During a Bout

One-on-one. No weapons. No armor. No magic.
You lose if you leave the ring, submit, or fall to 0 HP.
Unarmed strikes deal half your largest hit die as a damage die. (Barbarians deal d6; wizards, d3).
Grappling is fine.
Roll initiative each round.
On the first turn, and the turn after neither fighter takes damage in a turn, each can do one of these actions. (Roll initiative normally during these turns.)

Call to the crowd - Any Charisma check. DC 15 - the bout number. Succeed, get d6 temporary HP. (Both you and your opponent can succeed. You "know" how many temp HP the other got.)
Circle - Dexterity (Acrobatics), DC 10 + your opponent's HD. If you succeed, advantage on your next attack, or disadvantage on their next attack. If both succeed, cancel out.
Rally - Spend a Hit Die (plus Con modifier).
Rush - Attack anyway, and your opponent makes checks for these other actions with disadvantage. (Generally considered unsporting.)
Size up - Wisdom (Insight), DC 10 + your opponent's HD. Succeed, get a d4 you can use add to or subtract from a roll (yours or theirs) after it is made (d20 or damage).

Between Bouts

Entry stake is 50 GP. Basic brackets. (There are lots of bracket-making tools online.)
Your pot doubles for every bout you win, and you lose it all if you lose. You can quit before you know your opponent to keep your current pot.
There's opportunity for a short rest.
If anyone forgoes a short rest to get drunk, in the next bout they have disadvantage on initiative checks, but start with d6 temporary HP.
The DM rolls any NPC's HD at the start of each bout. Each that comes up lower than the bout number is expended.

Some opponents for the PCs (current players, spoilers if you click that):