Finding a game from a given fixed point equation

Question

I have a non linear function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ such that its fixed point equation is $f(w) = w$ and there exists a solution to this equation. Can anybody suggest how to find the associated game (i.e. players, their strategies and most importantly the payoff functions) with this fixed point equation? Or maybe give some pointers? Thank you in advance.

Answer 1

In order for you to find a game that generates this function $f$. This function cannot be arbitrary. Let $f=(f_1,f_2,\ldots, f_n)$ be a function of $(x_1,x_2,\ldots,n_n)$. You need that $\frac{\partial}{\partial x_i}f_i =0$, that is the ith-entry of the function does not depend on the ith-variable. In this case you can interpret $f_i$ as a best response of player i to the actions of the others (players distinct from i).

Suppose you have that $\frac{\partial}{\partial x_i}f_i =0$ for all $i$. The $f_i$ is player $i$'s best-response, $x_i$ is i's action, and the payoffs are:

$$ u_i(x) = -\left (x_i-f_i(x)\right)^2 $$

Of course that you can find an infinite number of games such that $f$ is the best-response mapping. The answer is just one example.

Answer 2

Formulate the problem as a two player game in which both players have the action space $\mathbb{R}^n$.

The generic action of player 1 is denoted by $x$, the generic action of player 2 is denoted by $y$.

Player 1 wants to match the action of player 1:

$$u_1(x,y)=-\|x-y\|$$

Player 2 wants to match $f(x)$, the $f$-transformed action of player 1:

$$u_2(x,y)=-\|f(x)-y\|.$$

It is clear that if $x$ is a fixed point of $f$, then $(x,x)$ is a Nash equilibrium.

Conversely, if player 1 plays a best response, then $x=y$ and if player 2 plays a best response, then $y=f(x)$, so in every Nash equilibrium, $x=y=f(x)$ and we get a fixed point of $f$.

The argument is taken from here. A more sophisticated version of argument can be used to show the equivalence of the existence of Nash equilibria and the Kakutani fixed point paper, see From Imitation Games to Kakutani (working paper version) by McLennan and Tourky.