What do you mean here?
Basically a system for having battles of wits when you either don't want to just RP it or want the result to be rooted in the mechanics. E.g. making opposed War rolls for playing chess Gateway, perhaps adding some social combat with some Manipulation rolls to set traps or Presence rolls to psych him out. Or trying to get successes on a series of Lore checks equal to someone's Resolve to convince him you know more than him, with his successes subtracting from yours or with him trying to get successes equal to your Resolve first. Or making Bureaucracy rolls with a Wealth-related bonus to try to outcompete a rival's company and force them into bankruptcy. Or Bureaucracy rolls to jam up a Realm military base so that they're too busy arguing over payrolls to send anyone out to find the Anathema. Stuff like that.
I'm not sure it needs a full system like physical and social combat do, but I'd really

That's something I'm uncertain about, because having particular rules for small sub-systems makes the game messy. to the greatest possible extent, the rules should seek to be as encompassing and simple as possible.
I'm thinking something similar to the Chase rules in NWoD, where both parties roll [Athletics, Ride, Sail, whatever] and whoever gets to a certain threshold wins.
E.g. in a foot race you might have to get your opponent's Speed in successes before he gets your Speed in successes. Or both of you try to get to a certain number of successes based on the length of the race, and your Speed gives you a bonus.
Or for something like a basketball game, both of you try to get to a certain number of successes first, those successes representing points; in a team game, each team tries to get to a certain total, and some team members can use their actions to remove successes from an opponent (like blocking shots, etc) instead of trying to score more themselves.
Environmental obstacles can happen, Charms may be used, other skills may become necessary [e.g., a plesiosaur pops up in the middle of a boat race, and you have to fight it off while trying to maintain your lead], etc.
Nothing very complex or specific, just a system that lets you resolve contests non-arbitrarily and throw complications into them if you want.