A few of my fellow students at the UvA may know that I never make any annotations in my textbooks. Most of those books look like they have never been touched, even tough some of them have seen multiple courses over the past years. There is one exception: An Introduction to Game Theory, by Martin Osborne. To be fair, it wasn’t me that made any of these annotations. Lending out a book is always a risky business. But looking at the hard work that one of my friends put into this book to make sense out of it, reminded me of a truth in economic theory. Although the concept can be quite simple, the mechanisms behind that concept rarely are. And so it came to be that the mathematician John F. Nash, Jr. is now a household name for economics students.

This article will not be an explanation of the Nash equilibrium and its many extensions. The theory is certainly interesting enough to deserve multiple articles in Rostra, but I would hardly be the ideal person to write those articles. This article will also not go into depth on the life of Nash. Most of the news items on his death have started with a reference to A Beautiful Mind, a movie certainly worth seeing. Still, it is remarkable that someone who has had such a major influence on academics is best known through the interpretation of Russell Crowe. In this article, I will try to make clear why the development of the Nash equilibrium was so brilliant, and how it has influenced economics as we know it today.

### A Game of Theories

Game theory was established by Von Neumann and Morgenstern with their book in 1944, focused on cooperative zero-sum games. This type of game implies that players have to work together to divide the gains from the game. In this theory, the formation of coalitions is important. This allows players to change the dynamics of the game. The most important contribution of Nash to game theory focuses on non-cooperative games, a situation in which each player maximises his own gain without negotiating with the other players. This development was far from trivial, even though that was the classification reportedly used by Von Neumann. The beauty behind non-cooperative games is the dynamic use of rationality. More specifically, the incorporation of expectations on the choices of others into the calculation of your own choice.

Nash managed to construct a game theory where every player is blind to all the other players, while still resulting in an equilibrium. The concept is that the actions of every player conditional on the other players’ actions can be predicted before the game starts. If you can do that for every player, the equilibrium of the game can be calculated too. As a result, the game is in a steady state. Since all actions are calculated beforehand, no player will deviate from the predicted actions during the game.

Although this theory sounds abstract, it is actually very easy to test. The Nash equilibrium opened up many avenues for experimental economics to develop. Two basic parts of the theory are testable: (1) players correctly predict the actions by other players, and (2) players use this prediction when choosing their own actions. Also, the theoretical extensions of the first concept of the Nash equilibrium are almost endless. Think of imperfect information, exogenous shocks or partial cooperation. The concept is adaptable to many situations in economics and beyond. Any bargaining or auction situation involves game theory. The Nash equilibrium is even widely used in biology and evolution, which sparked a new economic discipline: evolutionary game theory. You could say that Nash started a chain reaction in academics.

Over the past days, many articles have appeared on how we use game theory in our daily lives. And the thing that strikes me the most is that such a characterisation is the opposite of what makes the Nash equilibrium so famous in economics. Economists always try to model the complex world into a simple equation or theory, and rarely do they succeed as powerfully as Nash did. His theory provides a description of human behaviour based on mathematical assumptions that are not at all reasonable. Not even the best poker players will continuously execute the complicated mathematical equations behind his equilibrium. However, a lack of valid assumptions is not always problematic, if the end result is so useful across the board. Perhaps Nash will continue to be most famous for the movie about his life. But academia, and certainly economists, will know him for the way in which he captured our behaviour in a simple set of strategies.