The beautiful mind that was John Nash

Sudipta Sarangi

“This man is a genius.” This one-sentence letter of recommendation got John Forbes Nash Jr (June 13, 1928-May 23, 2015), the man who revolutionised game theory, admission to both Princeton and Harvard.

On May 23, the world lost a beautiful mind. John Nash, returning from a trip to Norway to accept the Abel prize, died in a car accident in New Jersey.

Born in rural West Virginia, Nash displayed an aptitude for mathematics early on which was encouraged by his parents. They arranged for him to take advance courses in maths at the local community college.

He then moved to the Carnegie Institute of Technology (now Carnegie Mellon) where he changed his mind twice (starting with chemical engineering, then chemistry) before finally settling on mathematics and finishing up with a Masters. When he was applying for doctoral degree in mathematics, his CIT advisor just wrote the one-line letter of recommendation.

Nash ended up choosing Princeton over Harvard thanks to an attractive scholarship. It was also a time when very few other places could provide the kind of stimulation Princeton offered to a young and smart scholar. Albert Einstein, John von Neumann, Albert Tucker, and Kurt Gödel were all there.

Game theory was in its infancy, and both von Neumann and Oskar Morgenstern, who had jointly written the first book on the topic, were at Princeton. It was in Tucker’s game theory seminar class that Nash met these authors and wrote his now classic paper on game-theoretic bargaining.

Nash equilibrium

However, it was his second paper that changed the face of modern economics while also influencing disciplines such as political science and evolutionary biology.

In this paper Nash took game theory beyond zero-sum games, that is, games in which one person’s gain is the other person’s loss, though draws are also permitted. He provided a way to look for stable outcomes in more general situations.

All competitive sports qualify as zero-sum games but there is definitely more to life and economic situations than sports. Nash provided a way for studying the Cold War or the interaction between India and Pakistan as a game. He paved the means to study competition between rival firms — cellphone carriers or cola companies or political parties.

Nash took the definition of what von Neumann had called a normal form game and provided a solution concept for them. A normal form game is a situation in which the players make their moves simultaneously; this then determines the outcome of the game.

The formal analysis of such a game needs a well-defined set of players and a listing of the strategies that are available to all the players in the game. Once all players choose a strategy, we obtain an outcome of the game and the payoff of each player associated with each outcome needs to be defined a priori. An easy example of such a game is rock-paper-scissors.

A Nash equilibrium then is a situation where no player can do better by choosing an alternative strategy. Hence it is a situation from which no player will wish to deviate unilaterally.

A couple of examples

Any situation where the final outcome is determined by the actions of multiple individuals or players is a game. Consider, for instance, three players each with the option of choosing to drive either on the left or the right side of the road. If everyone drives on one side, everyone is happy. However, if some drive on the left and others on the right, then clearly everyone will be unhappy.

There are two Nash equilibria in this game — everyone driving on the left or everyone driving on the right. To check that everyone driving on the left is a Nash equilibrium, observe that if the other two players are driving on the left, then you cannot do better by switching from driving on the left to the right.

The game that I just described is called a coordination game and not an uncommon situation. For instance, if you are planning to work with a group of people, there must be coordination on which software to use. If countries are planning to make train travel possible across borders, then the tracks must be all of the same width. Similarly, international technology standards like the size of CDs or USB ports requires coordination among countries.

Nash’s contribution

John Nash fundamentally changed game theory and in doing so provided the social sciences with a significant analytical tool.

First, he provided the tools to analyse situations that are not merely zero-sum games. Second, he made a clear distinction between cooperative and non-cooperative game theory by shifting the emphasis to individual decision-making.

In doing so he provided us with a basis to formally study strategic behaviour. Imagine a business school course in strategy without the development of game theory.

For economics, it provides an alternative to thinking of everything in terms of the market mechanism where firms take prices as given and react to it. We could now build models where firms set prices, sell differentiated product and affect the market through different means such as advertising and discounts.

His work also set the stage for studying sequential moves games like tic-tac-toe or a price war between firms. Reinhard Selten shared the Nobel Prize for Economics in 1994 with Nash for providing a solution concept for such games.

The third 1994 winner was John Harsanyi who provided the means for dealing with uncertainty in games — situations where players may not know the strategies or payoffs of the other players, by developing the concept of Bayesian Nash equilibrium.

In showing that it is very important to clearly define the rules of the game, Nash’s work was also a precursor to the work on information economics. For example, a person buying health insurance or a borrower is better informed about their situation than the seller of insurance or the lender. Such asymmetry can clearly affect outcomes.

Of course because of its stark assumptions and predictions Nash’s work in many ways has also contributed to the rise of experimental economics and behavioural economics.

Nash equilibrium requires a high degree of rationality and it is often claimed that in practice people are not like this. However, rationality is an important benchmark since we can model irrational behaviour in a million different ways making it impossible to develop a common body of knowledge.

Rationality then is a fine assumption provided that we remember that actual people may not always behave according to it.

Nash equilibrium does not tell us how to play a game — it says that if somehow the players got to a Nash equilibrium they would stick to it. In fact experimental economics suggests that people only learn to play a Nash equilibrium over time. When the stakes are higher, they learn faster.

As communication makes the world smaller, interactions among multiple actors is becoming more common; therefore game theory is becoming more relevant. Also, as decision-making gets more automated, behaviour will become more rational or more game-theoretic.

Finally, it is okay to change your mind, not just once but even twice — even Nobel Prize winners do it!

(The writer teaches microeconomics and game theory at Louisiana State University)