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.
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.
I agree with your analysis, but you should update the rule; your present wording claims that the threshold should always be 50%.
Yes, it was not apparent from the survival rule alone that survival can still be achieved according to the threshold rule. It should be clear now.