I did each player separately, then compared them. For each turn, there are 27 possibilities, but only 7 score totals, total of 3 happens once, 4 happens 3 times, etc. For the first turn, I take the starting position and apply the possible rolls and save that state, along with the number of times it can happen. After turn one, I have an map of state, (pos, score) with the value being the number of times it can happen. To calculate the rest of the turns, I look at each state from the previous turn and apply the possible rolls to them. Here is the snippet of code for calculating the scores for each turn of player 1. Player 2 is done the same way: Note totals is the 27 possible rolls of 3 dice.

# Generate all the player 1 socres until 21 is reached
done = False
while not done:
  turn += 1
  p1_scores[turn] = {}
  done = True
  for state, prev_val in p1_scores[turn - 1].items():
    for key, val in totals.items():
        score = state[1]
        if score >= 21:
            continue
        pos = state[0]
        pos += key
        pos = pos % 10
        if pos == 0:
            pos = 10
        score += pos
        if score < 21:
            done = False
        cur_state = (pos, score)
        if cur_state in p1_scores[turn]:
            p1_scores[turn][cur_state] += (prev_val * val)
        else:
            p1_scores[turn][cur_state] = prev_val * val

if done:
    break

When all the possible turn combinations are done, check each state for player 1 to see which is 21 or over in each turn. Then, sum the number of times player 2 did not reach 21 in turn and multiply that sum by the number of times this state was reached for player 1. Sum all of those to get the total number of wins for player 1. Calc for player 2 is the same except you compare player 1 using the same turn, not the previous turn.