I am not a mathematician, so I am amply challenged by this issue. It may be very easy, or it may be impossible, indeed I have found some references but I have trouble with the jargon. Perhaps, if you could put things into context for me this may turn out to be a quick affair.
I am trying to find an equilibrium to a decision problem where $(1,...i,...n)$ actors have to choose a probability $P_i$ leading to a vector of probabilities $P=(P_i)_i^n$. The range of options is given by $Z$ which is discrete, indexed by $z$. The actors optimize an expectation such that $$B_i(P) = P_i^* = \text{argmax}_{P_i} \int v_i(z) - u_i(z,P)^2 dP(z)= \text{argmax}_{P_i} \int f_i(z,P) dP(z)$$ Note that the expectation depends on all choices of $P_{-i}=(P_k)_{k \neq i}$.
I can assume that the problem is bounded for $z$ and $P$. I can not assume that $f_i$ is monotone. I will comment on the continuity below.
Now, I suppose that collecting the $B_i$ into $B$ would reformulate the problem to finding a fixed point $P^*$ such that $B(P^*) = P^*$.
I can make two assumptions such that I get a solution easily. If there exists for each actor $i$ a set $Z_i^*$ such that $f_i(z,P)$ attains a "global" maximum for each $P_{-i}$, so that the individual choices of $P$ are independent, then I can use this answer here: Fixed point for probability measures and easily derive that the set of fixed points is such that $P_i^*=\{P(z)>0 \Leftrightarrow z \in Z_i^*\}$. This can be seen by dividing $f_i$ inside the integral by the $\text{max}_z f_i$ which leads exactly to the problem in the link. As far as I can see this implies the existence of fixed points, as long as I have some $z$ such that $f_i(z,P) = $const $\forall P$ which is fine (this is just non-participation then, giving the lower bound)
While I am fine with assuming a set of optimal points $z$, it should depend on the other choices $Z_i^*(P_{-i})$, because otherwise it isn't really an interesting problem.
This sounds to be as I should use Kakutanis fixed point theorem on the set of $n-$vector probabilities (sp?) over $Z$: $\mathbb{P}$. It would probably wise to use results from measure theory, but then I have the problem that Kakutani is not for measure spaces and $Z$ is not complete anyway. On the other hand, since $P$ is discrete, it basically is an Euclidian subset, where each $P_i$ is the set of all vectors with Euclidian norm equal to one. This seems to imply that it is convex (I mean the norm is always one), bounded and non-empty. I am not entirely sure how to show that it is closed, but it seems intuitively true if I think of converging sequences.
Now I must show that $B$ is upper-hemicontinous. Let $P_n\rightarrow P$ be a sequence of probabilities. To show is that if $P_n$ are optimal, then so is $P$.
Since $P_n \in B(.)$, we should have that $P_{ni}(z)>0$ only if $z \in Z_i^*(P_n)$. Intuitively, it's a convex combinations over indifferent outcomes, given $P_n$. Now usually one would use Berges Maximum Theorem (?) to show that the argmax is upper hemicontinous. Here, there is no specific constraint set, so the constraint correspondence is constant (can I do that?), the set of probabilities, and therefore continous. Now the question becomes if the value function is continous. Since the integral operation should be continous, it just depends on the function $u_i$.
If I write the optimization problem for each actor individually, taking $P_{-i}$ as given, then indeed $u_i$ is continuous for $P_{i}$ . But if $P_{-i}$ changes, then I am not sure. So therefore I think: If (and only if) I can show that indeed the function is continous for the whole $P$, then this would imply that the argmax is upper-hemicontinous, and a fixed point exists.
I have two questions:
- Is the approach taken so far correct?
- Is my intuition correct that all I am missing is to show that all $u_i$ are continous jointly in $P_i$ and $P_{-i}$?
- Is there are better way to show this? I can not use monotonicity, so Tarski is out. I think it is also not a contraction mapping, but I am not sure.
Disclaimer: I know this is probably trivial for most of you. I am not an intellectual heavy weight to say the least. I hope I formulated my problem and my attempts adequately for some feedback. If not I will take my hat and feel free to make fun of me.
Thanks!