Something I want to put into Mercenary, Liberator, Tyrant is sporting competition, given how important it was in Greek culture. Affluent men trained in the gymnasium, it was one of the three pillars of Greekness, and the best competed in the formal Panhellenic Games. There were four big ones, arranged so that there were games every year, though the Olympic Games were the most prestigious. That's without informal games that might being called at any large gathering of Greeks (such as when an army or fleet was mustered).
What I want is a similar level of structure to that which you have with combat, not just a single roll to see who won, but something that gives you some options and a little tactical play.
Some of the events don't need anything additional - there's already rules for unarmed combat and wrestling. But the rest do, to my mind.
First up, racing. Bear in mind in all these events, there are no rules against contact with the other runners, though it's considered bad form. No doubt the Running Proficiency gives significant advantages here, but I'd want to involve all three physical ability scores in some sort of assessment of athletic potential.
The stadion was a 200m-ish sprint along a straight line. I can see four parts to this; the start, as in reacting to the starter's order; the push off to get going; the main run; and the finish.
The diaulos was a 400m-ish run, often in two halves, out to a post, around and back. So you've got the same two-part start, the first straight, the turn around the post, the second straight and the finish.
The dolichos was a longer-distance event, covering about 3 miles. It started and ended in the stadium, but would wind it's way around the whole site and take in significant landmarks like temples. This is much more of a stamina-based event than the other two.
The last race was the hoplitodromos, a single or double diaulos in full armour, including shield. Lots of hazards in this one, even aside from people using their shields to knock people over or otherwise hinder them, falling over or dropping your shield and becoming an obstacle to anyone behind you was a constant threat.
Field events were the long jump (that could be resolved in a single roll to establish how good your jump was), discus throw (again a throw to say how far you threw) and javelin (same again). Though there might be a tactical element with how close to the line you're willing to go to extend your jump for the first.
Lastly, chariot racing. I'd call this an extension of whatever rules we came up with for racing, but with added control-of-horses and interaction with other drivers peril. There were two races, two-horse and four-horse. Both took in twelve laps around a track where they went around a post at either end.