If a set of four dice, a, b, c, and d, are numbered such that a (on average) beats b, b beats c, and c beats d, it's intuitively obvious that a will also beat d. Intuition is wrong: a statistician called Bradley Efron has invented a set of four dice numbered such that a beats b beats c beats d beats a.
Each die beats its neighbor 2/3 or 66·6% of the time. But those dice are square, or hexahedral, with six sides. Here are dice with the same property for the other four regular polyhedra (the dice were generated by brute force searches, so are not the best possible):
(1,1,4,4) beats (1,3,3,3) beats (2,2,2,4) beats (1,1,4,4).
Estimated % of wins: 1 = 57.14%; 2 = 56.25%; 3 = 57.14%
(1,4,4,4,5,7,8,8) beats (1,2,3,3,8,8,8,8) beats (2,2,5,6,7,7,7,7) beats (2,4,4,5,6,6,8,8) beats (1,4,4,4,5,7,8,8).
Estimated % of wins: 1 = 50.91%; 2 = 58.06%; 3 = 52.54%; 4 = 54.72%
(1,1,1,4,5,6,7,8,9,10,10,11) beats (1,3,4,4,5,7,7,7,7,7,9,11) beats (2,3,4,4,4,5,5,6,10,11,11,12) beats (1,1,3,4,4,6,8,8,9,10,11,11) beats (1,1,1,4,5,6,7,8,9,10,10,11).
Estimated % of wins: 1 = 52.67%; 2 = 50.38%; 3 = 52.67%; 4 = 52.34%
(2,3,4,6,7,7,7,7,8,9,9,10,10,12,14,16,18,19,20,20) beats (1,3,4,5,5,6,6,6,7,9,12,13,14,14,15,17,17,17,20,20) beats (2,2,2,3,3,5,6,8,9,12,12,12,13,14,14,16,18,18,19,20) beats (1,3,3,6,6,7,8,9,10,11,11,11,12,12,13,13,16,16,17,17) beats (2,3,4,6,7,7,7,7,8,9,9,10,10,12,14,16,18,19,20,20).
Estimated % of wins: 1 = 51.57%; 2 = 52%.
Octahedron
Dodecahedron
Icosahedron