A set of three 'balanced' non-transitive dice

The structure of two sets of three non-transitive go-first-fair dice - sets 1 and 5 below.

I've spent a small amount of time noodling around with different dice sets in the last few days with a view to constructing interesting sets for use in my recent game idea, Cube Whist. Rather than trying to think about a whole set of dice at once, I've been looking at smaller subsets as a starting point, and seeing if we can make sets with certain properties.

There are a lot of properties that a set of dice could potentially have, and I am by no means going to attempt to be exhaustive in listing them, but I list a few that I am interested in. I have tried to use standard, or at least pre-existing, terminology where I am aware of such things.

1. A set of dice is tieless if no two dice share any faces
2. A die is degenerate if all its faces are identical
3. A set of dice is nondegenerate if no dice in it are degenerate
4. A set of dice is completely distinct if no die has any repeated face
5. A set of dice is nontransitive if they can be ordered (and labelled A, B, C, ...) such that the probability A rolls a higher number than B is greater than ½, and similarly B rolls higher than C, and so on, with the last dice beating the first
6. A set of dice is equi-nontransitive if they are nontransitive, and the 'dominating probabilities' (i.e. the probabilities involved in the intransitive cycle) are all equal
7. A set of dice is go-first-fair if, when they are all rolled, they each have an equal probability of being the highest
8. A set of dice is place-fair if, when they are all rolled, they each have an equal probability of ranking anywhere in the set (first, last, second, etc.)
9. A set of dice is permutation-fair if, when they are all rolled, any ordering of the dice is equally probable

For my application, I initially want to look at tieless dice (kicking the can of what to do about ties a little down the road). I also want the set to be nondegenerate (as degenerate dice are kind of boring), and in fact completely distinct, to make sure I am 'maximally utilising' the structure of dice. Some of the other properties (nontransitivity, go-first-fairness) are actually unaffected by relabelling consecutive numbers on a single die with repetitions of one of the numbers, so we could insist on a slightly weaker condition than completely distinct without it affecting other things, but as I am planning to have these dice interact with additional dice in the final game, I think it is worth thinking of things in terms of fully-distinct faces.

I think it is interesting to also consider nontransitivity - this is something you cannot get with cards (well, of course you can, but not as straightforwardly, or cleanly), so I feel like it would be a good feature for the game. I wanted to take this a step further, so that (amongst this set) there is no 'best' dice - meaning that I also wanted them to be go-first-fair and equi-nontransitive. My plan was to see if this is possible before thinking about any further constraints, and initially I'm just looking at sets of three dice.

Results

There are 2,858,856 completely distinct sets of three dice — for other numbers of dice see the OEIS sequence A025038. Of these:

• 916 are go-first-fair
• 10,705 are nontransitive
• 1,910 are equi-nontransitive
• 18 are go-first-fair and place-fair
• 11 are go-first-fair and permutation-fair
• 16 are go-first-fair and nontransitive
• 7 are go-first-fair and equi-nontransitive

So the key category we are interested in has seven entries - this is probably too small to try and narrow by additional properties, but two things I might look at to pick particular examples from this are minimising the number of consecutive numbers on a single die, and maximising the non-transitive probability. In this case also, insisting on equi-nontransitivity and go-first-fair gives us place-fair for free.

Below is a table listing all the go-first-fair dice which are also either nontransitive or place-fair. But also, I want to highlight two of the equi-nontransitive go-first-fair sets as representatives:

• {1, 6, 9, 10, 13, 18}, {2, 3, 11, 12, 14, 15}, {4, 5, 7, 8, 16, 17}, one of the sets which has maximum probability for one die beating another [set 1]
• {1, 6, 8, 11, 15, 16}, {2, 4, 9, 12, 13, 17}, {3, 5, 7, 10, 14, 18}, one of the sets which has smallest number of consecutive faces (2 — 15,16 and 12, 13) [set 5]

For my game I want to extend this to sets of four dice, but I might try and use one or other of these sets in the meantime as part of a larger set of dice to see how this dynamic plays out.