Is the Nash Equilibrium example in a "Beautiful Mind" accurate?

Question

I was wondering if the Nash Equilibrium example shown in the movie A Beautiful Mind is accurate? and if not, what's wrong with it?

Thanks

Answer 1

It's not quite right (unless you assume Nash was intentionally misleading his friends via a plausible, but incorrect, argument in order to maximize his chances with the blonde). This has been analyzed here.

Answer 2

No it is not.

Loosely speaking, a Nash equilibrium is a strategy profile (specifying a strategy for each agent involved) with the property that no agent could unilaterally change his/her strategy and be better off.

The example in the movie suggests as a Nash equilibrium the strategy profile in which all bachelors ignore the prettiest girl, and instead go for her friends. This is not a Nash equilibrium because, assuming that everyone prefers the best looking gal (as is implicitly assumed in the movie), any one of the bachelors could unilaterally deviate by going for her. Presumably, since no one else is paying her any attention, she would accept him and he would be better off that according to the proposed strategies. Hence, the proposed strategies are not a Nash equilibrium.

Answer 3

I in left, friend on right.

1,1 - all get (not pretty) brunettes. 
1,5 - I get not pretty, friend gets pretty. 
5,1 - friend gets pretty, I get not pretty. 
0,0 - all get nothing.

0,0 is Nash equilibrium state (where they would end up, if all decided to go to blonde) ("no agent can unilaterally change his/her strategy and be better off" as pointed in other answer).

1,1 is Pareto optimal. (Pareto optimality is the state at which resources in a given system are optimized in a way that one dimension cannot improve without a second worsening)

Nash incentivized his friend/friends to choose 5, 1 (Nash gets pretty, friend gets not pretty) - the name of this state is ... I don't know

Answer 4

The movie doesn't actually claim that the strategy Nash proposes is a Nash Equilibrium. And it isn't. Any one of the agents could go for the blonde and be better off.

Nor is it ideal for all parties, as 1 of them picked as random being with the blonde is more ideal.

So no, it is not a Nash Equilibrium. It is however a practical solution to the problem.