Unintuitive examples of non-transitivity

A transitive relation is a relation r where if A r B and B r C, then A r C. A non-transitive relation is any relation that isn’t transitive.

The most famous example is rock-paper-scissors: rock beats scissors, scissors beats paper, but rock doesn’t beat paper. However, there are other ways to make nontransitive relations.

All of the examples I’ll give in this article will be shown non-transitive by finding a basic rock-paper-scissors cycle in an antisymmetric relation.

It’s possible to make three random variables such that A > B, B > C, and C > A, each with over 60% probability: Set A to 3, B to 61% 2 and 39% 5, and C to 61% 4 and 39% 1. A 4-cycle can allow for the winning probability to be 2 in 3. It is impossible for the winning probability in any cycle to be larger than 75% if the variables are uncorrelated (Past 75%, each one has to have a lower median than the previous, which can’t happen in a cycle). If the variables are correlated, the probability can be arbitrarily large (To get 99%, you need 100 variables.)

Now, consider the functions f(x) = 3+3*round(x/3), g(x) = f(x+1)-1, and h(x) = f(x+2)-2. For any x, f(g(x)) > g(f(x)), g(h(x)) > h(g(x)), and h(f(x)) > f(h(x)). This means that comparing functions by which application order makes the result greater is a non-transitive relation.

Now for a non-theoretical place where non-transitivity shows up:
Say you’re hosting a vote for which of three things to use, and people submit their votes as rankings of the three things. It turns out that it’s possible for the majority rule on individual pairs to lead to a non-transitive relation, e.g. if the votes are A > B > C, B > C > A, and C > A > B, then A > B, B > C, and C > A all appear in 2/3 the votes. This is called the Condorcet paradox.