Here in Birmingham, I think there are at least three bus companies and two of them appear to cover the same routes and have the same numbers with the same stops - one is cheaper than the other, but I don't know if it runs less often or not, and I don't know anything about the third one.
I use Travel West Midlands (which is I think owned by National Express now). It has two stage fares - the cheaper one is not worth it - two stops for £1. Full fare is £1.70, or £3.30 for a return ticket that lets you ride the buses all over the area all day. Since I work in the city centre and ride the bus five days a week, I get the monthly, paid-by-direct-debit pass that costs £38 and then I can ride on the weekends too if I need to...
They have one sort of odd all day pass - it allows up to 5 people to ride all buses all day for £6.50. I know this because my brother, his wife and two kids visited this summer, and throw my husband into the mix and we could all make use of the bus (which we did a few times) for that price and, of course, my bus pass...