My background: As a mathematician, probability theory is my field of research; please do not answer/comment just to refer to a course on probability theory.
I expose below an erroneous application of probability theory. It is summarized at the end, in the Summary section. The two probabilities $\Pr$ and ${\Pr}^\ast$ are explained just before. If you are not a bit specialist in probability theory, please do not downvote if you disagree with this point, unless you show an error in my reasoning at a precise point (after numerous edits, I believe I have enough detailed my explanation so that one can point a precise point).
The previous comments helped me to write a short summary on my blog (rather oriented in general towards a mathematical audience), about the prisoner dilemma.