Let $p:A \times B \to \mathbb{R}$ be a nonnegative real-valued function on $A \times B$, where $A$ and $B$ are arbitrary set. Assume $f:A \to B$ and $g:B \to A$ are such that \begin{align*} f(a) &= \operatorname*{arg\,min}_b~p(a,b) \\ g(b) &= \operatorname*{arg\,max}_a~p(a,b) \end{align*}
Does the map $f \circ g:B \to B$ have a fixed point?
Which conditions are necessary for this fixed point to exists?
This question arised while searching an equilibrium point for a game. For my game, I empirically noticed that repeated iteration of the map $f \circ g$ eventually gives me a fixed point. So I start looking for some theoretical justification of this observation. I dig hard, but I couldn't really find one.
My game is the fair pricing of an insurance product.
The set $A$ is a convex subset of $\mathbb{R}^n$, and is the set of action of the policyholder. While the set $B=[0,1]$ is the set of premium of the insurer.
The function $f$ represents the objective function of a fair insurer who wants the price to be zero. While the function $g$ is the objective function of a rational policyholder who wants to arbitrage the insurer.
The function $p$ is such that $p(a,b)=\operatorname{E}(X \mid \text{Action=a, Premium=b})$ where $X=|\text{claim}-\text{premium}|$ is the random variable of the product cash flow.
I looked into Banach fixed point theorem, Nash equilibrium existence theorem and Kakutani's fixed point theorem.
Any help or pointer appreciated.