I am not sure what was meant by this question but I will try to answer it.
Mini horses are like children. Some like to just play by themselves, others are very social and play with everyone, some will play with people like they would play with other horses. We have even had a few that needed so much mental stimulus that we tied 5 lead ropes on the fence with 5 different knocks for him to untie. But horses that smart are few and far between; in 20 years we have only breed 3.
I will always remember my husband taking a mini out to paster for the very first time. He walked up to the mare and put a hand around her neck, like I told him to, and lead her out to pasture. Then when he let her go she spun around in 180 on him and ran back to the barn with her tail straight in the air. She thought that she was pretty funny leaving him all alone behind in the dust. Sometimes horses just like to be with you. There is no better toy or game than you and your company.
Toys: all these toys will work for mini horses too.
Miniature horses fight for a few reason. In their natural setting, miniature horses will bite and kick as a means of establishing dominance within their herd. Like many animals Mini horses have a hierarchy and horses will always challenge other horses of the same age. Some mares never make any friends and will always be cast out from the herd.
Stallions will lower their heads to mares and make an almost snarling face to make the mare stay within the heard when she strays. The mare (if with baby) will often protect the baby which is mostly unneeded, but the mare only knows that she doesn't know the stallion and will kick at him. In other cases a stallion will try to put the moves on a mare that is not ready to breed and there for she kicks and bits to let him know she isn't ready yet.
Example: http://www.youtube.com/watch?v=P3wT9ESxsTM&feature=em-comment_received&lc=dyRMftNUhsomgtpqYeuyMwtSzc1fn9dHJnZrx9D7xxc&lch=email
