April 22, 2002
From the desk of Jane Galt:It's here! It's here! It's
It's here! It's here! It's finally here! It's. . .
The Nash Equilibrium and the Coase Theorem: Two Great Tastes that Taste Great Together
A reader sends in this request for clarification on two important concepts: the Nash Equilibrium, and the Coase Theorem, authored by, respectively, John Nash and Ronald Coase, and who says you don’t go into academia to become famous?
I think I must have missed your explanation of comparative advantage, but I think I followed your explanation of CAFE. Now, here's another challenge for you. What is the relationship between Nash equilibrium and the Coase Theorem? I ask because I have been told that Nash equilibrium concerns situations that wind up being stable because any change makes at least some people worse off, and the Coase Theorem concerns situations where property gets used in some particular way (the use of the property is stable, in other words), no matter who initially owns it--because otherwise at least some people are worse off. I figure that even if there is not much connection, when you show why I might actually come to some understanding of both of those concepts. Thanks. And I liked your comments on "Jurismania."
And since you asked no nicely, and flattered me outrageously, I shall attempt to answer.
The Nash Equilibrium actually demonstrates how certain situations arrive at equilibrium at the point where neither party can make themselves better off given what the other party is already doing. This can be illustrated by taking a very simple case. Say there are two executives of a firm - call it Ronin Enterprises - who have some bad information they don’t want investors to know about. If they can get to the end of the quarter without investors finding out, they think they’ll be fine.
Now let’s suppose that these executives - call them Ken and Jeff - have large chunks of stock in the company. If either of them sells this large chunk of stock, it will trigger an investigation into the financials of the company. Once investors find out about the financial problems, the stock will be worthless. Creditors will descend. Congress will convene an investigation. General havoc will ensue, possibly including a Larry King special. No one wants that.
However, if they can hold out to the end of the quarter, they think that the situation will reverse itself and they can both cash out for a tidy sum.
So if one of them sells now and the other doesn’t, whoever sells will make a lot of money, while the other person is wiped out along with a once-proud firm. If both of them sell, they’ll both be wiped out. And if neither of them sells now, they both make money. We can chart the various possible outcomes like this (numbers are purely arbitrary and not meant to hold any relation to actual values of such stock as may have been held by any pair of executives named Ken and Jeff):
As you can see, if they both sell, it's a disaster. The best thing for all concerned -- the outcome that maximizes overall value -- is for both of them to hold their stock until the analysts go back to sleep.
But that's not what's going to happen.
Let's look at why. Let's say they both start out committed to not selling. In that case, they each get a return of 100 on their stock. 100 what? I hear you say. Millions of dollars, coupons good for one large soda with any purchase of a Superburger and Cross-cut Spicy Fries at Burger Boy -- it doesn't really matter 100 what. It's a model.
Anyway, so there they are, each grimly determined to sit on their stock and take their 100 when the time comes.
But then one of them notices that if he sells now, he could clear 200 soda coupons instead of 100. The little devil sitting on his shoulder says "Go ahead -- it's a dog-eat-dog world out there. You've got to look out for #1."
Perhaps at first he's committed to staying the course. But pretty soon all he can think about is those extra hundred sodas he could be drinking. His every waking moment is dominated by feverish images of all that sugary goodness pouring down some drain somewhere instead of into his throat.
And then the little devil speaks up. "Hey, Jeff," it says. "What if Ken sells?"
Jeff thinks about that. If Ken sells, he doesn't just lose those 100 extra sodas -- he loses the hundred he's already got. In fact, he's going to owe his investors an extra 300 sodas. He'll be working at Burger Boy for years to pay them all off. And they won't even let him work the Fry-O-Lator until he's been there for six months.
'Nuff said. Jeff sells like ice cream in August.
Now let's look at Ken's position. Maybe he was honorably determined to hold his stock until the deadline; maybe he was planning to do unto Jeff as Jeff just did unto him, only Jeff was a little quicker off the mark. Doesn't matter, because no matter what his intentions were, Ken is now looking at a long stretch down at Burger Boy making sodas for Jeff.
Now remember, while it might be better for him to pull them back to the position where both aren't selling, he can't do that -- only Jeff can choose whether or not Jeff sells. And given that Jeff is selling, there's only one thing that Ken can do to improve his position.
Ken sells too.
Now they're both down at Burger Boy getting fitted for paper hats. Ken's term before he pays off his debt is slightly shortened, at the expense of an equally long term for Jeff. Notice that this is the outcome that minimizes value; the total value of this outcome to the two participants is -500, versus -100 if one sells and one doesn't, and +200 if they both hold on.
It's also what's known as the Nash Equilibrium. It is the only outcome where neither participant can better their position by changing what he himself is already doing, given what the other participant is already doing.
Now let's travel six months forward in time, when Ken and Jeff are settling down to their new home at the Winnemaka State Correctional Facility. Unaware of the lingering resentments stemming from their stock market contretemps, the warden makes them roommates. And as they are unpacking, Jeff tries to leaven the dark mood in the cell by pulling out his portable turntable and playing one of his classic big band records.
Well, if there's anything that Ken hates more than double-crossing, back-stabbing business partners, it's big band music. The sound of clarinet music, frankly, makes him break out in hives. And there on the other side of the cell is Jeff, unpacking a couple of cubic feet of Ella Fitzgerald, Benny Goodman, and Duke Ellington. Let's just say that things are a little tense as the first bars of "Sing! Sing! Sing!" float through the reinforced-concrete halls of Cell Block C.
