Marble box game with payoffs that shrink

Players $A$ and $B$ each own $100$ marbles. Simultaneously and secretly, each chooses how many of their own marbles to put into a shared box, anywhere from $1$ to $100$; say $A$ contributes $a$ and $B$ contributes $b$.

Marbles are then drawn from the box twice, with the drawn marble returned before the second draw. Every time a marble belonging to $A$ is drawn, an outside party pays $A$ the amount $100-a$; every time one of $B$'s marbles is drawn, $B$ is paid $100-b$.

If both players act optimally, what is $A$'s expected total payout over the two draws?

$67$. Per draw $A$ earns $\tfrac{a}{a+b}(100-a)$; the best response is $a=\sqrt{b^2+100b}-b$, whose symmetric fixed point is $a=b=\tfrac{100}{3}$. Among integers, $33$ is the best response to $33$, so both put in $33$ marbles, and $A$ earns $\tfrac12\cdot 67$ per draw, i.e. $67$ over two draws.