How do I rescue Josie Thorpe?
I managed to do it by sympathizing/selecting empathy with him three times. Logic seems to piss him off, and humble doesn't do much.
However, after selecting empathy, he seemed to hesitate, agreed with what I said, and backed off a little bit. So after seeing that reaction, I just stuck with it for the other 2 choices, and it worked.
After having done the "persuasion" minigame a few times by trial and error, you can see that there are multiple ways to get finished: the conversational tree weaves around the central "you win" thread. You can tell that you've "hit" the person you're talking to...They visibly react. And if they back off, or get calmer, you're on the right track.
On the other hand, they also visibly react in a negative way, and you need to watch for that too, because your screw ups (and your successes) will color the whole conversation. I had a lot more trouble with the later boss conversation (protip: Don't go in defensive. If you get in his face early, he will chew you up and spit you out), and I ended up running through it three times, and some of the "telling" shots showed up in winning and losing runs.
I think it must run on a "point" system, where you get points for choosing the right way, and lose points for choosing the wrong way, and if your point total is high enough, you win.
I can't remember the exact sequence I used to free Josie. I mucked around in the bottom 2 (not aggressive or humble, but logic and the other one) until he started trying to leave with her, and then I hit him with humble, and he let her go. I honestly thought several times I was screwing it up, but it worked out, and I got "silver tongue" credit for it.
I definitely think you need to not be too hard, or too soft on him early. I think that either one of those will drive him the wrong way. Sarif was similar. If you call him out early, he slaps you down, but if you never call him out, he doesn't give you anything...You've got to be cool and rational, and then WHAM! Nail him below the belt when the shot opens up.
I did it by doing each of the three options Rational, Empathize and Humble in that order (I think those were the names).