I can think of two very similar anime that match your description, both of which involve tiny human-like robot girls who are owned by humans and fight each other for sport.
The first to come out (2012) was Busou Shinki.
It featured four main shinki (its word for the robot girls) and their male owner, in a world in which shinki are very common and owned by almost everyone, with there being established tournaments for shinki fights, though most of the episodes aren't about fighting but instead slice-of-life plots. One of the main plot points is that of the four main shinki, one of them has only recently arrived and is more committed to professional fighting: she often has personality clashes with the owner's original three shinki who are more laid back and prefer only to use their mech-suits to help out (or not - lots of slapstick comedy) with the slice-of-life stuff.
The spiritual successor to Busou Shinki (coming out in 2017) was Frame Arms Girl.
In Frame Arms Girl only the main character owns robot girls as she's being sent prototypes to try out. A new frame arms girl arrives every few episodes and all of them enjoy fighting, so most of the episodes include fights between the various girls as they test each other out.