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No Pure Strategy Is Stable

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## The Thought

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The only problem with the dynamics as I've described them is that the state curves back toward the AC corner before it reaches the edge of population space. Strictly speaking, TT ought to be favored over AC as long as there's even the tiniest amount of AD present. This, however, is where my idea comes in.

Suppose you're a kid growing up in a world of TTers. You gradually develop your own strategy by observing how others interact, and what you see is that they always cooperate with each other. What conclusion would you draw? Wouldn't you become an ACer rather than a TTer?

(It's true that kids interact with each other as well as with adults, and that they often fail to cooperate; I hope this doesn't completely invalidate my idea.)

Here's a more formal way of saying the same thing. If we add to the system the idea that strategies are learned rather than innate, a new force enters the dynamics, one that acts to push the state from TT toward AC. As a result, TT becomes unstable and the system oscillates—or possibly spirals into an equilibrium containing all three strategies.

The problem of learning the strategy TT reminds me of the problem of learning the color “grue” in The Grue-Bleen Paradox. (And no, there's no connection to Zork.)

“Grue” can be defined in English as follows: If something is green before midnight, December 31, 1999, and blue thereafter, then it is grue.

:

Many philosophers believe that no one could really learn the Gruebleen language as their first language. Sure, parents would point to the grass and say “grue,” point to the sky and say “bleen.” But there is more to grue and bleen than that.

:

At some point, a parent or teacher has to sit a Gruebleen-speaking child down and tell him the facts of grue and bleen.

All that doesn't even touch on the paradoxical aspects, by the way.

Another good way of looking at the situation comes from the book The Evolution of Cooperation, which I'm finally reading.

What accounts for TIT FOR TAT's robust success is its combination of being nice, retaliatory, forgiving, and clear.

The book goes into more detail, of course, but all I need here, I think, are a couple of definitions.

This is the property of being nice, which is to say never being the first to defect.

:

A rule can be called retaliatory if it immediately defects after an “uncalled for” defection from the other.

As a kid, I was certainly taught to be nice and forgiving, but I don't remember anyone ever explaining that I should be retaliatory (or clear). Quite the opposite, in fact … consider, for example, Matthew 5 : 38–39, which even appears under the heading “Concerning Retaliation”.

“You have heard that it was said, ‘An eye for an eye and a tooth for a tooth.’ But I say to you, Do not resist an evildoer. But if anyone strikes you on the right cheek, turn the other also; …

But, being retaliatory is exactly what distinguishes TT from AC. Thus, not only was I not taught to distinguish the two (as grue from green), I was also actively taught to use AC. Go figure!

I'm also reminded of what I said about being intolerant of foolishness. I was certainly taught to be tolerant; could that be the same thing as not being retaliatory?

In closing, here's a thought (from Bartlett's) I find appropriate, even though it's not about retaliation per se.

Eternal vigilance is the price of liberty.

Alleles and Loci
Notes (Game Theory)

@ November (2000)