The first mover advantage in Tit-for-Tat is not the same as the business strategy notion of first mover advantage.
We can see the hesitation in the start of peace talks when both sides are seeking to gain a military advantage. By definition this means the violence is still being used as a tactic to win and not as part of a de-escalation. Only by tempering the response to a reciprocal one (proportionate but not to be taken as a patsy) will progress be made, someone has to start.
Part 3 - Consider - On Power - Circumspect Use of Power