Roll d20, add attack bonus. Whomever gets higher is winning. Ties = no advantage either way. Crit fail = make a fool of yourself. Crit success = accidentally escalate an extra category.
Fights go from bruised -> bleeding -> screaming -> broken -> maimed/dead
If you win the round, describe how you make your opponent bruised/bleed/scream/etc and you get +3 on your next roll. If you lose, choose whether to tap out or escalate to the next category.
Example - The mighty Blax Jax (ftr 2) is facing off against Jadi Amar (ftr 1). Blax Jax rolls d20+1d4+1 (this is DCC, fighters get variable attack bonuses, plus his strength bonus) and Jadi Amar rolls d20+1d3. First round, Jadi gets 13 and Blax Jax gets 11. Blax Jax gets punched in the face, bruising him. He obviously isn't gonna take that lying down, so he escalates to bleeding. Next round Jadi Amar rolls 9 with his attack bonus, and Blax Jax gets an 11. However since Jadi was already dominating the fight at that point, he gets a +3 bonus which wins out again, busting his opponent's lip and spattering blood on the pit floor. Blax Jax is frustrated and escalates again, this time rolling a natural 20 and accidentally going straight from bleeding to broken, snapping Jadi's arm. Jadi concedes at this point, to nobody's surprise.
Sure, we could have a drawn out fist-fight using d2 damage for each attack and chipping away at HP, but this is wayyyy faster, and far more entertaining.
Who's your opponent?
Fighter level = PC level +-1 (adjust for non-fighter PCs)
Tap out on round: 1d3+1 (bruised = 1, bleeding = 2, etc)
As an aside - after trying to find an illustration of folks duking it out in a fighting pit, I am wondering if fighting pits might just be an anachronistic recasting of animal pits, such as this one used for rat baiting.