Thirty-sided versus twenty-sided die duel

Philip rolls a fair 30-sided die and Brandon rolls a fair 20-sided die. The higher roll wins, with ties going to Brandon, and the loser pays the winner the number showing on the winner's die.

(a) What is Philip's expected profit?

(b) Now Brandon may look at his roll and choose to roll his die once more (keeping the new result) before Philip's roll is revealed. Assuming Brandon plays optimally, by how much does Philip's expected profit change?

(a) $8.15; (b) it falls by about $2.72, to $5.4275. (a) When Philip rolls 1–20 the win/loss cases cancel except ties, costing $\tfrac23\cdot\tfrac1{20}\cdot10.5$; when he rolls 21–30 he always wins, averaging 25.5, so $E=\tfrac13(25.5)-0.35=8.15$. (b) Brandon should re-roll whenever his first roll leaves Philip more than 8.15 in expectation, which is any roll of 11 or less; computing exactly gives 5.4275.