Consider two bidders $A$ and $B$ bidding for a box of 10 coins. The auction is performed in a first-price sealed-bid manner. The winner gets $k$ dollars if there are $k$ heads in the box. The probability for each coin to be head is $1/2$. Obviously, both $A$ and $B$ know the fair value of the box is 5 dollars so both $A$ and $B$ bidding 5 dollars is a pure strategy Nash equilibrium.
Now consider a twist. Assume $A$ has the advantage to see whether the first five coins in the box are heads or not. $B$ cannot see the coins but knows $A$ can see. They then both bid once in a first-price sealed-bid manner for the box.
My question is what the Nash equilibrium is for the second game, if there exists any? I think there does not exist any pure strategy Nash equilibrium but I don't know how to find the mixed strategy Nash equilibrium.