One paid reroll of a die costing a dollar

Alice rolls a fair six-sided die and looks at the result. She may either keep it, receiving that many dollars, or pay $1 to roll a second time, in which case she receives the value of the second roll instead. She can reroll at most once. If she plays optimally, what is her expected net payout?

$\tfrac{23}{6}$. Rerolling is worth $3.5-1=2.5$ on average, so she keeps any first roll of $3$ or more and rerolls a $1$ or $2$. That gives $\tfrac13\cdot 2.5+\tfrac23\cdot 4.5=\tfrac{23}{6}\approx 3.83$.