I am looking to find the probability of winning a basic dice roll-off using a 6 sided die if one of the players can re-roll their die. The main thing that is throwing me off is that player 2 can re-roll the die but doesn't have to, and if the first roll or the re-roll equals player 1's roll then the process restarts.
Example 1: Player 1 - Rolls a 2 Player 2 - Rolls a 4 (win)
Example 2: Player 1 - Rolls a 2 Player 2 - Rolls a 1, re-rolls and gets a 5 (win)
Example 3: Player 1 - Rolls a 5 Player 2 - Rolls a 5 At this point player 2 can call it a tie and start fresh, or use his re-roll to attempt and roll a 6, although this doesn't seem to be his best option to win.
Example 4: Player 1 - Rolls a 4 Player 2 - Rolls a 4 At this point player 2 decides to call it a tie, and then they both re-roll. Player 2 still has the ability to then re-roll his result in this round. Player 1 - Rolls a 3 Player 2 - Rolls a 2, re-rolls and gets a 1 (loss)
I gather that only the last roll of player 2 "counts."
If player 2 is losing after her first roll, she will certainly roll again. If player 2 is winning after her first roll, she will certainly not roll again. If player 2 ties after her first roll, she will roll again if she has a 1 or 2 or 3 (since there is a greater chance she'll tie or win on her second roll) and she will not roll again if she has a 4, 5 or 6 (since there is a greater chance she'll lose on her second roll).
One can enumerate all these results in a decision tree and simply count the wins and losses.
Player 1: ${\bf 1}$
Player 2:
So if Player 1 rolls a ${\bf 1}$, the Player 2 has the following probabilities:
Tie: $1/6$
Win: $5/6$
Continue through other cases of player 1 rolling ${\bf 2}$ through ${\bf 6}$, and be sure to weight each result by the probability of the rolls.
Then add up the results!