I would like to know how the following game is known in the literature and, possibly, to have references for related papers.

**Description of the game:** Fix a space $X$ and two Borel probability measures $\mu$ and $\nu$ over $X$. There are two players, $A$ and $B$. They both know $\mu$ and $\nu$. Player $A$ chooses between $\mu$ and $\nu$. Player $B$ can not observe Player $A$'s choice. Say $\mu$ is chosen by $A$. Then an element $x\in X$ is randomly chosen in accordance  with $\mu$. Now player $B$, looking at $x$, must guess the choice of $A$, i.e., Player $B$ must say "you piked $\mu$", or "you picked $\nu$".  Player $B$ wins if their guess is correct. Player $A$ wins otherwise.


**How to formalize the game:**

 1. The strategies for Player $A$ can be formalized as (randomized) choices over the two element set, i.e., as elements in $[0,1]$.
 
 2. A strategy for Player $B$ can be formalized as a map $\sigma: X\rightarrow [0,1]$: if $x$ is the outcome, then guess $\mu$ with probability $\sigma(x)$ and $\nu$ with probability $1-\sigma(x)$. 

 3. Since Player $B$ can not observe Player $A$'s choice, the game can be consider as played concurrently.


I believe the game has an optimal equilibrium and its value  is a function of $\displaystyle \bigsqcup_{B\ Borel} | \mu(B) - \nu(B) |$, i.e., of the total variation distance between $\mu$ and $\nu$.


Thank you in advance for any information.

Matteo