There is no such thing as a 'dating game'. It is not a game, at least for me. If you are interested in someone, you could wait for the guy to initiate something - but what if he's shy and hoping you will make a first move? If both parties are equally shy, then the chance anything will happen is practically zero. And what if you two would fit perfectly? Imagine the love that could follow?
Girls like this 'old-fashioned' rule, but that's mostly because they never risk being turned down. It's easier if the guy makes the first move, giving you the power to choose whether or not to go out with him. Why would HE have to make the first move, and not you? When he does it first, he'll also feel like a dumbass if you don't like him. While sticking to this old-fashioned stuff, they demand emancipation - so they want to advance in one area, but not in the other. Believe me, I have nothing against emancipation and would only encourage women.
Trust me, if you like a guy and he's not making the first move - even after you tried some flirting - ask him directly. Some guys won't notice you immediately, or don't pick up flirting well. The way a guy reacts says a lot about his personality and you'll soon enough find out if he's a jerk (in which case you now know you don't want him as a boyfriend anyway) or if he's nice (in which case you're out of luck if he says no).
I consider myself to be pretty shy, and hardly ever asked a girl out. I only did so when I was really interested because I was really sure I liked the girl. In other cases I was interested, but afraid it would not work out because I still had to get to know her more. That's why I never asked the girl out and I'll never know if something could have been - I do regret that. But had she asked me out first, I would have said yes.
Personally I think it's attractive when a girl takes the first move, it shows some confidence. I'm bad in picking up flirting, which is a reason I do always go out with a girl that makes the first move - even though I don't like her that way, but I do make that clear right away so the choice is up to her if she would still like to have a coffee with me or something.
In the end, I never chase a girl too much. It can be fun for a short period of time, but when she continues seeing it as a game that needs to played I stop - no matter how much I like her. A girl that plays with me isn't worth it, because I'm serious about it.