Survival Rule:
For A in this scenario not to go extinct, it must score at least equally high against X as X does against itself, and if it doesn’t score higher, it must score at least equally high against itself as X does against itself while not losing direct encounters.
A population with a majority of DefectBots and a minority of TFT is a counterexample.
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.
This statement is wrong:
A population with a majority of DefectBots and a minority of TFT is a counterexample.
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.