 10 times, and statistically one more person is dead because of you. And you have gained only 10 milliQALY, roughly the value of $300 to a typical American. Would you kill someone for $300?

Peter Singer has argued that we should in fact think of it this way—when we cause a statistical death by our inaction, we should call it murder, just as if we had left a child to drown to keep our clothes from getting wet. I can’t agree with that. When you think seriously about the scale and uncertainty involved, it would be impossible to live at all if we were constantly trying to assess whether every action would lead to statistically more or less happiness to the aggregate of all human beings through all time. We would agonize over every cup of coffee, every new video game. In fact, the global economy would probably collapse because none of us would be able to work or willing to buy anything for fear of the consequences—and then whom would we be helping?

That uncertainty matters. Even the fact that there are other people who could do the job matters. If a child is drowning and there is a trained lifeguard right next to you, the lifeguard should go save the child, and if they don’t it’s their responsibility, not yours. Maybe if they don’t you should try; but really they should have been the one to do it.
But we must also not allow ourselves to simply fall into apathy, to do nothing simply because we cannot do everything. We cannot assess the consequences of every specific action into the indefinite future, but we can find general rules and patterns that govern the consequences of actions we might take. (This is the difference between act utilitarianism, which is unrealistic, and rule utilitarianism, which I believe is the proper foundation for moral understanding.)

Thus, I believe the solution to the Tragedy of the Commons is policy. It is to coordinate our actions together, and create enforcement mechanisms to ensure compliance with that coordinated effort. We don’t look at acts in isolation, but at policy systems holistically. The proper question is not “What should I do?” but “How should we live?”

In the short run, this can lead to results that seem deeply suboptimal—but in the long run, policy answers lead to sustainable solutions rather than quick-fixes.

People are starving! Why don’t we just steal money from the rich and use it to feed people? Well, think about what would happen if we said that the property system can simply be unilaterally undermined if someone believes they are achieving good by doing so. The property system would essentially collapse, along with the economy as we know it. A policy answer to that same question might involve progressive taxation enacted by a democratic legislature—we agree, as a society, that it is justified to redistribute wealth from those who have much more than they need to those who have much less.

Our government is corrupt! We should launch a revolution! Think about how many people die when you launch a revolution. Think about past revolutions. While some did succeed in bringing about more just governments (e.g. the French Revolution, the American Revolution), they did so only after a long period of strife; and other revolutions (e.g. the Russian Revolution, the Iranian Revolution) have made things even worse. Revolution is extremely costly and highly unpredictable; we must use it only as a last resort against truly intractable tyranny. The policy answer is of course democracy; we establish a system of government that elects leaders based on votes, and then if they become corrupt we vote to remove them. (Sadly, we don’t seem so good about that second part—the US Congress has a 14% approval rating but a 95% re-election rate.)

And in terms of ecology, this means that berating ourselves for our sinfulness in forgetting to recycle or not buying a hybrid car does not solve the problem. (Not that it’s bad to recycle, drive a hybrid car, and eat vegetarian—by all means, do these things. But it’s not enough.) We need a policy solution, something like a carbon tax or cap-and-trade that will enforce incentives against excessive carbon emissions.

In case you don’t think politics makes a difference, all of the Democrat candidates for President have proposed such plans—Bernie Sanders favors a carbon tax, Martin O’Malley supports an aggressive cap-and-trade plan, and Hillary Clinton favors heavily subsidizing wind and solar power. The Republican candidates on the other hand? Most of them don’t even believe in climate change. Chris Christie and Carly Fiorina at least accept the basic scientific facts, but (1) they are very unlikely to win at this point and (2) even they haven’t announced any specific policy proposals for dealing with it.

This is why voting is so important. We can’t do enough on our own; the coordination problem is too large. We need to elect politicians who will make policy. We need to use the systems of coordination enforcement that we have built over generations—and that is fundamentally what a government is, a system of coordination enforcement. Only then can we overcome the tendency among human beings to become apathetic and short-sighted when faced with a Tragedy of the Commons.

The Prisoner’s Dilemma

JDN 2457348
When this post officially goes live, it will have been one full week since I launched my Patreon, on which I’ve already received enough support to be more than halfway to my first funding goal. After this post, I will be far enough ahead in posting that I can release every post one full week ahead of time for my Patreon patrons (can I just call them Patreons?).

