Thank you for pointing out what you perceive as an error.

However, this is not an actual counterexample to the rules. If TFT falls below a certain threshold as explained in the threshold rule, it will actually go extinct in this case. Let me shed a little more light on the issue of thresholds, which I have largely ignored in my post because of the cumbersome equations involved.

If A doesn't score higher against X than X does against itself, there trivially exists a minimum amount of A-strategies for which the point gain from A vs A encounters becomes irrelevantly small compared to point loss from A vs X, since the latter is at least 1 while the amount of A-strategies can be arbitrarily low with point gain from A vs A approaching 0. Knowing this, we can calculate the threshold in question; here exemplarily done for TFT (A) vs DefectBot (X) for 10-turn matches:

• Points for A against itself (aa) = 40
• Points for A against X (ax) = 9
• Points for X against A (xa) = 16
• Points for X against itself (xx) = 10

We define n as the threshold percentage of A in the pool, in which case the following is true (assuming a large enough pool):

• n * aa + (1 - n) * ax = n * xa + (1 - n) * xx
• n * (aa + xx - ax - xa) = xx - ax
• n * 25 = 1
• n = 0.04

So if TFT makes up less than 4% of the pool, it will actually go extinct. The main reason why this threshold is so low is because xx - ax is only 1, the smallest possible value.