The tipping point concept of non-transitivity

(above - picture of a set of 3 non-transitive dice from Grand Illusions website.)

What I'm seeing is not that people can't "think big", because they can. The US President can go from tying his shoe to considering Armageddon in a heartbeat. We all are free to consider BIG problems or TINY problems and the "auto-zoom" feature of our brains makes whatever we're considering fill out mental screen.

So, it's easy to be misled by small examples into thinking they're BIG issues. We don't seem to come with "ground wires" that keep our feet on the same ground.

That's probably a lot of what goes on in my favorite Snoopy cartoon where he's lying on top of the doghouse and thinking:

Did you ever notice
that if you think about something at 2 AM
and then again at noon the next day
you get two different answers?

But this morning I'm focused on why it is that a loop is so surprisingly hard for people to grasp.

I think it's not the wider view or scale, because people can do that "zoom" so effortlessly they don't even see it happen.

I think its that

• The value of "constants" changes with scale, and
• the relative ordering is non-transitive.

People aren't overly baffled when what looks like a short-term great idea turns out, in the long term, to be a terrible idea. As Dennis the Menace said, standing in a corner for punishment, "How come dumb ideas look so great while you're doing them?"

But each time people run into this, it's like suggesting to a Labrador Retriever that it might be time to go for a walk. "Oh, my God! Yes! A Walk! What an astonishing idea!" (Thank you Dave Barry for that thought about Labs.) The idea is visible, and logical, and sensible, but somehow it fades away to nothing between uses. We keep forgetting it.

The most likely reason I can imagine for that is that there is a larger idea, a context idea, that this change-with-scale property violates or offends, and, as soon as our conscious mind lets go of it, the cleanup crew in our brain looks for where to put it back and, mystified by it, decides it must be trash, because it doesn't fit anywhere with something bigger we preserve.

That's the easy one.

The loop thingie is ten times harder for people to grasp, even once. Even when people see it, touch it, play with it, some part of their brain rejects the concept as "clearly false" and is preparing to disassemble and discard it as soon as possible to restore sanity and normalcy.

And the problem isn't with a loop. People grasp the concept "circle." People don't run screaming from a "hula hoop" toy. It's more subtle.

It's more like the sense when you put a twist in a loop of paper, ending up with a Mobius strip. This does not feel right. This is uncomfortable, and barely tolerable, regardless how many times you've played with them or tried to cut one apart lengthwise and failed.

But, no, it's even worse than that. It's an M. C. Esher type loop, with a twist in a dimension that we don't even recognize as a dimension when we TRY to focus on it with our full attention.

It's a property of the children's game "rock paper scissors" - where there are three rules:
rock smashes (beats) scissors
scissors cuts (beats) paper
paper covers (beats) rock

So, there is no "best" one. This turns out to be a much more widespread phenomenon that we would prefer. We see it but reject it. For most things with multiple dimensions, the term "best" is meaningless, but we're so attached to it, we want to make it true anyway. We can't get resolution if we admit that there is no "best mate" or "best house" or "best job" or "best employee" or "best candidate" or "best football team". If you compare them by pairs, each pair seems to have a "better", but if you make a map of "better" it has no top or "best", but instead goes in a loop, or more than one loop. It's uncomfortable and a little scary. Things we thought we could rely on turn out to be shaky. We try to forget it, and succeed. Over and over.





Here's the classic example - the "non-transitive dice" that Martin Gardner described decades ago, and that Ivars Peterson attributes to Bradley Efron, a statistician at Stanford University.


You can read about these, but you just have to buy a set, or build a set out of construction paper, and even then you can see it but you can't believe it. There is no best one of these 4 dice, or of the 3 at the top of this post. A beats B, B beats C, C beats D, and D goes around the end of the barn and comes back and beats A. It's a loop and it just seems wrong.

(So, warning, don't try to win money with these, because the loser will be convinced you must have cheated.)

Well, as always, you must be wondering what this all has to do with health care or the problems of the world. So, back a few days ago I posted an analysis I did of why so many airlines are running late these days. Included in that was this loop diagram, that I made up, that you can click on to zoom up to readable size.



This one is a circle of "blame", where the blame is "non-transitive." Each set of people, in their local world, can blame the next group down the chain for the problem, and is clearly "right" -- which would be OK except that the list of blamee's goes in a full circle back to the "blamers."

Again, if you view this one box and it's neighbors at a time, it seems fine and makes sense. But if you put them all together in a circle, something seems to have gone terribly wrong.
Like this Esher print I love (from Wikipedia)


Or this one of stairs from Wikipedia.

There is wrongness there. But the wrongness is subtle.

That happens a lot more than we hold in our heads to be true.

So, where this comes down to Earth is the following conclusion. If people are going to learn about system dynamics and feedback loops, we need to get them past the point where very simple loops like the ones shown above, are perfectly sensible and acceptable, instead of where they are now, which suffers the mental version of tissue-rejection.

The problems will not come to us. We must go to them.

There is no way to make a circle into a line, regardless how "linear" a little part of the edge is if we simply elect to ignore the parts that go out of sight on each side of a narrow field of view.

Three facts seem to be true:

• Closed circles of causality make us queasy.
• Closed circles of "blame" make us and our legal system very uncomfortable.
• Closed circles of "blame" that show that what's happening to us is our own fault coming back to haunt us with a lag time and amplification are just intolerable thoughts and are rejected out of hand instantly. That's crazy talk.
  

We need to learn to be able to see BOTH lines and circles of causality without becoming queasy and needing a drink.

Suggestions welcome as to how to do that.

Wade

-------
from Ivar Peterson's MathTrek

Gardner, Martin. 1987. Nontransitive paradoxes. In Time Travel and Other Mathematical Bewilderments. New York: W.H. Freeman.

______. 1983. Nontransitive dice and other probability paradoxes. In Wheels, Life, and Other Mathematical Amusements. New York: W.H. Freeman.

One possible source of nontransitive dice is toy and novelty collector Tim Rowett. He offers a set of "Magic Dice" along with rules for several games at http://www.grand-illusions.com/magicdice.htm. You can find out more about Rowett's collection at http://www.grand-illusions.com/tim/tim.htm.