"And are you really "exploiting" an "irrational" opponent, if the party "exploited" ends up better off? Wouldn't you end up wishing you were stupider, so you could be exploited - wishing to be unilaterally stupider, regardless of the other party's intelligence? Hence the phrase "regret of rationality"..."

Plus regret of information. In a mixed population of classical decision theory (CDT) agents and Tit-for-Tat (TFT) agents, paired randomly and without common knowledge of one another's types, the CDT agents will imitate the TFT agents until the final rounds, even when two CDT agents face each other. However, if the types of the two CDT agents are credibly announced then they will always defect and suffer great losses of utility.