There are two people playing a game. Each player has a fair 6 sided die. The first player to get 20 points within 10 rounds wins. The player with the higher roll gains the difference in points for that round. For example, I roll a 5 and you roll a 3, I gain 2 points that round. (Rolling same number is no points gained or lost | Game ends if no one gets 20 points in 10 rounds) Given this information, what is the expected number of points gained per round by the winning player?
I tried approaching this multiple ways. Calculating expected value of winning in 4 rounds, 5 rounds, etc, but it seemed to be many calculations. I then tried an approach mapping the probabilities of points gained/lost each round but could not seem to get anywhere. Is there an efficient way to solve this problem?