It’s actually fitting that today’s topic is the Prisoner’s Dilemma, for Patreon is a great example of how real human beings can find solutions to this problem even if infinite identical psychopaths could not.

The Prisoner’s Dilemma is the most fundamental problem in game theory—arguably the reason game theory is worth bothering with in the first place. There is a standard story that people generally tell to set up the dilemma, but honestly I find that they obscure more than they illuminate. You can find it in the Wikipedia article if you’re interested.

The basic idea of the Prisoner’s Dilemma is that there are many times in life when you have a choice: You can do the nice thing and cooperate, which costs you something, but benefits the other person more; or you can do the selfish thing and defect, which benefits you but harms the other person more.

The game can basically be defined as four possibilities: If you both cooperate, you each get 1 point. If you both defect, you each get 0 points. If you cooperate when the other player defects, you lose 1 point while the other player gets 2 points. If you defect when the other player cooperates, you get 2 points while the other player loses 1 point.

P2 Cooperate P2 Defect
P1 Cooperate +1, +1 -1, +2
P2 Defect +2, -1 0, 0

These games are nonzero-sum, meaning that the total amount of benefit or harm incurred is not constant; it depends upon what players choose to do. In my example, the total benefit varies from +2 (both cooperate) to +1 (one cooperates, one defects) to 0 (both defect).

The answer which is “neat, plausible, and wrong” (to use Mencken’s oft-misquoted turn of phrase) is to reason this way: If the other player cooperates, I can get +1 if I cooperate, or +2 if I defect. So I should defect. If the other player defects, I can get -1 if I cooperate, or 0 if I defect. So I should defect. In either case I defect, therefore I should always defect.

The problem with this argument is that your behavior affects the other player. You can’t simply hold their behavior fixed when making your choice. If you always defect, the other player has no incentive to cooperate, so you both always defect and get 0. But if you credibly promise to cooperate every time they also cooperate, you create an incentive to cooperate that can get you both +1 instead.

If there were a fixed amount of benefit, the game would be zero-sum, and cooperation would always be damaging yourself. In zero-sum games, the assumption that acting selfishly maximizes your payoffs is correct; we could still debate whether it’s necessarily more rational (I don’t think it’s always irrational to harm yourself to benefit someone else an equal amount), but it definitely is what maximizes your money.

But in nonzero-sum games, that assumption no longer holds; we can both end up better off by cooperating than we would have been if we had both defected.
Below is a very simple zero-sum game (notice how indeed in each outcome, the payoffs sum to zero; any zero-sum game can be written so that this is so, hence the name):

Player 2 cooperates Player 2 defects
Player 1 cooperates 0, 0 -1, +1
Player 1 defects +1, -1 0, 0

In that game, there really is no reason for you to cooperate; you make yourself no better off if they cooperate, and you give them a strong incentive to defect and make you worse off. But that game is not a Prisoner’s Dilemma, even though it may look superficially similar.

The real world, however, is full of variations on the Prisoner’s Dilemma. This sort of situation is fundamental to our experience; it probably happens to you multiple times every single day.
When you finish eating at a restaurant, you could pay the bill (cooperate) or you could dine and dash (defect). When you are waiting in line, you could quietly take your place in the queue (cooperate) or you could cut ahead of people (defect). If you’re married, you could stay faithful to your spouse (cooperate) or you could cheat on them (defect). You could pay more for the shoes made in the USA (cooperate), or buy the cheap shoes that were made in a sweatshop (defect). You could pay more to buy a more fuel-efficient car (cooperate), or buy that cheap gas-guzzler even though you know how much it pollutes (defect). Most of us cooperate most of the time, but occasionally are tempted into defecting.

The “Prisoner’s Dilemma” is honestly not much of a dilemma. A lot of neoclassical economists really struggle with it; their model of rational behavior is so narrow that it keeps putting out the result that they are supposed to always defect, but they know that this results in a bad outcome. More recently we’ve done experiments and we find that very few people actually behave that way (though typically neoclassical economists do), and also that people end up making more money in these experimental games than they would if they behaved as neoclassical economics says would be optimal.

