# What is John Nash's game theory

## John Nash in brief

Table of Contents

Version from 10/27/2006 (09/25/06)

**John Nash** is the inventor of the Nash equilibrium and the portrayed person in the film "A Beautiful Mind". He received the Nobel Prize in Economics in 1994 (together with John Harsanyi and Reinhard Selten). His full name is **John Forbes Nash Jr.**

John Nash's most important contributions are the formulation of the Nash equilibrium and the Nash bargaining solution and the distinction between cooperative and non-cooperative games.

John Nash was born in West Virginia on June 13, 1928 and received his PhD from Princeton University.

### The genius of John Nash

**In some ways, John Nash shares the same fate as Steven Hawking:** Both suffer from a very noticeable disease and are therefore probably more famous than they would be without this disease. But there is one difference: John Nash has the Nobel Prize, Hawking does not.

Perhaps most amazingly, while most professors take pride in their long list of publications, John Nash only has a handful of them. But three essential publications are enough for a genius. And with these he did nothing less than to shape game theory as we know it today and to completely turn the social sciences upside down.

His dissertation is just 32 pages long, the formulas are hand-written and the only example of use is the game of poker. But his work is visionary. In three hundred years from now, if anyone wonders what the 20th century has produced in social science research, I venture to predict that the Nash equilibrium will be high on that list; maybe it will be the only item on the list.

You may be wondering why I keep talking about the social sciences when John Nash was a mathematician and it is sometimes claimed that he received the Nobel Prize for his mathematical research. This is simply because, although he dressed the whole thing in a mathematical guise and published it in mathematical journals, the mathematical side of his work is comparatively simple. More brutally, any advanced undergraduate math student could have produced this evidence. But this evidence is not important, it is the basic idea that counts. And their meaning in real life. The math is just a decorative accessory. (Incidentally, this also applies to many of the other Nobel Prize winners in game theory, with the possible exception of John Harsanyi and Robert Aumann.)

For a long time I had wished to get hold of the original John Nash’s dissertation, but I have not found a copy anywhere. It hasn't gotten easier with taking it in your hand, but if you feel the need to read it, you can do it comfortably with a bag of chips in your armchair these days by clicking here: John Nash's dissertation on "Non-cooperative Games" from May 1950.

### What did John Nash win the Nobel Prize for?

But by now at the latest you want to finally know what the curious mathematician John Nash has achieved so great.

Without a doubt its most important idea is the Nash equilibrium. I'll spare you the details and tell you why this Nash equilibrium is so important:

- It is the mathematical (i.e. general) formulation of a very general principle that appears in various places in the real world. John Nash’s equilibrium describes not only the behavior of rational decision-makers, but also that of biological systems, that of molecules and that of crowds.
- The Nash equilibrium describes (this is shown correctly in the film A Beautiful Mind), why Adam Smith’s hypothesis is not always correct that the egoism of the individual leads to the benefit of all. In this respect, John Nash turned economics upside down. The social sciences fared no differently: By analyzing what is collectively possible when the individual has to participate, the Nash equilibrium has put the collectivist approaches in as dire straits as the individualistic economics. We are only beginning today to understand all the implications of the simple idea.

In addition, John Nash clarified a few other little things (or at least made them known, because these ideas were already buzzing in the air back then, provided you were hanging around at Princeton University or MIT):

- He introduced the mixed strategy including its ideas of interpretation (purifications).
- By referring to population games, he prepared the analysis of emergence (i.e. the appearance of macroscopic phenomena from the behavior of many individual beings, such as in a swarm).
- John Nash coined the distinction between cooperative and non-cooperative game theory.
- He also defined the Nash negotiation solution for cooperative games, which describes how negotiating partners can share an additional profit that they generate together.
- With the Nash equilibrium, he introduced a solution concept into game theory that was finally also convincing for non-zero-sum games, whereby the originally almost military-style game theory became a real social science and was able to free itself from the stranglehold of zero-sum games.
- He introduced the static interpretation, in which the players do not come to the rational solution through a sequence of restricted rational considerations, but jump straight there. (Do you remember the blonde fellow student in A Beautiful Mind?)