Now say that Ken, upon demanding to see the warden, finds that Jeff has a perfect right to listen to his big band records between the hours of 8 am and 8 pm, according to prison rules. Stone walls may not a prison make, but Jeff is allowed to transport Ken to his own personal vision of hell for 12 hours a day, and there is nothing he can do about it.
Or can he? Let us imagine that Ken's wife, in whom all of his non-stock wealth was vested, has lavishly funded his account at the prison commissary to the tune of $1000. It occurs to Ken, being the smooth-wheeler dealer that he is, that, much as it galls him to do so, he could pay Jeff not to play his music.
Now, we're told time and time again that such things as music are priceless, but this is not true, as you will find out very quickly if you attempt to take some from Tower Records without paying. If you'll follow the simple excercise below, you'll find that sooner or later, we can put a price on just about anything.
Think of something you hate, something you'd never, ever do, like -- eating a bug. Would you do it if I paid you a dollar? No, you say? Never? How about if I paid you a billion dollars? $1,000,0000,000 in the bank just for eating one, harmless little bug. A billion dollars could buy a lot of therapy. Now open up and let batman fly into the cave. . .
Most things, even silence, can be bought -- if the price is high enough.
So let's imagine that Jeff, although devoted to swing, is also devoted to Mars bars, his supply of which is inadequate due to a stingy wife. He realizes, privately, that for as little as $600 he would be willing to give up his swing records and content himself with the sound of roasted peanuts wrapped in luscious caramel and creamy milk chocolate crunchiing between his teeth.
Ken, on the other hand, just quit smoking and went on the Atkins diet. He'd be more than willing to fork over his entire commissary account for the privilege of not listening to one more rendition of "Chattanooga Choo-Choo".
What's the solution? Obviously Ken pays Jeff some sum between $600 and $1000 -- how much is a matter of negotiation between the two of them -- and the cell echoes with golden silence.
But what if Jeff didn't have the right to listen to his music? What if rather than saying that all inmates could listen to music from 8-8, the warden had said that they could do so only if they got their cellmate's permission? Suddenly Ken has all the power. What would the outcome be?
The outcome would be the same -- silence. Jeff isn't willing to pay Ken enough to be allowed to listen to Ella and Benny.
That's the heart of the Coase Theorem: regardless of who has the rights, common areas will end up being put to the highest-valued use.
Suddenly you're a little confused. What do you mean by highest valued? Jeff values his use of the common space to listen to music at $599 (the most, we assume, that he would be willing to pay to hear his music, given that he will sell his right to do so for $600), while Ken values his use, not listening to music at $1000. Silence wins because in the marketplace of the cell, with these two consumers, it has a higher value ($1000) than music ($599). And where there is a market for rights, the highest valued allocation of those rights wins.
Another way to look at it is to think of the right to pollute (in the broadest possible sense) as an asset. It doesn't matter who initially owns the asset; it will end up with the person who values it the most.
That's the beauty of the Coase Theorem. Think of any situation with conflicting rights: the environment, noise pollution, the right not to see leather pants on those who don't have the kind of body that makes leather pants attractive. It doesn't matter whether the polluter (chemical, noise, or visual) has the right to pollute, or you have the right not to be polluted; the outcome will be the same, because whoever values their outcome more will invariably win. It's a revolutionary idea.
It's also simplistic, as all revolutionary ideas are. For one thing, it assumes that there are no transaction costs, which is to say the costs, monetary and non-monetary, associated with doing a transaction on top of the fee exchanged. Let's take a form of negative externality with which we're all familiar: spam. I find spam so annoying that I would probably be willing to pay some amount of money to be left alone. But in order to do that I would have to find all the spammers and pay each of them to leave me alone, which might push my costs, in time and money, much higher than I would be willing to pay just to be free of daily exhortations to lose weight, grow hair, or sample the brand-new webcams of Barely Legal Teenage Girls.
It also assumes perfect enforcement. I might pay all those spammers just to find that they've moved their servers to Taiwan, changed their names by one letter, and resumed their torrent of unwanted invitations.
And it assumes that you're allowed to make a market. Much as I dislike the kid next door dealing crack in the yard, I might be willing to live with it for a percentage of the take. But the cops take a dim view of this sort of thing. And the EPA won't let me take money to let International Paper dump dioxins in my percentage of the water table.
Nor does it cope with the justice of the allocation of rights -- should I have to pay not to have some old fogey blasting "Rum and Coca Cola" into my ears at all hours? And conversely, shouldn't I be able to enjoy a little music now and again? Coase only tells us how things will be allocated; not, in some metaphysical way, how they should. That's worrysome, if you're the type of person who seeks metaphysical justice from the law.
Nonetheless, the Coase Theorem has changed the way economists view property rights and externalities sufficiently for them to give him a Nobel Prize for it. And well deserved, if you want my opinion, which I'm pretty sure nobody does.
So that is the Nash Equilibrium and the Coase Theorem. And to answer your question, gentle reader, there's really not much relationship between them, except that they're both named after Nobel Laureates, and they're both really cool. And, of course, that hopefully you now understand both of them.
Posted by Jane Galt at April 22, 2002 8:32 PM | TrackBack | Technorati inbound links