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## Russell Crowe was wrong

Yesterday I taught the first of five algorithmic economics lectures in my undergraduate AI course. This lecture just introduced the basic concepts of game theory, focusing on Nash equilibria. I was contemplating various ways of making the lecture more lively, and it occurred to me that I could stand on the shoulders of giants. Indeed, didn’t Russell Crowe already explain Nash’s ideas in A Beautiful Mind, complete with a 1940’s-style male chauvinistic example?

The first and last time I watched the movie was when it was released in 2001. Back then I was an undergrad freshman, working for 20+ hours a week on the programming exercises of Hebrew U’s Intro to CS course, which was taught by some guy called Noam Nisan. I didn’t know anything about game theory, and Crowe’s explanation made a lot of sense at the time.

I easily found the relevant scene on youtube. In the scene, Nash’s friends are trying to figure out how to seduce a beautiful blonde and her less beautiful friends. Then Nash/Crowe has an epiphany. The hubbub of the seedy Princeton bar is drowned by inspirational music, as Nash announces:

If we all go for the blonde, we block each other. Not a single one of us is gonna get her. So then we go for her friends, but then they give us the cold shoulder because nobody likes to be second choice. Well, what if no one goes for the blonde? We don’t get in each other’s way, and we don’t insult the other girls. That’s the only way we win. … Adam Smith said the best result comes from everyone in the group doing what’s best for himself, right? … The best result would come from everyone in the group doing what’s best for himself and the group!

But if no one goes for the blonde, wouldn’t a player gain from deviating to the blonde, without making others worse off? This game has Pareto efficient Nash equilibria (one player goes for the blonde, the others go for her friends), but the strategy profile advocated by Crowe is not Pareto efficient, nor an equilibrium.

Nash concludes by proclaiming “Adam Smith was wrong!” Maybe, but so was Russell Crowe.

### 17 Responses

1. That scene was not explaining Nash equilibrium but the Nash Bargaining Solution. A big difference.

• I don’t think this interpretation works either, because the point of bargaining solutions is to reach Pareto efficient outcomes, and the outcome proposed by the movie is not Pareto efficient.

• Why isn’t it Pareto efficient. Who would be made better off without others worse off here?

• The player who deviates to the blonde would be better off while the others can still go for her friends.

• That;s Nash equilibrium not Nash bargaining or Pareto optimality. There are two possible outcomes: (i) they race for the Blonde or (ii) they don’t. They chose (ii). They were both on the Pareto frontier but negotiated to one of them. It is basic cooperative game theory. Why do you keep applying non-cooperative game theory here?

• I wanted to look at a noncooperative game to increase the chances of mapping at least one of Nash’s 1950 contributions to this game… In any case, whether you model the game as a cooperative or noncooperative game, IMHO the most natural interpretation is to include the outcomes where one player gets/goes for the blonde and the other player(s) get/go for her friend(s); these outcomes Pareto-dominate the ones that you mentioned. We may be over-analyzing this though 🙂

2. This is why Russell Crowe is definitely an Australian (when he does something right, we call him a New Zealander) 🙂
http://en.wikipedia.org/wiki/Russell_Crowe

3. Is it possible that he was thinking within the framework of cooperative game theory rather than non-cooperative game theory? It seems to me that he was proposing a group strategy that maximised their collective interest. Of course in reality it is difficult to enforce agreements, which is probably the main reason that the temptation to deviate cannot be ruled out unless in abstract analysis where a coalition is assumed to be well coordinated towards collective goals.
BTW, your class seems to be extremely interesting! I hope the students appreciate the thought and energy you put into your teaching. Best wishes for the rest of the semester..

4. a quick follow-up in response to your question “But if no one goes for the blonde, wouldn’t a player gain from deviating to the blonde, without making others worse off?”:
I think the players who didn’t deviate will be worse off since they can easily imagine themselves in the place of the player who deviated and are unlikely to appreciate the fact that they actively passed on their chance for a higher payoff. Consider the following payoff structure for a two-player game as a representation of the situation in the movie:

for player 1: pi(D,C) >= pi(C,C) >= pi(D,D) >= pi(C,D)
for player 2: pi(C,D) >= pi(C,C) >= pi(D,D) >= pi(D,C)

As illustrated above, mutual cooperation is both players’ second choice, whereas unilateral defection is the defector’s best and the cooperator’s worst payoff. Thus, I think there is a type of stability (due to relative fairness) in the group strategy proposed in the movie which is absent in the scenario described in your question.

• I agree, if we think of envy as a negative externality then the movie’s outcome is efficient, and can potentially be the Nash bargaining solution given appropriate payoffs. This may not be the most direct interpretation of the game though 🙂

5. In a class at U Freiburg I used this XKCD comic to explain this: http://xkcd.com/182/

6. This is the most efficient outcome among envy-free solutions! It is stable in this sense.

• I think one can randomly select a guy to go for the blonde and others go for the friends. I think it is the most effcient outcome in envy-free solutions.

7. Maybe Russel Crowe just (sort of) defined the Price of Anarchy independent of Koutsoupias and Papadimitriou 😀

8. As long as only one of them comes to your conclusion, then all is well. Otherwise…

9. Deviating by switching to the blond girl might result in getting her cold shoulder because she might not want to be second choice either.

So Crowe was probably having a two step game in Mind: All guys who go for the same girl in the first step will block each other and loose. Whoever tries to switch in the second step will insult his second-choice-girl (and possibly the first one) and looses, as well.

There are a some undefined outcomes (what happens if some guys block each other in the first step, but just one of them stays true to his chosen girl in the second step?), but it seems to me that this game has a lot equilibria avoiding the blond girl.

There is, however, a point in signalling towards Crowe’s strategy-set during the planing discussion (step zero) and then deviating during the execution of step one. But Crowe could not know about signalling back then. One of his fellows anticipates it, though.

10. The scene couldn’t have been inteded as an explanation of Nash equilibrium anyway, because it came *before* Nash discovered it.
But it could be a good example of something he might have thought while thinking about related problems.