In each of the games, both players can choose between two strategies, labeled A and B.

Pure-strategy Nash equilibrium

A pure-strategy Nash equilibrium is a strategy set with the property that no single player can obtain a higher expected payoff by deviating unilaterally and playing an alternate strategy

In game 1, if they choose different strategies (A,B) or (B,A), both get payoffs of 0. If they both choose strategies A, they both get a payoff 2. If they choose strategies B, they both get a payoff 1. The strategy sets (A,A) and (B,B) hence result in Nash equilibria, as deviation of a single player would result in a lower payoff for that player. In game 2, if they choose different strategies (A,B) or (B,A), player 1 gets a payoff of -1 and player 2 a payoff of 1. If they both choose A or both choose B, player 1 gets the payoff of 1 and player 2 the payoff of -1. There are no pure-strategy Nash equilibria in this game, because in each strategy set, one of the players stands to gain from deviating.

Mixed-strategy Nash equilibrium

One of Nash’s results was to show that there must exist at least one Nash equilibrium point in all finite games. Since no pure-strategy Nash equilibrium exists for game 2, there must exist one in mixed strategies:

A mixed-strategy Nash equilibrium is a strategy set with the property that at least one player is playing a randomized strategy and no player can obtain a higher expected payoff by deviating unilaterally and playing an alternate strategy

In cases such as game 2, instead of choosing a single strategy, players can instead choose probability distributions over the set of strategies available to them. In equilibrium, each player’s probability distribution makes all others indifferent between their pure strategies. For instance, as player 1 we can play A half the time and B half the time, and let the flipping of a coin decide when we play which. Player 2’s only rational response will have to be to do the same. As such, a mixed-strategy Nash equilibrium in the matching pennies game exists if both play A and B with equal probability simultaneously.

Interpretations

Nash in his thesis proposed two ways of thinking about his equilibrium concept:

One based on rationality; and

One based on statistical populations;

In the rationality interpretation, players are perceived as rational and they have complete information about the structure of the game, including all of the players’ preferences regarding possible outcomes, where this information is common knowledge. Since all players have complete information about each others’ strategic alternatives and preferences, they can also compute each other’s optimal choice of strategy for each set of expectations. If all of the players expect the same Nash equilibrium, and the game is played only once, then there are no incentives for anyone to change their strategies.

In the interpretation according to statistical populations, Nash states that “[i]t is unnecessary to assume that the participants have full knowledge of the total structure of the game, or the ability and inclination to go through any complex reasoning processes”. This because “What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. If there is a stable average frequency with which each pure strategy is employed by the average member of the appropriate population, then this stable average frequency constitutes a mixed strategy Nash equilibrium.” (Nash, 1950c).

As Harold Kuhn would later write:

"The Nobel selection committee apparently took the two interpretations that are contained in the thesis seriously. The rational interpretation could have been argued by Cournot, but the statistical interpretation, which is so important for biological games, is wholly original. Although the nature of non-cooperative games is explained in all three of these papers, only the thesis contains an exposition of these two interpretations. When asked at the Nobel seminar why the interpretations were not included in the Annals paper. Nash responded, "I don't know whether it was just pruned down in style for the Annals of Mathematics." - Excerpt, "The Essential John Nash" by Kuhn et al (2002)

Discovery

Rather than what was depicted in the movie, as his biographer Sylvia Nasar writes, Nash came upon the idea when he was an graduate student at Princeton University, researching the mathematical modeling of games of strategy and bargaining between economic actors. As Nasar writes,

“A few days after the disastrous meeting with von Neumann, Nash accosted David Gale. “I think I’ve found a way to generalize von Neumann’s min-max theorem,” he blurted out. “The fundamental idea is that in a two-person zero-sum solution, the best strategy for both is … The whole theory is built on it. And it works with any number of people and doesn’t have to be a zero-sum game!” - Excerpt, "A Beautiful Mind" by Sylvia Nasar (1998)

The conversation between Nash and David Gale was recounted by Gale himself to Nasar in 1995. Nash was at the time working on the so-called ‘bargaining problem’ where two individuals have the opportunity for mutual benefit, but no action taken by one of the individuals unilaterally (without consent) can affect the well-being of the other. Think of the classic “divide and choose protocol” of two people trying to divide a cake evenly, where one carves and the other chooses which piece he or she wants, providing a so-called envy-free cake-cutting procedure.

Characteristically, as Nasar writes, Gale was less enchanted by the possible applications of Nash’s new result than the mathematics, stating in 1995 that “The mathematics was so beautiful. It was so right mathematically.”

“Gale realized that Nash’s idea applied to a far broader class of real-world situations than von Neumann’s notion of zero-sum games. “He had a concept that generalized to disarmament” - Excerpt, "A Beautiful Mind" by Sylvia Nasar (1998)

Gale also helped Nash claim credit for the result as soon as possible by drafting a note to the National Academy of Sciences. Solomon Lefschetz submitted the note on their behalf, and the result appeared in less than a single page entitled Equilibrium points in N-person games in the 36th volume of the Proceedings of the National Academy of Sciences in January of 1950.

Nash (1950b). Equilibrium Points in N-person Games. Proceedings of the National Academy of Sciences 36 (1).

Epilogue

Nash’s thesis would eventually spawn three journal papers and a Nobel Prize in Economics (1994).

Journal papers

The three articles contain three different proofs of the existence of Nash equilibria. The first, entitled Equilibrium Points in N-person Games (1950b) is the note Nash and Gale drafted for the Proceedings of the National Academy of Sciences. The second, called Non-Cooperative games (1951) was published in the Annals of Mathematics Vol. 54 (2). In Two-person cooperative games (1953), published in Econometrica 21, Nash extended his work on the bargaining problem (Nash, 1950a) to a wider class of situations in which threats can a play a role (Kuhn et al, 2002).

Nobel Prize

Several weeks before the 1994 Nobel prize in economics was announced on Oct. 11, two mathematicians — Harold W. Kuhn and John Forbes Nash Jr. — visited their old teacher, Albert W. Tucker, now almost 90 and bedridden, at Meadow Lakes, a nursing home near here. Mr. Nash hadn’t spoken with his mentor in several years. Their hour-long conversation, from which Mr. Kuhn excused himself, concerned number theory. When Mr. Nash stepped out of the room, Mr. Kuhn returned to tell Mr. Tucker a stunning secret: Unbeknownst to Mr. Nash, the Royal Swedish Academy intended to grant Mr. Nash a Nobel Prize for work he had done as the old man’s student in 1949, work that turned out to have revolutionary implications for economics. The award was a miracle. — Nasar, 1994.

On the 11th of October 1994, Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel announced that the 1994 Nobel Prize in Economics would be awarded to Dr. John Forbes Nash, Jr. for his pioneering analysis of equilibria in the theory of non-cooperative games:

John F. Nash introduced the distinction between cooperative games, in which binding agreements can be made, and non-cooperative games, where binding agreements are not feasible. Nash developed an equilibrium concept for non-cooperative games that later came to be called Nash equilibrium.