This question relates to a card/dice game by Stuart Ashen which unfortunately I cannot name here because its name is Norfolk slang for male genitalia.
This game is played in several phases, but the question relates to the final "battle phase" which is played using 6-sided dice. The previous phases of the game serve to determine the number of Health Points a player has in the battle phase, which can (and usually will) be different for each player.
The dice portion of the game is then played as follows.
- Both players roll a fair 6-sided dice.
- If either player rolled 1-3, their opponent loses 5 Health points unless the opponent rolled 4-5, in which case nothing happens.
- If both players roll 4-5, a fair coin is flipped and the loser of the flip loses 5 Health points.
- If either player rolls a 6, their opponent loses 10 Health points no matter what. The effect of the opponent's roll still applies.
Thus the full table of possibilities is:
- Both roll 1-3: both lose 5 points.
- X rolls 1-3 and Y rolls 4-5: nothing happens.
- X rolls 1-3 and Y rolls 6: X loses 10 points and Y loses 5.
- Both roll 4-5: flip a coin and the loser loses 5 points.
- X rolls 4-5 and Y rolls 6: X loses 10 points.
- Both roll 6: Both lose 10 points.
This process repeats until one player runs out of points, at which point the other player wins. If both players run out of points at the same time, the winner is determined randomly (the rules in the actual game for a draw are slightly more complex than this but they more or less come down to randomly).
If player A goes into the battle phase with x points, and their opponent has x+d points, is there a single formula for the probability of player A eventually winning? And as a corollary, is there some threshold value of d, calculated in terms of x or otherwise, where player A effectively cannot win?
Based on some trials-based simulations the probability of the underdog winning drops dramatically when d increases (as you'd expect), but it also rises quite quickly when x increases as a result of more rounds being played. The majority of dice combinations maintain the status quo but do affect the speed at which the base x points are drained.