Nontransitive, sometimes intransitive or non-transitive, dice are a fascinating concept in probability. It concerns dice such that, in head to head matches, instead of having a neat ranking of "Die A will beat Die B which will beat Die C" and so on when rolled against each other, loops occur. For instance, this post concerns the three set problem, where when paired, A rolls higher then B, B rolls higher than C, and C rolls higher than A.
But not only does it ask that, it asks a very specific extension: how favorable can you make the odds? The answer turns out to be 7:5; in the highest voted answer, eight dice sets are given, four with symmetry, such that each beats the next one seven times out of twelve.
My question is this: suppose I want a set of four nontransitive dice with six faces each, A, B, C, D, such that A beats B, B beats C, C beats D, and D beats A. Among those matchings, there should be no ties possible. However, if the pairings are A v C or B v D, it should be evenly matched. If the odds for the loop have to be equal and maximized, how far from even can they get?
A further question might be, for five dice, can you have two loops, A>B>C>D>E>A and A>C>E>B>D>A such that all probabilities are the same? And again, how unfair can you make it?