From a historical perspective, his work laid the foundation for game theory to leave the field of mathematics and be applied to real scientific issues.

In order not to repeat myself too much, I just give the tip here to surf around my pages a little, then you will find the Nash equilibrium (and the other ideas of John Nash) in the most diverse game variants and tastes - from application examples with political to philosophical background as well as explanations of the mathematical principles. Or - of course - read my game theory book.

### The road to John Nash's Nobel Prize

After his first big throws, there was no further publication by John Nash for a long time (namely from 1966), but anyone who has dealt with game theory since the 1980s found it a shame that he would probably never get the Nobel Prize because he was "crazy" (a visiting professor who came back from America reported seeing John Nash with a bottle of red wine on the Princeton campus. We all believed that at the time because his exact illness was largely unknown at the time).

The trouble was, it didn't make sense to have another Nobel Prize in game theory as long as John Nash didn't have one. That was very regrettable, because game theory had already passed through wide areas of economics (still unnoticed by many) and was just about to play an increasingly important role in political science. Even then it became clear that practically the entire theoretical literature of economics and social sciences would be an application of game theory. And yet no Nobel Prize for the whole area - that was becoming a problem.

In the early nineties, my colleagues and I noticed that the name John F. Nash was suddenly popping up shamefully in various places (for example on the editorial board of one of the most respected economic journals). It was obvious what that meant: John Nash should finally get the Nobel Prize. But nothing happened.

A little later there was a highly confidential letter from the chair where I was assistant at the time. I was also not allowed to see or read anything about it, but at least the sender was clear enough: the Nobel Committee. It was also noticed that my boss at the time was writing a mysterious report and was certain that the first Nobel Prize would be awarded to a German economist at the end of the year. That also met the expectations of those who had not received a confidential letter from Sweden. Because our theory was that the Nobel Committee would not dare to give the award to John Nash alone, but would also honor the other two natural candidates: Reinhard Selten (from the University of Bonn) and John Harsanyi, who with their Additions to the Nash equilibrium had made non-cooperative game theory really applicable.

But still nothing happened. Despite all secret reports and the appearance of the name John F. Nash, the Nobel Prize went to someone else that year. Likewise in the following year. Slowly we began to forget the whole thing again, and in addition Reinhard Selten (who was "our" contact person on matters of game theory) was always careful that nobody would even use the word Nobel in his presence.

So we continued to tinker with game theory problems or interpretations that, outside of our very small circles, really did not interest anyone, to put it cautiously. (It was not infrequently said about us that the state of mind of the greatest living game theorist was a role model for everyone else who deals with it.

Then, on a sunny autumn day, I went to lunch and when I got back my answering machine was full of messages. After the first message it was clear why: John Nash, Reinhard Selten and John Harsanyi had received the Nobel Prize. The first news back then was that the prize had been awarded for “non-operative game theory”, and there were quite a few people who at the time wanted to know what it was supposed to be. Since my game theory textbook had been on the market for two years and was considered to be the most understandable book on the subject (which hopefully still applies today), I had the honor of explaining to numerous confused journalists that it was not non-operational, but non- Cooperative game theory means that, to make matters worse, there can still be cooperation in the non-cooperative part. (Even then was the point in time when some game theorists began to dislike my representations because I did not see game theory as a mathematical exotic, but as a theory that also has practical and even political consequences. But at least I was allowed to For a few weeks back then, feeling like I had received the Mini Nobel Prize.)

### John Nash’s illness

Would you like to find out more about John Nash’s person and life? Others have already portrayed it better than I could: Watch the film A Beautiful Mind - very exciting, and it impressively shows the two sides - genius and madness - of John Nash. If you prefer the longer version, you can also read the book on John Nash and his illness. It's hard to believe that a mathematician's life can turn out to be a real thriller.

Of course, you won't find out that much about his theories there, but that's why there is my website www.spieltheorie.de, where you can browse a little further (for example by using the Google search function above or by clicking on the hyperlinks). I have covered numerous examples of the Nash equilibrium.

Or check out my game theory book - there you'll find John Nash's theories with a lot or little math, depending on your taste.

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