A few things work for me. If I purposely tell myself not to give a fuck about whether I have or don't have a girl things change dramatically in my favor.
Also, and this isn't lying or running game, but girls I find attractive that I see on a regular basis I make a point of throwing in compliments for them. Some girls I might say they have nice hair or I like the new hair cut. I notice things like that. Maybe I'll compliment a ring they are wearing or earing. The good part is I will only say something like that if it indeed does catch my eye and I mean it because A. any girl I'm attracted to I see more than just a sex object and B. I'm a horrible liar.
Really attractive girls I might throw out an aggressive compliment like you have a nice ass or you're looking damn good in those jeans today and leave it deadpan at that or just totally change to a different subject. It all depends on the girl. I typically save compliments like that for really hot girls who's personality I kinda know. Probably my favorite thing to do is make eye contact with a girl, look her up and down and say damn you look good and then smile. Almost tied for first would be eye contact with a grin and wink.
Some girls I might be sorta neutral with and others I might be a little aggressive with. Most girls who's interest I have attracted either was from me being quite aggressive with them and basically letting them know I want them and won't be shy about it or when I'm just totally relaxed and not stressing about it.
And I'm one of those guys with limited game, but this stuff works for me.
I know this doesn't address that particular situation, but I couldn't give an honest answer as to how I would respond in that scenario.
I was walking to the main entrance of a local store today and a quite attractive woman in her early 30's with a cart was heading past me and I looked her way and saw she was looking at me and kept my eyes focused on hers and just lightly grinned. She gave me a shy kinda grin as she went by.
But, yesterday at another store I caught a girl looking at me and made eye contact and held it until she broke it off. No grins, nothing.
I know some girls would say this isn't true, but unless she is an unusual chick, never break eye contact first.
Oh ya, the last and probably most important thing I can add that is tops is if I like a girl I try to make her feel good whenever around her. Whether that be from physical or material compliments to jokes and/or good humor. This is over complicating things, but the way I see it, the best thing you can do is have a girl associate you with positive feelings.
Just my opinions.