# Mean Field/Differential Game and Measurability

Consider the following scenario. There is a continuum of players in a population, with population measure normalized to $1$. Each player has a type $\theta \in [0,1]$ and we suppose that $\theta$ is distributed according to a (nice) c.d.f $H(\theta) \sim [0,1]$ with full support. Write $\mathbb{E}_{H}\big[\theta\big] = \mu$ (where I am abusing notation somewhat by treating $\theta$ as an R.V.).

Each (infinitesimal) player may choose a random variable $X_{\theta}$ distributed according to cdf $F_{\theta}$ with support on $[0,1]$ subject to the constraint that the expectation, $\mathbb{E}_{F_{\theta}}\big[X_{\theta}\big] = \theta$. If we think of the realization $x_{\theta}$ as type $\theta$'s resulting type (which remember is random), then given the choices of strategies (R.V.s) by the other players, there will be a new associated population distribution of types $G$.

(EX 1:) For example, suppose $H(\theta) = \theta$ is simply the uniform distribution on $[0,1]$ and let each player choose the Bernoulli distribution with mean $\theta$. Then it is clear that $G$ is itself a Bernoulli distribution with mean $\mu$.

I have several questions:

1. How to formalize the aggregation of the independent random variables $X_{\theta}$?

From, https://math.stackexchange.com/questions/414966/a-continuum-of-independent-random-variables , https://www.math.lsu.edu/~sengupta/7360f09/kolmogorov.pdf , and http://www.its.caltech.edu/~kcborder/Notes/Kolmogorov.pdf it seems like the Kolmogorov Extension is what I want, no?

1. Given a distribution $H$, what properties must $G$ have?

For instance, it is clear that if $F_{\theta}$ consists simply of a choice of $\theta$ with probability $1$ $\forall \theta$, then $H = G$. Furthermore, if $H$ consists of a Bernoulli distribution with mean $\mu$ it seems equally clear that $G$ must be the same (we could never have a $G$ with support on the interior of $[0,1]$ since there are just two types $\theta = 0$ and $1$).

As EX 1 illustrates it seems like I can always "push" the distribution out, which reminds me of the notion of a balayage or sweeping. Hence, my intuition suggests that any $G$ such that $G$ is a mean-preserving-spread of $H$ is achievable. Any thoughts?

I am also worried that there may be measurability issues that I am glossing over.

• My intuition is correct for number 2. Simply use the realization + 0 mean noise definition" of Second order Stochastic Dominance
– user11305
Sep 17 '18 at 16:01