Off one’s own bat = on one’s own initiative;  without being asked to do by anyone else.

• Whatever she did she did off her own bat.
• No one asked him to help; he did it off his own bat.
• Rozanne didn’t wait for her parents to tell her to take out the garbage.  She did it off her own bat.

Play ball = co-operate in order to achieve something ( usually used in negative context)

• Because he refused to play ball with these criminals, he was shot and seriously injured.
• If you play ball, we will be reducing your sentence from 8 years to just two.
• Just play ball and do what they say, and nobody will hurt you.

On a level playing ground/field =  in a fair situation where everyone has an equal chance of succeeding.

• All applicants have to agree to the same conditions for the interview in order to ensure a level playing ground.
• They are still not providing a level playing ground in terms of opportunities for women.
• The new legislation is intended to create a level playing field for students of all backgrounds.