Let me repeat that: People make more money than they would if they acted according to what’s supposed to be optimal according to neoclassical economists. I think that’s why it feels like such a paradox to them; their twin ideals of infinite identical psychopaths and maximizing the money you make have shown themselves to be at odds with one another.

But in fact, it’s really not that paradoxical: Rationality doesn’t mean being maximally selfish at every opportunity. It also doesn’t mean maximizing the money you make, but even if it did, it still wouldn’t mean being maximally selfish.

We have tested experimentally what sort of strategy is most effective at making the most money in the Prisoner’s Dilemma; basically we make a bunch of competing computer programs to play the game against one another for points, and tally up the points. When we do that, the winner is almost always a remarkably simple strategy, called “Tit for Tat”. If your opponent cooperated last time, cooperate. If your opponent defected last time, defect. Reward cooperation, punish defection.

In more complex cases (such as allowing for random errors in behavior), some subtle variations on that strategy turn out to be better, but are still basically focused around rewarding cooperation and punishing defection.
This probably seems quite intuitive, yes? It may even be the strategy that it occurred to you to try when you first learned about the game. This strategy comes naturally to humans, not because it is actually obvious as a mathematical result (the obvious mathematical result is the neoclassical one that turns out to be wrong), but because it is effective—human beings evolved to think this way because it gave us the ability to form stable cooperative coalitions. This is what gives us our enormous evolutionary advantage over just about everything else; we have transcended the limitations of a single individual and now work together in much larger groups. E.O. Wilson likes to call us “eusocial”, a term formally applied only to a very narrow range of species such as ants and bees (and for some reason, naked mole rats); but I don’t think this is actually strong enough, because human beings are social in a way that even ants are not. We cooperate on the scale of millions of individuals, who are basically unrelated genetically (or very distantly related). That is what makes us the species who eradicate viruses and land robots on other planets. Much more so than intelligence per se, the human superpower is cooperation.

Indeed, it is not a great exaggeration to say that morality exists as a concept in the human mind because cooperation is optimal in many nonzero-sum games such as these. If the world were zero-sum, morality wouldn’t work; the immoral action would always make you better off, and the bad guys would always win. We probably would never even have evolved to think in moral terms, because any individual or species that started to go that direction would be rapidly outcompeted by those that remained steadfastly selfish.

Monopoly and Oligopoly

JDN 2457180 EDT 08:49

Welcome to the second installment in my series, “Top 10 Things to Know About Economics.” The first was not all that well-received, because it turns it out it was just too dense with equations (it didn’t help that the equation formatting was a pain.) Fortunately I think I can explain monopoly and oligopoly with far fewer equations—which I will represent as PNG for your convenience.

You probably already know at least in basic terms how a monopoly works: When there is only one seller of a product, that seller can charge higher prices. But did you ever stop and think about why they can charge higher prices—or why they’d want to?

The latter question is not as trivial as it sounds; higher prices don’t necessarily mean higher profits. By the Law of Demand (which, like the Pirate Code, is really more like a guideline), raising the price of a product will result in fewer being sold. There are two countervailing effects: Raising the price raises the profits from selling each item, but reduces the number of items sold. The optimal price, therefore, is the one that balances these two effects, maximizing price times quantity.

A monopoly can actually set this optimal price (provided that they can figure out what it is, of course; but let’s assume they can). They therefore solve this maximization problem for price P(Q) a function of quantity sold, quantity Q, and cost C(Q) a function of quantity produced (which at the optimum is equal to quantity sold; no sense making them if you won’t sell them!):


As you may remember if you’ve studied calculus, the maximum is achieved at the point where the derivative is zero. If you haven’t studied calculus, the basic intuition here is that you move along the curve seeing whether the profits go up or down with each small change, and when you reach the very top—the maximum—you’ll be at a point where you switch from going up to going down, and at that exact point a small change will move neither up nor down. The derivative is really just a fancy term for the slope of the curve at each point; at a maximum this slope changes from positive to negative, and at the exact point it is zero.



This is a general solution, but it’s easier to understand if we use something more specific. As usual, let’s make things simpler by assuming everything is linear; we’ll assume that demand starts at a maximum price of P0 and then decreases at a rate 1/e. This is the demand curve.


Then, we’ll assume that the marginal cost of production C'(Q) is also linear, increasing at a rate 1/n. This is the supply curve.


