Intransitive Orderings

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?

Rock, Paper, Scissors is an example of a nontransitive game.

My favorite is “Penney Ante” (Penney’s Game). But you might find nontransitive dice more interesting.

Rock Paper Scissors Lizard Spock
From the Big Bang Theory

Numberphile once did a video on three groups of lizards that exist in RPS-style equilibrium.

Interesting! It’s not three different species; it’s three different “morphs” of the male of one species competing for sexual partners. (Apparently the female of this species also has two distinct morphs.)

I wonder if one could form an analogy using three different types of human male. :cool:

Nah, we’re unified in our approach, and in our hierarchy of only two (men vs. off), we are victorious every time. Off doesn’t beat men; men beat off.


Nontransitive dice

Technically speaking, if there is no transitivity, then you can’t really put the elements in order, can you. Which is precisely what makes things interesting, as there is no best element, best language, or best die to pick.