I vaguely recall in an AP or module that Paizo has already written these kinds of rules (for competition at a festival), including an archery competition specifically.
I can't remember what the rules were, or which AP/module it was in but hopefully somebody else has a better memory than I do.
I think you're thinking of Wardens of Wildwood. I know Mythspeakers has a festival as well.
I would suggest using the general purpose Victory Point system. It is a bit less detailed than a custom system for each carnival game, but that also means that you only need to explain the rules to the players once for all of the games.
Keep each carnival game short and designed to be handled by one character.
So for your examples, an Archery game would be 3 rounds of ranged attack rolls to gain points. If they gain 2 points, they progress to the semi-finals. Semi-finals could be a roll-off against an opponent (since this is just a carnival game after all) or another short VP system with a couple of rounds.
A drinking game could be a VP system using fortitude save bonus for checks. That could be set up as a diminishing point system, so they can keep rolling until they fail their save 3 times. They get rewards based on how many rounds they roll.
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I vaguely recall in an AP or module that Paizo has already written these kinds of rules (for competition at a festival), including an archery competition specifically.
I can't remember what the rules were, or which AP/module it was in but hopefully somebody else has a better memory than I do.
That would be the Rushlight Festival in the Kingmaker AP.
There was an archery contest, a log cutting contest, a public performance contest, and a joust, IIRC. And I think the joust was supposed to be a drunken (or at least tipsy) joust.
I think Spore War had some of this as well. It's definitely a good place for a Victory Point system. Archery could use attack rolls against various DCs (higher DCS for more distant/moving targets that are worth more points), and skill checks for other types of competitions.
For something like a grappling tournament you can do best two out of three type matches where the PC rolls and if they beat the DC two out of three rolls they advance to the next round. Have say four rounds and see how well they do.
Best way to handle it in my experience is to mot make the rolls overly complicated. Do a roll, narrate how it goes, and such. Let the players come up with creative ideas to try different skills or to get bonuses, and have fun with it.
I vaguely recall in an AP or module that Paizo has already written these kinds of rules (for competition at a festival), including an archery competition specifically.
I can't remember what the rules were, or which AP/module it was in but hopefully somebody else has a better memory than I do.
That would be the Rushlight Festival in the Kingmaker AP.
There was an archery contest, a log cutting contest, a public performance contest, and a joust, IIRC. And I think the joust was supposed to be a drunken (or at least tipsy) joust.
That's the one!