midwesterner, thanks for the correction!
Yoma 22-23: first episode is when one kills another in anger (and father of the victim is trying to keep the knife tahor..). This did not lead to the prohibition as this was clearly an issue of anger after competition. Second episode that I referred to when one pushed the other was clearly during the competition and lead to banning the practice. So, it seems that Beit Din did not mind people getting competitive, just not getting physical. Btw, mussar recommends to parents who can’t hold themselves from murder (shouting at children) should use positive rewards and competitions between children. Try saying “quietly” to your spouse that the other kid two years ago did that homework easily.