Now we can graph the supply and demand curves from these equations. But the monopoly doesn’t simply set supply equal to demand; instead, they set supply equal to marginal revenue, which takes into account the fact that selling more items requires lowering the price on all of them. Marginal revenue is this term:


This is strictly less than the actual price, because increasing the quantity sold requires decreasing the price—which means that P'(Q) < 0. They set the quantity by setting marginal revenue equal to marginal cost. Then they set the price by substituting that quantity back into the demand equation.

Thus, the monopoly should set this quantity:


They would then charge this price (substitute back into the demand equation):


On a graph, there are the supply and demand curves, and then below the demand curve, the marginal revenue curve; it’s the intersection of that curve with the supply curve that the monopoly uses to set its quantity, and then it substitutes that quantity into the demand curve to get the price:


Now I’ll show that this is higher than the price in a perfectly competitive market. In a competitive market, competitive companies can’t do anything to change the price, so from their perspective P'(Q) = 0. They can only control the quantity they produce and sell; they keep producing more as long as they receive more money for each one than it cost to produce it. By the Law of Diminishing Returns (again more like a guideline) the cost will increase as they produce more, until finally the last one they sell cost just as much to make as they made from selling it. (Why bother selling that last one, you ask? You’re right; they’d actually sell one less than this, but if we assume that we’re talking about thousands of products sold, one shouldn’t make much difference.)

Price is simply equal to marginal cost:


In our specific linear case that comes out to this quantity:


Therefore, they charge this price (you can substitute into either the supply or demand equations, because in a competitive market supply equals demand):


Subtract the two, and you can see that monopoly price is higher than the competitive price by this amount:


Notice that the monopoly price will always be larger than the competitive price, so long as e > 0 and n > 0, meaning that increasing the quantity sold requires decreasing the price, but increasing the cost of production. A monopoly has an incentive to raise the price higher than the competitive price, but not too much higher—they still want to make sure they sell enough products.

Monopolies introduce deadweight loss, because in order to hold the price up they don’t produce as many products as people actually want. More precisely, each new product produced would add overall value to the economy, but the monopoly stops producing them anyway because it wouldn’t add to their own profits.

One “solution” to this problem is to let the monopoly actually take those profits; they can do this if they price-discriminate, charging a higher price for some customers than others. In the best-case scenario (for them), they charge each customer a price that they are just barely willing to pay, and thus produce until no customer is willing to pay more than the product costs to make. That final product sold also has price equal to marginal cost, so the total quantity sold is the same under competition. It is, in that sense, “efficient”.

What many neoclassical economists seem to forget about price-discriminating monopolies is that they appropriate the entire surplus value of the product—the customers are only just barely willing to buy; they get no surplus value from doing so.

In reality, very few monopolies can price-discriminate that precisely; instead, they put customers into broad categories and then try to optimize the price for each of those categories. Credit ratings, student discounts, veteran discounts, even happy hours are all forms of this categorical price discrimination. If the company cares even a little bit about what sort of customer you are rather than how much money you’re paying, they are price-discriminating.

It’s so ubiquitous I’m actually having trouble finding a good example of a product that doesn’t have categorical price discrimination. I was thinking maybe computers? Nope, student discounts. Cars? No, employee discounts and credit ratings. Refrigerators, maybe? Well, unless there are coupons (coupons price discriminate against people who don’t want to bother clipping them). Certainly not cocktails (happy hour) or haircuts (discrimination by sex, the audacity!); and don’t even get me started on software.

I introduced price-discrimination in the context of monopoly, which is usually how it’s done; but one thing you’ll notice about all the markets I just indicated is that they aren’t monopolies, yet they still exhibit price discrimination. Cars, computers, refrigerators, and software are made under oligopoly, a system in which a handful of companies control the majority of the market. As you might imagine, an oligopoly tends to act somewhere in between a monopoly and a competitive market—but there are some very interesting wrinkles I’ll get to in a moment.

