If you want to understand the Nash equilibrium and why it is so important to our society and our lives, the one thing that you should not do (unless you are a very serious mathematician indeed) is to attempt to infer these qualities from Nash’s incredibly short original paper, published in the Proceedings of the National Academy of Sciences of the U.S.A., Vol. 36 (1950) pp. 48-49 (http://web.mit.edu/linguistics/events/iap07/Nash-Eqm.pdf). It is a superb display of top-flight mathematical thinking, but undoubtedly needs translation for the rest of us.

Fortunately, although mathematics was required for the proof, the basic idea is easy to grasp in non-mathematical terms, as are its frightening ramifications.

What is a Nash equilibrium? Put simply, it is a position in a situation of competition or conflict where both sides have selected a strategy, and where neither side can then independently change their strategy without ending up in a less desirable position. If we’re walking toward each other on a narrow sidewalk, for example, and we both step aside to squeeze past, we’ll find ourselves in a Nash equilibrium because, if either of us independently changed our mind and stepped back, we would only come face-to-face again, with the consequent merry little dance that most of us have experienced.

In probably the shortest scientific paper ever to win its author a Nobel Prize, Nash used a combination of symbolic logic and advanced mathematics to prove that there is ** always **a Nash equilibrium waiting to trap us in “non-cooperative” situations – that is, situations where the parties are either not able or not willing to communicate. Cooperation may serve us well, but if we cannot negotiate successfully from a position that may be bad for both parties, any attempt by either party to improve their position will result in a (perfectly logical) reaction by the other party where both end up in a worse position than before.

Just look through any newspaper, or any celebrity gossip magazine, and you’ll find examples of the sort of logic that Nash was talking about. Think of two people involved in an acrimonious divorce, or two nations on the brink of war. It would pay both parties to compromise, but so long as one refuses to compromise, it is not worth the other party’s while to give way. They become trapped in a Nash equilibrium where they both lose out. In the case of a divorcing couple, the loss is through emotional stress and the money paid to lawyers. In the case of nations confronting each other, the results can be much more serious.

Before Nash’s discovery, our frequent failure to cooperate with others for mutual benefit was most often thought of in terms of our psychology and/or our morality. The significance of his discovery is the demonstration that a genuine logical paradox, rather than emotional or moral factors, lies at the heart of many of our most serious problems, from resource depletion and global warming to corruption, collapsing societies and war. Cooperation could help resolve such problems, but Nash’s trap blocks us again and again.

Adapted from Chapter 1 of my book *Rock, Paper, Scissors: Game Theory in Everyday Life.*

I don’t think you adequately explain why the Nash equilibrium matters. It’s just putting a name or label to something. The example you give (two people on a narrow sidewalk) does not illustrate what happens without cooperation or change of strategy on one side or both. Let’s say we meet on a narrow sidewalk and my policy is ALWAYS to stop in my tracks, neither moving forward nor back. That forces you (and 100% of the time this is what happens in real life) to sidle past me. After you pass, I carry on. Neither or us has compromised. I merely force you into an automatic reaction that you would not otherwise have exhibited — that is, make yourself smaller so you can squeeze by. With or without any scientific or mathematical theory or translation or analysis of what just happened (ie. with or without Nash), life goes on.

So how does the Nash equilibrium matter, whether in respect to conflicts between arms-bearing nations or to life’s happenstances? I don’t think putting a label on something matters unless one is affected by labels and phony prizes.

The point of the Nash equilibrium is that BOTH sides have independently decided on their optimum strategies. So if BOTH have decided not to give way (a not uncommon situation in real life; just think of what can happen in acrimonious divorce cases), then if one eventually does, and steps down into the gutter to get past … … .