The famous game theorist John Nash and his wife Alicia were killed in a traffic accident yesterday in New Jersey. He was 86.

I met him and had lunch with him when he came to speak at the Naval Postgraduate School in Monterey a few years ago.

The movie about him, *A Beautiful Mind*, is very misleading, but Sylvia Nasar's book by the same title seems highly accurate.

Here's his bio in *The Concise Encyclopedia of Economics*.

An excerpt from that bio:

A simple example of a Nash equilibrium is the prisoners' dilemma. Another example is the location problem. Imagine that Budweiser and Miller are trying to decide where to place their beer stands on a beach that is perfectly straight. Assume also that sunbathers are located an equal distance from each other and that they want to minimize the distance they walk to get a beer. Where, then, should Bud locate if Miller has not yet chosen its location? If Bud locates one-quarter of the way along the beach, then Miller can locate next to Bud and have three-quarters of the market. Bud knows this and thus concludes that the best location is right in the middle of the beach. Miller locates just slightly to one side or the other. Neither Bud nor Miller can improve its position by choosing an alternate location. This is a Nash equilibrium.

HT to Michael Munger.

Nasar's book isn't accurate on the economics. She shows repeatedly that she doesn't understand Nash Equilibrium. The best example of this is her depiction of the first laboratory game. This took place at Rand I think in the 1950s, with Armen Alchian playing a prisoner's dilemma game against someone else for a fixed number of periods. Armen played the right strategy: always defect! Nasar depicts him unkindly, as confused and unaware that he and the other player could both be better off if only they both cooperated. For me, 99% of the value of Nasar's book is the transcript of this game, which has great historical value. After each period, the participants jotted down their thoughts on a card. Armen's notes provide brilliant insights into this brilliant economist.

I never really understood the use of the prisoner's dilemma when describing Nash Equilibrium. The prisoner's dilemma has a Nash Equilibrium, but not a very interesting one, because it is also a dominant strategy equilibrium. It therefore doesn't show what's special about Nash Equilibrium. All DSEs are also NEs, but not all NEs are DSEs, and it's the games that have NEs but not DSEs that really show Nash's contribution. If I recall correctly, von Neumann came up with DSEs years before Nash came up with Nash Equilibrium (though von Neumann was focused on zero-sum games).

The Battle of the Sexes is a better example for demonstrating what is special about Nash Equilibrium.

This confusion between dominant strategy equilibrium and Nash equilibrium also afflicts the bar scene in the movie. It's great at demonstrating a prisoner's dilemma-style game, but not at showing what Nash Equilibrium adds.

Thanks, AlexR. I never read the whole of

A Beautiful Mind, which is why I said itseemshighly accurate. It was the mental health parts that I found interesting. I thought I recalled various game theorists saying that Sylvia got it right, but maybe I recalled wrong.Also, I thought I recalled Nash thinking Armen was stupid because he

didn’tdefect but instead tried to cooperate. Are you sure you’re right about this?Thanks also, Mike Hammock. My recollection was that the bar scene even screwed up on the prisoner’s dilemma. That’s what I had in mind when I said the movie was misleading. Am I recalling incorrectly?

@AlexR,

My marked-up copy of

A Beautiful Mindburned in my 2007 fire. So I went from memory above. But I did a search and here’s a quote from the book:Though Williams and Alchian didn’t always cooperate, the results hardly resembled a Nash equilibrium. Drescher and Flood argued, and von Neumann apparently agreed, that their experiment showed that the players tended not to choose Nash equilibrium strategies and instead were likely to “split the difference."So are you, saying, Alex, that Sylvia is misreporting here?

Wouldn't always cooperating be a better solution in an iterated Prisoner's Dilemma?

David,

Nash was critical of the Rand game design, as I recall. Apparently the designers believed that running the prisoner's dilemma game for a large number of rounds (100, I seem to recall) created scope for an equilibrium with cooperation. Nash explained that this is wrong, and Armen likewise intuited that, with a fixed and known number of periods, cooperation unravels. What's remarkable to me is that, again as I recall, the game took place shortly after publication of Nash's paper, and I don't think Armen was aware of it.

First, I'd like to say that the acting in the bar scene is terrible. It's really hammy and hard to watch.

Second, I agree that the scene is misleading, or at least confusing, in terms of explaining the prisoner's dilemma. If Nash (in the movie) is saying that the equilibrium is that none of them go for the blonde (and let's leave aside the questionable treatment of women in this whole bit), that's clearly wrong. If none of them go for the blonde, then each of them wants to deviate and go for the blonde. That's not an equilibrium.

