## Leonhard Euler

A Beautiful Mind Review

Hüseyin Şahin

The movie that we have chosen to review for The Math Chronicles features the life of a very influential mathematician of the 20th century who has won Nobel Prize and Abel Prize for his contributions in mathematics and economics. The movie is directed by Ron Howard and was inspired by Sylvia Nasar’s Pullitzer winning book of the same name. The movie stars Russel Crowe, Ed Harris, and Jennifer Connely.

John Nash, the brilliant mathematician, who was diagnosed with schizophrenia at a young age is very popular with what is known as Nash equilibrium. The movie “A Beautiful Mind” starts to tell Nash’s story from his university years and portrays his brilliant side along with his struggles considering his delusional episodes.

The movie starts with Nash’s years in Princeton University, and the emphasis is on his will to publish an original idea. This is portrayed by demonstrating that all his friends are close to publishing their thesis, but Nash hasn’t even started as he is seeking for an original idea, and indeed when he discusses with his friends how to approach a group of women at a bar, he develops a new concept of governing dynamics which is now known as the Nash Equilibrium. Nash argues that a cooperative approach would work better although his friend Adam Smith argues that Nash’s approach would decrease their chances.

Figure 1: Release Poster of “A Beautiful Mind”

Nash later mathematically formulates this concept and publishes it which provides him with the opportunity to continue his works at MIT. In the movie, it is also shown that Nash is recruited mysteriously by William Parcher to work for Pentagon and crack ciphers. He then falls in love with one of his students Alicia Larde who is the first to notice Nash’s schizophrenia symptoms. Dr. Rosen whom Nash thinks is a Soviet agent diagnoses Nash with schizophrenia which Nash doesn’t believe at first. Nash then learns to ignore his hallucinations as he realizes that the ages of the people he imagines stays constant.

Figure 2: Nash when trying to formulate the Nash equilibrium

The movie was nominated for eight academy awards of which it won four including Best Picture, Best Director, Best Adapted Screenplay, and Best Supporting Actress.

The concept of Nash equilibrium is mentioned in the movie and explained although not very extensively. In the movie Nash explains it as follows: “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.”

Formal definition of a Nash equilibrium though can be given as follows.

“Let (S,f) be a game with u players; Sᵢ is the set of strategies for player i; S = S₁ x S₂ x ... x Sᵤ is the set of strategy profiles, and f(x) = (f₁(x),...,fᵤ(x)) is its payoff function evaluated at x ∈ S. Let xᵢ be a strategy profile of player i and x₋ᵢ be a strategy profile of all players except player i.

When each player i ∈ {1,...,u} chooses a strategy xᵢ, resulting in a strategy profile x = (x₁,…,xᵤ), then player i obtains payoff fᵢ(x). Note that the payoff depends on the strategy profile chosen, i.e. on the strategy chosen by player i as well as the strategies chosen by all the other players.

A strategy profile x* ∈ S is a Nash equilibrium if no unilateral definition in strategy by any single player is profitable for that player, that is

∀i,xᵢ ∈ Sᵢ : fᵢ(x*ᵢ, x*₋ᵢ) ≥ fᵢ(xᵢ,x*₋ᵢ)” (Veisdal 2019)

All in all, I think the movie “A Beautiful Mind” is worth watching considering how it inspires people and almost accurately portrays one of the most brilliant mathematicians of the 20th century.

Hüseyin Şahin

References:

Veisdal, Jorgen (2019). https://medium.com/cantors-paradise/the-nash-equilibrium-explained-c9ad7e97633a. Date Accessed: 10 February 2020.

Figure References:

[1]“A Beautiful Mind.” https://www.imdb.com/title/tt0268978/. Date Accessed: 10 February 2020.

[2] Von Tunzelmann, Alex. “A Beautiful Mind Hides Ugly Truths.” https://www.theguardian.com/film filmblog/2012/dec/19/a-beautiful-mind-john-nash. Date Accessed: 10 February 2020.