Thirty-sided versus twenty-sided die with one reroll

Alice rolls a fair $30$-sided die and Bob a fair $20$-sided die, and the higher number wins, with ties going to Bob. Without seeing Alice's result, Bob may choose to roll his die a second time, in which case the new roll replaces the old one. If Bob plays optimally, what is the probability that Alice wins?

$\tfrac{17}{30}$. Keeping a value $b$ wins for Bob with probability $\tfrac{b}{30}$, while a reroll wins with probability $\tfrac{10.5}{30}$, so he rerolls on $1$–$10$. His win chance is $\tfrac1{20}\sum_{b=11}^{20}\tfrac b{30}+\tfrac12\cdot\tfrac{10.5}{30}=\tfrac{13}{30}$, leaving Alice $\tfrac{17}{30}$.