John Nash submitted his thesis to the Math Department at Princeton in 1950, introducing non-cooperative game theory and Nash equilibrium to the world and to economics. The period was one where there were rapid advances in game theory. Here’s Robert Aumann, who later won the Nobel Prize as well for his work on repeated games,
The period of the late 40's and early 50's was a period of excitement in game theory. The discipline had broken out of its cocoon and was testing its wings. Giants walked the earth. At Princeton, John Nash laid the groundwork for the general non-cooperative theory and for cooperative bargaining theory. Lloyd Shapley defined the value for coalitional games, initiated the theory of stochastic games, coinvented the core with D.B. Gillies, and together with John Milnor developed the first game models with a continuum of players. Harold Kuhn reformulated the exten sive form of a game, and worked on behavior strategies and perfect recall. Al Tucker discovered the Prisoner's Dilemma, and supported a number of young game theorists through the Office of Naval Research.
Interestingly, even in economics, leaving aside business, it took a long time for game theory to become an essential tool. The path from Homo Economicus to Homo Ludens was long and winding.1 Finally, Nash and his work found their way to popular culture through Sylvia Nasser’s book, later made into a movie A Beautiful Mind, with John Nash played by Russell Crowe.
The movie was excellent, but the game theory depicted is utterly wrong. See the video below where Nash has an epiphany, in a bar no less, and applies his concept to Nash’s friends, choosing which woman to pursue.
Here is the quote by Nash from the movie
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.
As my daughter would say, it is all a bit cringe. But let me cut them some slack. After all, this is what a director in 2001 thinks how things played out in the 1950s.
So why is this wrong?
The strategy suggested by Nash is not a Nash equilibrium. Let me demonstrate. One of life’s great joys is simplification so for the bar game, I shall assume
Nash’s friends, Sol and Hansen are the two players
They are choosing to approach either Becky (played by Tanya Clarke) or one of her two friends (Friend)
Both Sol and Hansen prefer Becky to Friend
We can write the game as follows
The payoffs follow from Nash’s description. If both pursue Becky, they “knock” each other out, go home alone, and get 0 each. If Hansen chooses Becky and Sol chooses Friend, Hansen gets 2 and is better off than Sol, who gets 1. Symmetrically, if Hansen chooses Friend and Sol chooses Becky, Hansen gets 1 and Sol gets 2. If they both choose Friend, they get 1 (recall Becky has two friends).
The game has two Nash equilibria: (Becky, Friend) and (Friend, Becky).2 Nash suggests they should both choose Friend. But then either can deviate to Becky and be better off. What Nash is suggesting is not a Nash equilibrium. By the way, you should be able to see that the game has a first-mover advantage.
Moral: Come for the movie; skip the game theory!
Change the game
Let me make Sol jealous/envious. Earlier, when Sol chose Friend and Hansen chose Becky, his payoff was 1. Now, all green and Hulkish, he gets -1. The game looks like this.
In this game, Sol has a dominant strategy: Becky. When Hansen realizes that Sol will always choose Becky, his best-response is to choose Friend. The unique Nash equilibrium is the bottom-left box (Friend, Becky). Note that if they both choose Friend, there is no reason to be envious, so the payoffs remain the same.
Moral: Envy as a strategic advantage!
More changes
Finally, let’s make both jealous. When either chooses Friend, and the other chooses Becky, they get -1. The game now looks like this.
Now, both Sol and Hansen have a dominant strategy: Becky. The unique Nash equilibrium is the top-left box where they choose Becky, and both get 0. This, of course, is the famous Prisoner’s Dilemma game.
This is the scenario that Nash has in mind and underlies his recommendation. His friends would be better off ignoring Becky and choosing one of her friends. It is a Pareto superior outcome.
Moral: Choosing the individually rational strategy leads to bad outcomes for all!
A note of caution
Game theory is a fun and often useful way to think about the world. I think of it as one of the many lenses we should use. But by no means the only lens. The usefulness of game theory is that it imposes rigor on our thinking. We need people to think systematically about decision problems and not just in a shooting-the-breeze intuitive way. Game theory provides insights into human behavior that are not obvious and often counterintuitive. But translating insights from the highly constructed world of games to the real world requires judgment, attention to context, and, above all, a deep understanding of the baggage and biases we all carry.
One big reason was that this research was financially supported by the Department of Defense. Therefore, its initial applications were to tactical military problems. Later, the focus shifted to the Cold War, the arms race, and nuclear deterrence.
There is another Nash equilibrium in mixed strategies, but that is for a different post.