My granddaughter plays pretty much every day after school, but not a formal "playdate." I just don't really do the "playdate" thing. She can go outside and play with the neighborhood children - they are all out after homework time. She does on occasion go to friend's house after school - I guess that's a play date but we don't call it that. She's going to a friend's house after school. I only allow that every once in a while, however,. because I like her to get home and get on her homework.
She is an only child in my household, although she has one sister and three brothers (they do not live with me). She is good about entertaining herself; I am not her social director. If she says she's bored, we make suggestions, but it's up to her to find what she wants to do. Some examples of things she does are computer time, play in her room with her dollhouse, play with her DS, art projects, roller blade, bike ride, or just hang out outside.