What information does $g^x$ reveal about $x$?
You state


  One way to formalise this question would be: Let $A \subset \mathbb{Z}_P^*$. If Player 2 is honest, we have
  $$
\mathbb{P}[x \in A] = \frac{|A|}{p-1}.
$$
  Can Player 2 increase this probability using the above strategy? 
  
  (Naturally, $A$ can be defined by "the set of all numbers $x$ in $\mathbb{Z}_p$ such that $g^x$ has a '1' in the last digit" and then Player 2 can easily make this probability equal 1, so this formalisation is not very good. Suggestions?)


One way to address this issue, let's call it the encoding of $A$ issue is to hide the obvious properties of $A.$ Thus the players can interact with a randomized encoding of $A.$ Specifically, for your case, we can define $A$ as the set of all numbers $x$ in the integers modulo $n$ such that $H(x)$ has a certain bit pattern $(a_1,a_2,\cdots, a_t)$ in its leading $t$ bits, where $H$ is a secure hash function.

So the game can be played on the hashed image of the relevant quantities.