Third, If he's saying that the equilibrium is that they go for the blonde, but they're all worse off than if they had coordinated--and unavoidably stuck with this situation--this is indeed like the prisoner's dilemma. But as I said before, it's a dominant strategy equilibrium, as well as a Nash equilibrium, and therefore not a good demonstration of Nash's contribution.

Fourth, I'm not sure that model describes the scenario presented in the movie. Let's simplify it to two players. I can't draw a standard form game here, but you can draw it yourself using the following information:

Player 1 (row) and player 2 (column) are choosing between two strategies, blonde and brunette. If they both choose blonde, they both get zero (because the fight over the blonde means neither gets her, and the brunettes don't want to be second place). If player 1 goes for the blonde, and player 2 goes for the brunette, player 1 gets a payoff of 2, and player 2 gets a payoff of 1 (because they each get a woman, but the blonde is "better" than the brunette). Conversely, if player 1 goes for the brunette, and player 2 goes for the blonde, they get payoffs of 1 and 2, respectively.

Put this in a standard form game box, and it looks like there are two Nash equilibria: (Blonde, Brunette) and (Brunette, Blonde). In other words, if one of the leering grad students gets the blonde and the other gets the brunette, neither wants to deviate. They would both want to deviate from the other positions (Blonde, Blonde and Brunette, Brunette). The problem, of course--and what makes Nash Equilibria interesting--is that it's hard to figure out what they will do if they can't communicate or bargain (which is the standard assumption in these games). It would have been great if the movie had said that these were the equilibria of the model, since it would have genuinely shown Nash's insight.

Finally, it's really not fair to Adam Smith, as Smith didn't say that people should always and everywhere pursue their self-interest.

Here's the bar scene.

The guy who is smoking and asks "Have you remembered nothing," is simply painful to watch. Or maybe I've watched this scene too many times.

How embarrassing! I left out something important from my game above.

If both players choose Brunette, they each get a payoff of 1 (because they've each got a brunette, but apparently they consider them inferior to the blonde).

So the payoffs in the top row are (0,0) and (2,1), and in the bottom row they are (1,2) and (1,1).

@AlexR,

I’m still not clear. Are you saying that Sylvia Nasar misreported about Armen?

David,

It depends on what you mean by "misreporting." I don't recall the name of Armen's opponent in the game, but I do recall that Nasar thanked him in her acknowledgements, whereas Armen isn't thanked. Presumably Armen declined to be interviewed for the book. So Nasar reported the opponent's view of events, presumably accurately. If you're still confused, do a Google Book search--I wouldn't recommend buying another copy of the book! The key point is that Alchian was trashed unjustly. Maybe it was an honest mistake, maybe deliberate. Journalists are notorious for kindly depicting those who cooperate with them and trashing those who don't.

David,

I just pulled up the relevant passage from the current edition using Google Books, and it's far better than I recall. So either my recollection was seriously faulty or Nasar cleaned up her text in later editions (or a combination of both). But the current edition still describes Williams (Armen's opponent) as "understanding" that they should cooperate and "punishing" Armen when he defected. I won't strain my recollection further, but would encourage anyone to read the transcript of the game and draw their own conclusions about what Armen did and didn't understand about the game.

@AlexR,

I found it. And it turned out that when Armen died, I did reference it.

Here’s the running transcript.

Here’s my post.

Thanks David, that's a great post! But I feel compelled to elaborate a bit in further defense of Armen. It's clear from the running script that Armen's initial expectation was that both he and Williams would play "Defect" (D) in every round. This is the unique undominated strategy. Nash knew it, and so did Armen. Every one else, even von Neumann, got it wrong! After, to Armen's surprise, Williams persisted in playing C for several rounds despite Armen's D, it became clear to Armen that he wasn't playing against a rational optimizer. What to do then? It looks like Armen experimented for awhile to figure out William's type. Such experimentation is clearly valuable and rational! Eventually, Armen figured out that Williams wasn't trying to "trick" him, but was just eager to cooperate. What's the optimal response to such a type? Keep cooperating with him and try to guess when the light bulb will come on over the other guy and he'll realize that cooperation must unravel toward the end. Then try to time it so you defect in the period immediately before this realization hits. That's what Armen said he was doing in the notes! And he succeeded. The light bulb never came on for Williams, and Armen defected in the next-to-last period. So, to my mind, it's clear that Armen was acting rationally throughout. He understood the prisoner's dilemma game better than his opponent and better than the game's creators.

AA does seem to think the goal is to maximize the relative payout. The "sure win" in the first comment shows that. Also, the "he won't share" doesn't make much sense.

If he does think "winning" is scoring more than the other player, then he doesn't understand the game.