Essentially, a way of constructing a formal challenge, with the victor gaining a particular benefit. Challenges could take place in Adventure mode, Fort mode, or Legends. They would basically be a way of getting something from someone without killing them or beating them into submission.
Creating a challenge would involve opening up a screen similar to a trading screen, with the player defining the nature, time and place of the competition, and request a reward they would get for winning and a price they would pay if they lost. The challenged party would evaluate the challenge based on their own perceived chance of victory, their personality or values (people with high confidence or pride would be less likely to back down from a challenge), as well as what they would gain or lose for winning or losing, and similar to trading, could make counter-offers (modifying the terms of the challenge to be more favorable to them, or asking for a prize they would prefer) or simply refuse.
Issuing blatantly unfair or frivolous challenges could impact the player's reputation.
Competitions could be related to any skill. Combat duels would be one form of competition, naturally, but you could have music or dance competitions, writing, crafting, hunting, and more. Depending on the type of challenge, more units could be involved as well, or the challenger (or the challenged) could pick another unit(s) as proxies. For example, one might challenge a warlord for a non-lethal battle between their two strongest party members and two units of the warlord's own choosing.
Competitions could be fought over a particular artifact, a government position, the loser's fealty, exile or death. Or challenges could be offered simply for fun.
If a unit cheated, or if they lost and failed to uphold their part of the agreement, they would gain a reputation as a cheater and this would make it harder for them to create or join competitions. Depending on the details, they could also be attacked by units in the offended civilization. (For example, if they lost a duel to the death and managed to escape, they would be branded as an outlaw and could be killed by any unit in the civilization without repercussions, or if they were exiled and tried to return other civ members might spit at them or refuse them service.)
One could also create an "open tournament" for a particular reward, which would spread as a rumor, and units that considered themselves skilled could enter the tournament. This would be common in worldgen but the player could also issue one - for instance, if one wanted to find a strong companion, they could offer a combat tournament for a cash reward, and then try to recruit the winner into their party.
Hunting competitions could also be "races" to kill a particular creature. Integrating this system into Adventure Mode would also allow leaders to issue challenges to kill a target for a reward, instead of just asking the player to do it for "glory".