Say you were to “host” a single elimination tournament of 8 dice, each with a different number of sides: 4, 6, 8, 10, 12, 20, 24, and 30. In each round, the dice that rolls the highest wins (re-roll both on ties). All dice are numbered normally from 1 to the number of sides except the 10-sided, which is 0-9. The dice are seeded and placed into a bracket for the tournament (30-sided is 1-seed, 24-sided is 2-seed, etc.) (10-sided is the 5th seed, and 8-sided is the 6th seed). Here’s what it would look like:
The “l”s are used as spacing.
Assume all dice are perfectly fair and have an equal chance of rolling each number. What is the chance each dice has of winning the tournament?
(If you’re wondering why I’m asking this, my friend gave it to me as a problem to see if I could solve it. I couldn’t and went back to him asking for the answer only to find out he didn’t know either)
In a more general sense, if you have an 8x8 matrix that contains the probability that one dice/team will beat another (based on seed, so row 1 col 2 is the chance that the 1 seed beat the 2 seed), what is the chance of each seed winning the tournament?
Feel free to just answer the specific case (with the dice) if the general case is too complicated.