Variations of the Nash Equilibrium

Factual Questions
1

First let me say that my mental health may not be perfect. Okay, somtimes in the past, I have been delusional and sometimes I hallucinate. Also, I have an aptitude towards mathematics. I knew who John Forbes Nash was before he was a movie. When I worked in quality control I read some of his stuff on game theory, not that there was much of it I actually understood. I read some interpretations of his Nobel Prize winning work but it was a long time ago and I have been brain damaged since then. That knowledge is gone where my calculus has gone and I am now able to try and recover it. I am curious, though, does the Nash Equilibrium account for the results, if just one of the participants is a raving lunatic and has wild hair…

2

I think I know what you’re driving at.

  1. A Nash equilibrium is simply a state where given what everyone else is doing each player wants to keep doing what they’re doing. The existence of Nash equilibria doesn’t say that one will be achieved, or explain how. It doesn’t even say it will be maintained, since I might think I can do better by changing my behaviour if I believe others will change theirs in response. The beauty of the Nash equlibrium concept is that cuts through the apparent indeterminacy of the strategic situation (where what is best for me depends on what you do and vice versa).

  2. I think what you’re getting at is whether rationality is required to reach a Nash equilibrium, and whether what might happen if some players are irrational has been a matter that has been studied. Players will reach a (unique) Nash equilibrium if there is common knowledge of rationality. If our lunatic is calculating and raves consistently, that’s no problem.

Imperfect rationality can be introduced into these sorts of games. The “trembling hand” idea (that the other player is rational, but might make a mistake) was one of the ways in which theorists tried to show how mostly rational players might depart from Nash behaviour in the finitely iterated prisoners dilemma game - the theorists being uncomfortable with the idea that rational behaviour led to a bad outcome for everyone in this simple strategic situation.

Having wild hair and foaming at the mouth is one way of advertising that you’re not all that rational. Being committed or otherwise difficult changes the game. In The Godfather for example, there is a family that makes its living by being hostage guarantors: they have the reputation of seeking revenge irrationally. Various people have explored such situations (eg Schelling, Elster, Frank), but that’s getting away from game theory and towards decision theory. Game theory tends to work with utility maximising agents.