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When two people make choices that affect each other, such as two shops setting prices or two students deciding whether to share notes, the right move for one depends on what the other does. John Nash gave mathematics a precise way to describe where such situations settle. This post covers what he proved, how to find a Nash equilibrium in a small game by hand, the mistakes students make with the idea, and what you can borrow from how he worked.
Who was John Nash?
Nash was born on 13 June 1928 in Bluefield, West Virginia, and died on 23 May 2015 in New Jersey, according to his Nobel Prize facts page. The same page says he was diagnosed with paranoid schizophrenia in 1959, spoke openly about mental illness, and died with his wife in a car accident.
In his Nobel autobiography, Nash describes entering Carnegie Tech as a chemical engineering student on a scholarship and switching to mathematics. An elective course in international economics there gave him the idea behind his paper "The Bargaining Problem". He chose Princeton for graduate study, where the game theory work became his PhD thesis, which he was prepared to see rejected, so he also had other results ready to submit in its place.
What did Nash actually prove?
The 1994 Nobel press release explains the background. John von Neumann and Oskar Morgenstern had built game theory around two-person zero-sum games, where one player's gain is exactly the other's loss. Nash distinguished cooperative games, where binding agreements can be made, from non-cooperative games, where they cannot, and defined an equilibrium for the second kind that works for any number of players.
His thesis, *Non-cooperative Games* (1950), led to a two-page paper, Equilibrium Points in N-Person Games, in the Proceedings of the National Academy of Sciences in January 1950, and a longer article in the Annals of Mathematics in 1951. A footnote to the PNAS paper thanks David Gale for suggesting Kakutani's fixed-point theorem to simplify the proof.
The press release states the central result: for every game with a finite number of players, an equilibrium exists in mixed strategies. A mixed strategy means choosing among your options with set probabilities rather than always picking the same one. In a Nash equilibrium, each player's strategy is an optimal response to the others', so no one gains by changing only their own choice.
Nash shared the 1994 prize with John Harsanyi and Reinhard Selten "for their pioneering analysis of equilibria in the theory of non-cooperative games". In 2015 the Abel Prize went to Nash and Louis Nirenberg for work on nonlinear partial differential equations. The Nobel page notes he is the only person to have won both.
How do you find a Nash equilibrium by hand?
Take two shops, A and B, each choosing a high or low price. The numbers below are sample profits made up for this example, written as (A's profit, B's profit):
Both high: (5, 5)
A low, B high: (7, 1)
A high, B low: (1, 7)
Both low: (3, 3)
Check A's optimal response to each of B's choices:
If B goes high, A gets 5 by going high or 7 by going low. Low is better.
If B goes low, A gets 1 by going high or 3 by going low. Low is better again.
The game is symmetric, so B also prefers low whatever A does. The only pair where neither shop wants to switch alone is (low, low), with profits (3, 3). That is the Nash equilibrium, even though (high, high) would give both shops 5. This is the shape of the famous prisoner's dilemma: the equilibrium is stable, not necessarily the outcome everyone would prefer.
Now a game with no stable pure choice. In matching pennies, each player shows heads or tails; player 1 wins a point if they match, player 2 wins if they differ. Suppose player 2 shows heads with probability q. Player 1's expected score from heads is q − (1 − q) = 2q − 1, and from tails it is (1 − q) − q = 1 − 2q. Player 1 only has no reason to prefer one side when 2q − 1 = 1 − 2q, which gives q = 1/2. The same argument fixes player 1 at 1/2. Both mixing evenly is the equilibrium, exactly the kind of mixed-strategy point Nash showed must exist. If expected values feel shaky, revise probability distributions of a random variable first.
What mistakes do students make with Nash equilibrium?
Thinking equilibrium means the ideal outcome. The shops example shows it need not be. It means no single player can improve alone.
Checking only one player. You must test every player's optimal response; one player being content is not enough.
Forgetting mixed strategies. If no cell is stable, the game still has an equilibrium, found by making the other player indifferent.
Assuming it is unique. The press release notes that many games have several equilibria, which led to later refinements by Selten and others.
What can a student learn from how Nash worked?
Nash took an idea from a single elective course and turned it into a precise definition, then proved it always applies. He also hedged: by his own account, he kept a second set of results ready in case the thesis was not accepted. Both habits translate well. When a new idea catches your interest, write down exactly what it says in one sentence before you use it, and when you prepare for something that matters, keep a fallback plan alongside your main one.
Small games are a good place to practise this kind of careful case-checking. If you get stuck checking each player's options, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
Draw the four-cell shop table on paper tonight, change one profit number, and check whether the equilibrium moves.
