Nash equilibrium concept and fixed poind theorems

Question

I am searching about the intuition of the fixed point theorems that are used in some game theoretic problems to find the Nash equilibrium. In some cases, I also see some other games that they do not provide a solution based on some fixed point theorem. I want to understand if the fixed point theorems a necessary and sufficient condition for Nash equilibrium? If yes, why in some cases it is not used or mentioned to define the equilibrium problem? If no, then the fixed point therem has to do with the model and the type of the game? Where can I find some details? The last but not least, I am searching for some kind of Bayesian Nash equilibrium concept with fixed point theorem, does anybody have some mathematical model in mind for sometheing like this?

Answer 1

Every Nash equilibrium of a game is a fixed point in the properly defined strategy space of that game.

The intuition behind the Nash equilibrium is that once we we are in equilibrium then no player has incentives to deviate from the prescribed strategies. To see it more formally let $\beta^{\ast}=\{\beta^{\ast}_i\}_{i=1}^N$ denote the equilibrium strategy profile where $i$ is the index over players. Then $\beta^{\ast}$ is an equilibrium if each player $i$ finds it optimal to use $\beta^{\ast}_i$ when all players $j\neq i$ play according $\beta^{\ast}_j$. Thus, equilibrium strategies are self-confirming in the sense that if we start with players thinking that everyone will use equilibrium strategy and allow each player (separately) to re-optimize then each player will choose the equilibrium strategy we started with. In that sense, intuitively, you can see that equilibrium is a fixed-point of a properly best-response mapping.

Fixed point theorems are mostly invoked in game theory to prove existence of equilibrium. They are less useful for finding equilibrium explicitly. In many cases, one can solve the model by simply solving explicitly each player's problem. Finally, you can have hybrid approaches.

Regarding Bayesian Nash equilibrium (BNE) there are plenty of books that explain it in detail. But note that these are mostly graduate level textbooks. The most popular are:

1. Fudenberg and Tirole "Game Theory"
2. Osborne and Rubinstein "A Coure in Game Theory"
3. Myerson "Game Theory: Analysis of Conflict"

All these books are very rigorous and cover a lot of material starting with simplest settings.