Cocktails and haircuts are sold in a different but still quite interesting system called monopolistic competition; indeed, I’m not convinced that there is any other form of competition in the real world. True perfectly-competitive markets just don’t seem to actually exist. Under monopolistic competition, there are many companies that don’t have much control over price in the overall market, but the products they sell aren’t quite the same—they’re close, but not equivalent. Some barbers are just better at cutting hair, and some bars are more fun than others. More importantly, they aren’t the same for everyone. They have different customer bases, which may overlap but still aren’t the same. You don’t just want a barber who is good, you want one who works close to where you live. You don’t just want a bar that’s fun; you want one that you can stop by after work. Even if you are quite discerning and sensitive to price, you’re not going to drive from Ann Arbor to Cleveland to get your hair cut—it would cost more for the gasoline than the difference. And someone is Cleveland isn’t going to drive all the way to Ann Arbor, either! Hence, barbers in Ann Arbor have something like a monopoly (or oligopoly) over Ann Arbor haircuts, and barbers in Cleveland have something like a monopoly over Cleveland haircuts. That’s monopolistic competition.

Supposedly monopolistic competition drives profits to zero in the long run, but I’ve yet to see this happen in any real market. Maybe the problem is that conceit “the long run”; as Keynes said, “in the long run we are all dead.” Sometimes the argument is made that it has driven real economic profits to zero, because you’ve got to take into account the cost of entry, the normal profit. But of course, that’s extremely difficult to measure, so how do we know whether profits have been driven to normal profit? Moreover, the cost of entry isn’t the same for everyone, so people with lower cost of entry are still going to make real economic profits. This means that the majority of companies are going to still make some real economic profit, and only the ones that had the hardest time entering will actually see their profits driven to zero.

Monopolistic competition is relatively simple. Oligopoly, on the other hand, is fiercely complicated. Why? Because under oligopoly, you actually have to treat human beings as human beings.

What I mean by that is that under perfect competition or even monopolistic competition, the economic incentives are so powerful that people basically have to behave according to the neoclassical rational agent model, or they’re going to go out of business. There is very little room for errors or even altruistic acts, because your profit margin is so tight. In perfect competition, there is literally zero room; in monopolistic competition, the only room for individual behavior is provided by the degree of monopoly, which in most industries is fairly small. One person’s actions are unable to shift the direction of the overall market, so the market as a system has ultimate power.

Under oligopoly, on the other hand, there are a handful of companies, and people know their names. You as a CEO have a reputation with customers—and perhaps more importantly, a reputation with other companies. Individual decision-makers matter, and one person’s decision depends on their prediction of other people’s decision. That means we need game theory.

The simplest case is that of duopoly, where there are only two major companies. Not many industries are like this, but I can think of three: soft drinks (Coke and Pepsi), commercial airliners (Boeing and Airbus), and home-user operating systems (Microsoft and Apple). In all three cases, there is also some monopolistic element, because the products they sell are not exactly the same; but for now let’s ignore that and suppose they are close enough that nobody cares.

Imagine yourself in the position of, say, Boeing: How much should you charge for an airplane?

If Airbus didn’t exist, it’s simple; you’d charge the monopoly price. But since they do exist, the price you charge must depend not only on the conditions of the market, but also what you think Airbus is likely to do—and what they are likely to do depends in turn on what they think you are likely to do.

If you think Airbus is going to charge the monopoly price, what should you do? You could charge the monopoly price as well, which is called collusion. It’s illegal to actually sign a contract with Airbus to charge that price (though this doesn’t seem to stop cable companies or banks—probably has something to do with the fact that we never punish them for doing it), and let’s suppose you as the CEO of Boeing are an honest and law-abiding citizen (I know, it’s pretty fanciful; I’m having trouble keeping a straight face myself) and aren’t going to violate the antitrust laws. You can still engage in tacit collusion, in which you both charge the monopoly price and take your half of the very high monopoly profits.

There’s a temptation not to collude, however, which the airlines who buy your planes are very much hoping you’ll succumb to. Suppose Airbus is selling their A350-100 for $341 million. You could sell the comparable 777-300ER for $330 million and basically collude, or you could cut the price and draw in more buyers. Say you cut it to $250 million; it probably only costs $150 million to make, so you’re still making a profit on each one; but where you sold say 150 planes a year and profited $180 million on each (a total profit of $27 billion), you could instead capture the whole market and sell 300 planes a year and profit $100 million on each (a total profit of $30 billion). That’s a 10% higher profit and $3 billion a year for your shareholders; why wouldn’t you do that?

