The Even/Odd game is defined as follows:
Both players pick a number in $(0,1]$ for positive $y$. Even's number is called $E$. Odd's number is called $O$. The scorer number is $\Big \lceil \dfrac{E+y+1}{O+y} \Big\rceil$. If it is even, Even gets a point. If it is odd, Odd gets a point. I conjecture that Even would be winning in the long run. What is the expected value for each player, assuming perfect play for a given $y?$ How does the behavior of perfect players change with different values of $y?$ There is no Nash equilibrium for $y = 1.$