Non transitive dice.
They are a thing. See http://www.futilitycloset.com/2013/10/23/nontransitive-dice-2/
So, this page is a place to explore the game-design potential of a set of non transitive dice.
See also discussion on my fb page at https://www.facebook.com/nemothorx/posts/10151955159793216
Regarding non-transitive dice. I hope to write up something for your wiki page, but it will have to wait a few weeks. But I thought you'd like this cool result. Of all the possible combinations of triplets of 6 sided dice using 6 numbers, there is just one triplet that is symmetric and has the maximum "strength" - such that each dice beats the next with the same probability, and that probability is the largest of all symmetrical options. (I hope that makes sense). Unfortunately, they still aren't equally likely in a three-way roll. The set is: (1,2,2,5,5,6), (1,3,4,4,4,4) and (3,3,3,3,4,6). The likelihood that one beats the next is 4/7.
Non-transitive dice are a set of dice, which have the property that, on average, the first dice beats the second, the second beats the third, the third beats the fourth, etc, and finally (and critically), the last dice beats the first.
Non-transitivity is an interesting property in any competitive venture. Almost anyone that has ever played competitive sport will have encountered the counter-intuitive situation where person A beats person B, B beats C, but C beats A, leaving you wondering ‘who is the best?’
This property is termed non-transitive, based on the mathematical notion of https://en.wikipedia.org/wiki/Transitivity
As it turns out, it is possible to make many sets of dice that have this property. The possibility of designing board games specifically for such sets of dice is explored here.
One desirable property for non-transitive dice for use in board games is that the probability of each die beating it’s neighbour is the same, so that no dice has an overall advantage in the gameplay. Such a set may be termed ‘symmetric’. Designing such a set of dice is difficult analytically, however a brute force approach may be employed. Such an approach has shown that for a set of three, six-sided dice, each confined to display only the numbers 1-6, there are 129 different sets of symmetric non-transitive dice. Of those 129 sets, there is a unique set which has the additional property that the probability of each dice beating its neighbour is a maximum (over the 129 sets). This set is: (1,2,2,5,5,6), (1,3,4,4,4,4) & (3,3,3,3,4,5). For this set, the chance that each dice beats its neighbour is 4/7.
Another desirable property for non-transitive dice in game-play would be that if all three dice were rolled simultaneously, then there would be an equal chance of any player winning. Unfortunately, for a set of three six-sided dice, confined to display the numbers 1-6, such a combination is impossible.
One interesting property of a set of non-transitive dice is that any set can be transformed into another set (which is sometimes identical to the original) by swapping each number with its higher or lower pair (ie. swap 1 & 6, 2 & 5, 3 & 4. This is effectively equivalent to simply redefining the winning condition as having the lower number, rather than the higher number, and could be an interesting mechanic in game-play.