Whose responsibility is it?

I'm sort of old-fashioned when it comes to this, and I like it much better when the guy initiates the conversations. It makes me feel far more comfortable, by showing I'm not putting myself out there too much and making a fool of myself. But in general, the guy should initiate most of the conversation, but the girl should do it every so often as well.

I'm sorry, but I don't know the answer and I have never heard of the terms 'relationship' or 'responsibility'.

Perhaps you could search on Bing or another search engine, or look up this topic on Wikipedia.