Basically, non-transitive dice allow you to model a rock-paper-scissors like asymmetry where one die can be better than a second (statistically), the second can be better than the third, but the third can nonetheless be better than the first! Here's a concrete example of a set of three six sided non-transitive dice:
Die A: 1,4,4,4,4,4
Die B: 3,3,3,3,3,6
Die C: 2,2,2,5,5,5
(A beats B 25 out of 36 times, B beats C 21 out of 36 times, and C beats A 21 out of 36 times)
The most obvious application that comes to my mind is in modeling unit asymmetries in wargames, but my guess is that's just the beginning. Any thoughts on other applications? Also, do people think that incorporating modifiers into non-transitive dice (for example adding or subtracting 1 from a roll) would diminish their elegance?
Here's a link to an article on wikipedia about non-transitive dice (with in depth examples!), as well as a link to a store that sells two different sets of them:
http://en.wikipedia.org/wiki/Nontransitive_dice
http://www.grand-illusions.com/acatalog/Non_Transitive_Dice_-_Set_1.html
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What you're saying makes a lot of sense, but I believe the real complexities of these non-transitive dice become more apparent when you look at situations that are not one-on-one. For example, given the same set of dice referenced above:
Die A: 1,4,4,4,4,4
Die B: 3,3,3,3,3,6
Die C: 2,2,2,5,5,5
Imagine 1,000 As vs. 1 B. There would be at least a 1/6 chance of the B die surviving as all it has to do is roll a 6 and it doesn't matter what those 1,000 As roll! (Note: this would not be the case with 1,000 Cs vs. 1 A.) Conversely, if you have 1,000 Bs vs. 1 A it would be statistically amazing if those Bs didn't destroy that A as all they would need is to roll a single 6! In a strange way it could be argued that the B dice become more powerful than the A dice in this situation in contrast to the one-on-one situation where the A dice are clearly superior.
My statistical and computational abilities are limited, but it seems like there is a lot of room for the unexpected to evolve out of these dice. Of course if we're going to talk about situations that are not one-on-one, we'll need to get clear just how the dice interact. For example, if two players each roll ten dice, how do we figure out how the dice are assigned to one another? For example, does the defender (if it's a wargame) simply assign the dice, or do you match highest number to highest number and then work your way down (kind of like Risk), or should some other totally different method be used?