Nash equilibrium - A Beautiful Mind?

Question

In the film "A Beautiful Mind", there's a scene where four guys are in the bar and five women walk in - one incredibly hot. Russell Crowe says:

"If we all go for the blonde and block each other, not a single one of us is going to get her. So then we go for her friends, but they will all give us the cold shoulder because no on likes to be second choice. But what if none of us goes for the blonde? We won't get in each other's way and we won't insult the other girls. It's the only way to win. It's the only way we all get laid."

Do you think this is a true case of Nash Equilibrium? If not why not, and if you do why? I have to present this and need some rationale!!

Thanks!

Answer 1

The Nash equilibrium is for all of the men to go for the girl. Of course only one will succed and the other 3 will fail to get any of the other girls. If one person goes for the girl and the other 3 men go towards the non blond girls, how do the men decide who goes towards the blond? Surely if one man tries toget the blond girl then all the other men will try as well because they want to have the blond. So in this case the Nash equilibrium is for one man to get a girl and he will get the blond girl. Lets say the payoffs are 2 for the man who gets the blond girl, 1 for the payoff for getting another girl and 0 for being rejected by all the girls. So there are 4 different Nash equilibrium payoffs (2,0,0,0) (0,2,0,0) (0,0,2,0) (0,0,0,2) which will occur whenever 1 man decides to go for the blond.

However, what happens if they do not welfare maximise, then they will not end up at a Nash equilibrium because in a Nash equilibrium no one can be better off. If one man gets the blond, it does not matter what the others do because they will be rejected by the blond and then by the other girls. Therefore if they all prefer to get a girl than not getting a girl then they can cooperate and make sure none of them go for the blond girl. Assuming they will then not be rejected by the other girls because of personal tastes then the welfare of the men will be (1,1,1,1). Social welfare (the sum of all individual welfare) may not be higher because these payoffs do not represent monetary value. So the Nash equilibriums (2,0,0,0) (0,2,0,0) (0,0,2,0) (0,0,0,2) may still be socially desirable but the 3 men who are rejected are better off if all of the men avoid the blond girl.

Answer 2

Funnily enough, I talk about this in an answer I made 3 weeks ago (link provided below). The question was about Adam Smith's views.

An earlier answer, by someone called iqstrike, said that Smith's view of preferred behaviour within an economy was played out in "A Beautiful Mind". It would be along the lines of every man following his self-interest, with every man going for the incredibly hot woman, and thus creating less wealth overall, as only one man got the hot woman and everyone else went away empty-handed (you can see iqstrike got it wrong: in the film, Nash warned against a situation where *every man* went home empty handed).

Nash equilibrium = if you hold everyone else's actions constant, then it's not in your interest to change your move.

I say in my answer that every man ignoring the incredibly hot woman is not a Nash equilibrium. But even when you have a situation where one man is going for the incredibly hot woman, and the others settling for the others, this may not be a Nash equilibrium either (one of the men may want to try and compete).

You can definitely say that the film's solution (ignoring the incredibly hot woman) is not a Nash equilibrium.

EDIT: refresing my memory about the Nash equilbrium on http://en.wikipedia.org/wiki/Nash_equilibrium . I read an interesting sentence about the Prisoner's Dilemma:

"What has long made this an interesting case to study is the fact 'both betray' is globally inferior to 'both remain loyal'. The globally optimal strategy is unstable; it is not an equilibrium."

With the prisoner's dilemma, what the other guy does can really change your outcome (if you keep quiet - ie remain loyal - and he betrays, then you get a very negative pay-off)....whereas with the "A Beautiful Mind" scenario that iqstrike turned into numbers, getting a regular gal got you 8...and whether or not someone got the perfect-10 didn't affect your 8....you could easily change the rules so that you would feel so frustrated if someone got the perfect-10, that it would turn your 8 into a 0....then you could describe a global optimum whereby all the men "remain loyal" to the plan, and go for the average women; but that it would be unstable as there would be the tendency to defect (and with that defection, all those 8s would turn to 0s).

Answer 3

Is this not an example of Pareto optimal point as well, as the only way one man can get better results is at the expense of another, having looked at the Wiki page, it seems like a zero sum co-odination game rather than a Nash equilibrium, but having said that I may be wrong on both fronts!

Answer 4

No it's not.

A Nash equilibrium means that no party has the incentive to deviate from that solution.

But each guy will think: well if the rest go for the blonde's friends, then I can go for the blonde and have her all for myself.

Then they all go for the blonde.