Transitivity and its failures. – Math with Bad Drawings

In life, we tend to expect transitivity. In other words: if A > B, and B > C, then A > C.

A jackal is heavier than a cobra. A cobra is heavier than a mongoose. So a jackal had better outweigh a mongoose, or else some weight-conscious animal has been editing brazen lies onto Wikipedia.

But weight is simple. A single measurement. Complex traits—like, say, fighting ability—can’t be so easily summarized. You’ve got to consider speed, strength, strategy, tooth sharpness, poison resistance, endorsement deals… with so many interacting factors, transitivity fails. In this case, a mongoose can defeat a cobra, which can defeat a jackal, which can defeat a mongoose.

The same is true of human combat: in a trio of famous fights, Joe Frazier beat Muhammad Ali, who beat George Foreman, who beat Joe Frazier.

You find a similar dynamic in another facet of everyday life. That’s right: the mating strategies of the male side-blotched lizard.

Some males (“monogamists”) stick close to a single mate. But they’re outcompeted by another kind (“aggressors”) who conquer a large territory, building a harem of many mates. Aggressors, too, have a weakness: a third kind of male (“sneakers”) who wait until the aggressor is away, then get busy with his unprotected mates. Yet the sneaker, in turn, cannot succeed against the watchful monogamist. Aggressor conquers Monogamist, who defends against Sneaker, who gets the better of Aggressor.

Next time someone proposes a game of Rock, Paper, Scissors, I urge you to counter-propose with a game of Lizard, Lizard, Lizard.

Political scientists boast their own version of non-transitivity: the Condorcet Paradox. In an election with multiple choices, it’s possible that the electorate will prefer Taft to Wilson, Wilson to Roosevelt, and Roosevelt to Taft. More than just another great a Rocks, Paper, Scissors replacement, this is a vexing challenge for political theorists. It means that seemingly innocent changes to the structure of an election may have dramatic effects on its outcome.

Indulge me one more example, a favorite of mathematicians. You place three special dice on the table for inspection, and allow your opponent to pick whichever one they want. You then pick one of the remaining two. Both dice are rolled, and the highest number wins.

The trick? Their strength is non-transitive. A usually beats B, which usually beats C, which usually beats A.

Whoever picks their die second can always seize the advantage.

Cool fact about this particular trio: If you bring out a second die of each kind, and compete by rolling a pair of same-color dice, then the cycle reverses direction.

As we’ve seen, transitivity holds in the simplest cases (6 > 5, and 5 > 4, so 6 > 4) but wilts under the breath of complexity. I’m afraid to report that real life is rather complex. Every decision we make could lead to a dizzying array of outcomes: some good, some bad, some likely, some not, and all of them contingent on forces beyond our control.

In one psychology study, students were asked to choose between pairs of fictional job applicants. Their preferences formed a non-transitive loop: A beat B, who beat C, who beat D, who beat E, who beat A. “I must have made a mistake somewhere,” one student fretted, when shown the non-transitivity of his choices. He hadn’t.

It’s just that transitivity is simple, and making decisions under uncertainty is not.

These thoughts circle my head any time I’m asked to rank anything. Sure, our world permits occasional clarity. The best gymnast is Simone Biles. The best Billy Joel album is “The Stranger.” The best squash to eat—not to mention to pronounce—is “butternut.”

But usually it’s not so simple. What’s your favorite Le Croix flavor? Who is the strongest student in your class? What writer has inspired and/or depressed you most? In such cases, there may be no right answer, just a non-transitive mess.


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