Nontransitive Dice

Definition A set of dice is nontransitive when the relation “beats” among them is not transitive. The relation comes from a game for two players. Each player chooses a different die from the set, the second player after seeing the first player’s choice. Each player rolls their die once, the two rolls are independent, and each die is fair, which means that each of its six faces is equally likely to land on top, even when two faces show the same number. The higher number wins the roll, and one die beats another when it is more likely to win the roll than to lose it. A standard set uses these three dice.

No number appears on two different dice, so a roll can never tie. Die B beats A, A beats C and C beats B, each with probability $5/9$. The resolution is that “beats” compares two dice at a time, each comparison with its own $36$ equally likely pairs of faces, and nothing forces these separate comparisons to fit into one ranking from strongest to weakest. So whichever die the first player chooses, the second player can choose one that wins a roll with probability $5/9$, taking B against A, C against B and A against C. That is an advantage, not a certainty, since the second player still loses a roll with probability $4/9$.

Definition A relation on a set says, for each ordered pair $(x, y)$ of members of the set, whether $x$ is related to $y$. The relation is transitive if, for all members $x$, $y$ and $z$, whenever $x$ is related to $y$ and $y$ is related to $z$, also $x$ is related to $z$. A relation that fails this for at least one choice of $x$, $y$ and $z$ is not transitive, also called nontransitive. When the members are ranked in a list without ties, the relation “comes earlier in the list” is transitive, so a relation that is not transitive cannot be read off from any ranking.

Definition For two dice $X$ and $Y$ rolled once each, independently, the win probability of $X$ against $Y$ is $P(X > Y)$, the probability that $X$ shows a strictly higher number. The three probabilities $P(X > Y)$, $P(X = Y)$ and $P(X < Y)$ add to $1$, so when ties are possible a win probability below $1/2$ can still be larger than the chance of losing. On this page $X$ beats $Y$ when $P(X > Y) > P(X < Y)$, and when ties cannot happen this is the same as $P(X > Y) > 1/2$. If both dice are fair and six-sided, each of the $36$ ordered pairs made of one face of $X$ and one face of $Y$ has probability $1/36$, so

where $N$ is the number of those pairs in which the face of $X$ shows the higher number.

Definition The expected value of a random number $X$ that takes the values $x_1, \dots, x_k$ with probabilities $p_1, \dots, p_k$ is

For a fair six-sided die it is the sum of the six face numbers divided by $6$, and over many independent rolls the average of the results tends to settle close to it. The expected value describes one die on its own, while a win probability depends on how the numbers of two dice fall between each other. So equal expected values do not make two dice equally strong against each other, and a larger expected value does not guarantee a win probability above $1/2$.