Allow me to make a few more suggestions.
- Variation in tournament events. Meaning team melee on foot and on horseback, duels, jousting, horse racing, archery, javelin throwing, etc.
- Lower, medium and higher stakes tournaments.
Lower would be for the common man looking to make a name for himself as a fighter or the one who has nothing to lose and just wants money, these would include archery, team melee and finally duels (one winner for each, meaning you could still be a winner of archery even if you sucked in team melee. Duels and other melee fights would give much more renown however, since that would be what the people would enjoy watching the most). These tournaments would be the most frequent, giving less renown and rewards, lasting only a day, and even bribery and cheating would be common (an opponent using a longer lance than yours, or using a real sword instead of a blunt one and what have you), you could try to call them out but that would make you lose relations with the city's thugs, which COULD lead to getting attacked randomly on the street aaaand I'm getting way too side-tracked.
Medium, for renowned fighters, knights and soldiers, minor nobles could also participate but all would require a certain amount of renown. These would last 2 or 3 days, and include more variation and games and celebration at the end of each tournament day. Bribery and cheating would be uncommon. Jousting and other advanced events would be present. One could partake in said celebrations and feasts in the city, talk and eat and drink with the people and if you won the tournament they could even compliment you, doing this would increase one's relation and popularity with the city overall, not necessarily with notables. Higher rewards and other benefits. Second most frequent tournaments after the lower ones.
Higher, of course, for the most renowned fighters of the realm and lords. These would take part in a much larger and open field, with nobles who aren't participating in the tournament watching from the higher seats where the nobility or the rich sit. Since knights and tournaments and courtly love, why not let me impress the lady I'm trying to marry that way? Bribery and such would almost be non-existent, except for when dishonourable and rich lords bribed the judges. Events would include: Team melee on foot and on horseback, jousting, duels (of course let the player choose his TYPE of weapon, but the weapons themselves will be given to the fighter by the tournament organizers) and archery. These tournaments would last from 4-6 days or even a week, with faction-wide celebrations and feasts and a massive renown gain (when compared with lower stakes at least) at the end of the tournament, when a champion is decided. Speaking of the champion, he would be rewarded with at least 10k gold, an ornamented piece of armour or arms, high renown gain, increase of relation with the people, and I would say rewarding land or property, say, a village but more insignificant villages would have to be added into the game for that to make any sense. These kinds of tournaments would take one or at most three times a year, and the news of one taking placing would be spread all over the faction. The participants would apply some 20 days before, and start training in the city in which the tournament would take place, most commonly the capital.
That's already a lot of suggestions there. But seeing as they didn't even bother to do away with those horrendous faces of the crowd or the extremely basic tournament system, I'm not sure how realistic any expectation of these being considered at all is.