Cockroaches

A   
wonderful thing has happened this decade which is comparable to the moon landing of '69.

The cockroach problem has been solved for all finite instances.

Which means that any building which is infested with cockroaches can be rid of them in a few weeks ... really ... no fooling. For the first time in the history of civilization, man has the upper hand in the cockroach wars. The deciding factor is new pesticides which began widespread use in the mid 90's. I have personally seen the results and confirm the efficacy of the "treatment".

Prior to this the only remedy was to "keep the place very clean". This tactic never worked because a cockroach's problem isn't food ... it only needs the tiniest amount ... a single crumb. What it must have to create an infestation is ... water.

Why? Because after everybody goes home (as from a business), the available water evaporates but the food just sits there, forever edible. Hence, you could never starve them out and if you had a leaky pipe ... the roach party never stopped.

Can we kill them all now? ... No, of course not. We can kill ~98% and that's a victory ... isn't it? What constitutes a victory?

The Mathematical Problem ...

When does an "inhabitation" become an "infestation"?

We know when a place is infested because ... they're everywhere. What does this mean quantitively? A place is inhabited when there's just some livin' here & there. What does this mean quantitively?

Is the earth inhabited by humans or infested by them?

Obviously, this constitutes an infestation. We are everywhere ... displacing all other large animals ... ruining the planet's fundamental natural environment (not because of greed or neglect, but because there are just tooooo many of us). But just when did the population become an infestation? Is there some dividing line which we can all agree on? ... Something with a coldly logical basis?

I have detected this problem before ... "When is a minority too small to be capable of effecting any change in the big picture?"

I mean here a "grass roots" minority. If an individual attains prominence and people follow his recommendations, he's a minority of one, right? Hmmmm ... No. That's not what I mean ... when many people follow him, those people + him = a much larger growing group.

What I really mean is ... How big must it get (as a percentage of the culture) before a grass roots group shows up on the general social radar? ... i.e. before it becomes an "infestation".

I suspect that the answer is less than 5% perhaps as low as 1/2%. That is, below this hypothetical level, the probablilty of such a group successfully influencing it's culture decreases precipitously. It's never zero ... but there must be a big dropoff at some identifiable point on a graph.

I believe this to be an important problem in the milieu of politics because it would determine just when people in such a group should be encouraged by their chances of political success and when they should patiently await sufficient growth in their ranks.

Here is the form taken by the graph of such an equation. The point of interest is the inflection point (actually the point of minimum radius of curvature) designated "A". The long gentle curve "B" is necessitated by the fact that at 100%, the probability of affecting the culture must be 1 by definition.

The curve cannot be convex here because it must get there without approaching 1 asymtotically. The two ends of the graph may even be inversions of each other (meaning that there is a precipitous upturn as we approach 100% ... which makes sense since the small percentage is the one which would be opposing the larger ... get it?).

Therefore, what is the exact equation? Is there only one or is there a family of such equations? ... Maybe one equation ... for each "n groups in a given culture"?

I don't know the answer to this. However, I suspect that it has already been done long ago. [any info appreciated]

Addendum 11/29/99:
This problem is actually one of "difference perception". That is, how big of a difference must be made until it becomes noticable to human perception?

Example: Playing cards

If you are accustomed to the "feel" of a given deck of cards, you can tell when only 1 has been removed by the difference in the thickness. That's about 2%.

But ... if you have a stack of loose leaf paper, say, 100 pages thick ... and subtract just 1 ... few could tell the difference. That would be about 1%.

So, our tactile difference perception is between 1 and 2 percent.

This may differ for other senses, but you get the general drift. The number I'm seeking above is around a percent or two. It's certainly not 10% because if someone lost ten percent of his height overnight ... everyone who knew him would notice immediately. But if only 1 inch out of, say, 70 ... they would say ... "Hmmm ... did you change something?", i.e. borderline.



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