Two people, $A$ and $B$, have a $30$-sided and $20$-sided die, respectively. Each rolls their die, and the person with the highest roll wins. ($B$ also wins in the event of a tie.) The loser pays the winner the value on the winner's die.
- What is expected value for player $A$?
- How does the expected value of the game for player $A$ change when player $B$ can re-roll?
- How much is it worth for player $A$ to get a re-roll in this scenario, where player $B$ can have the re-roll?
- If you remove player $A$ re-roll. How many re-rolls does player $B$ need in order for him to be a favorite in the game?
I took the average score for player $A$ to be $15.5$ and for player $B$ to be $10.5$. I am assuming this to be expected return for player $A$ so $15.5 - 10.5 = 5$. We also need to take into account when $X = Y$ when $A$ loses to $B$ on draw so $5 - (7/20) = 4.65$.
What I do not understand is how to factor in for when player $B$ can re-roll. I understand that $B$ would re-roll if he gets a value $< 10.5$ which happens $\frac 12$ of the time. Nor can I seem to grasp how to set up the follow-up questions.