Is anyone else intrigued by intransitive orderings?
The most well-known intransitive ordering may be
Rock beats Scissors, which beat Paper, which beats Rock.
I remember a claimed intransitive ordering among the boxers Ali, Foreman and Frazier, but was this refuted after rematches?
Another can arise in Hold’em Poker. Suppose there are only two players before the flop, and they’re “all in” so no further betting is possible. If one player has a pair of Deuces, you’d rather have Queen-Jack suited (52.7% to win) than Ace-Eight off-suit (47.1%), but Ace-Eight off is favored (53.0%) against Queen-Jack suited. (The figures shown assume no suit collisions other than the QJ suiting.)
I’ve been especially intrigued by a claim in diachronic linguistics. The natural tendency to relax grammar but pack maximal meaning into speech effort means that languages with fusional grammar (like Slavic) tend to evolve into isolational grammar (like Thai), then into agglutinative (like Turkic), then back to fusional. (A complete cycle may take several thousand years.)
Military strategy presents examples, for example medieval armies used three groups of warriors with pike-men beating cavalry, who beat archers, who beat pike-men.
What about intransitive orderings involving more than three elements? A fielded army will consist of several types of soldier; will different opposing mixes lead to a complicated ordering of the mixed arrangements?
Several intransitive orderings have been proposed for the five Chinese elements, for example: Fire melts Metal; Metal chops Wood; Wood parts Earth; Earth absorbs Water; Water extinguishes Fire.
Any other interesting intransitive orderings?