Well, think about what will happen when Airbus releases next year’s price list. You cut the price to $250 million, so they retaliate by cutting their price to $200 million. Next thing you know, you’re cutting your own price to $150.1 million just to stay in the market, and they’re doing the same. When the dust settles, you still only control half the market, but now you profit a mere $100,000 per airplane, making your total profits a measly $15 million instead of $27 billion—that’s $27,000 million. (I looked it up, and as it turns out, Boeing’s actual gross profit is about $14 billion, so I underestimated the real cost of each airplane—but they’re clearly still colluding.) For a gain of 10% in one year you’ve paid a loss of 99.95% indefinitely. The airlines will be thrilled, and they’ll likely pass on much of those savings to their customers, who will fly more often, engage in more tourism, and improve the economy in tourism-dependent countries like France and Greece, so the world may well be better off. But you as CEO of Boeing don’t care about the world; you care about the shareholders of Boeing—and the shareholders of Boeing just got hosed. Don’t expect to keep your seat in the next election.

But now, suppose you think that Airbus is planning on setting a price of $250 million next year anyway. They should know you’ll retaliate, but maybe their current CEO is retiring next year and doesn’t care what happens to the company after that or something. Or maybe they’re just stupid or reckless. In any case, your sources (which, as an upstanding citizen, obviously wouldn’t include any industrial espionage!) tell you that Airbus is going to charge $250 million next year.

Well, in that case there’s no point in you charging $330 million; you’ll lose the market and look like a sucker. You could drop to $250 million and try to set up a new, lower collusive equilibrium; but really what you want to do is punish them severely for backstabbing you. (After all, human beings are particularly quick to anger when we perceive betrayal. So maybe you’ll charge $200 million and beat them at their own conniving game.

The next year, Airbus has a choice. They could raise back to $341 million and give you another year of big profits to atone for their reckless actions, or they could cut down to $180 million and keep the price war going. You might think that they should continue the war, but that’s short-term thinking; in the long run their best strategy is to atone for their actions and work to restore the collusion. In response, Boeing’s best strategy is to punish them when they break the collusion, but not hold a grudge; if they go back to the high price, Boeing should as well. This very simple strategy is called tit-for-tat, and it is utterly dominant in every simulation we’ve ever tried of this situation, which is technically called an iterated prisoner’s dilemma.

What if there are more than two companies involved? Then things get even more complicated, because now we’re dealing with things like what A’s prediction of what B predicts that C will predict A will do. In general this is a situation we only barely understand, and I think it is a topic that needs considerably more research than it has received.

There is an interesting simple model that actually seems to capture a lot about how oligopolies work, but no one can quite figure out why it works. That model is called Cournot competition. It assumes that companies take prices and fixed and compete by selecting the quantity they produce at each cycle. That’s incredibly bizarre; it seems much more realistic to say that they compete by setting prices. But if you do that, you get Bertrand competition, which requires us to go through that whole game-theory analysis—but now with three, or four, or ten companies!

Under Cournot competition, you decide how much to produce Q1 by monopolizing what’s left over after the other companies have produced their quantities Q2, Q3, and so on. If there are k companies, you optimize under the constraint that (k-1)Q2 has already been produced.

Let’s use our linear models again. Here, the quantity that goes into figuring the price is the total quantity, which is Q1+(k-1)Q2; while the quantity you sell is just Q1. But then, another weird part is that for the marginal cost function we use the whole market—maybe you’re limited by some natural resource, like oil or lithium?

It’s not as important for you to follow along with the algebra, though here you go if you want:


Then the key point is that the situation is symmetric, so Q1 = Q2 = Q3 = Q. Then the total quantity produced, which is what consumers care about, is kQ. That’s what sets the actual price as well.


The two equations to focus on are these ones:


If you plug in k=1, you get a monopoly. If you take the limit as k approaches infinity, you get perfect competition. And in between, you actually get a fairly accurate representation of how the number of companies in an industry affects the price and quantity sold! From some really bizarre assumptions about how competition works! The best explanation I’ve seen of why this might happen is this 1983 paper showing that price competition can behave like Cournot competition if companies have to first commit to producing a certain quantity before naming their prices.

But of course, it doesn’t always give an accurate representation of oligopoly, and for that we’ll probably need a much more sophisticated multiplayer game theory analysis which has yet to be done.

And that, dear readers, is how monopoly and oligopoly raise prices.