The Pie Rule is Delicious
Many thanks to @BreadPuncher for pointing me towards this very cool idea from the classic era of board games. The Pie Rule is a method of balancing the first-player advantage present in many abstract games. It goes something like this:
If I'm the first player, I take my first turn. Then you, as the second player, have the choice of one of two actions:
You can let my turn stand, in which case the rest of the game proceeds according to normal turn order. OR...
You switch positions with me, so that whatever I did in that first turn is now what you did. That makes me the new "second" player, then I must take my new "first" turn.
The idea is that if one player gets to divide pie, the other player is the arbiter of whether the pie is divided fairly. If not, the arbiter can just take that slice instead. Knowing this, the cutter will try to keep her slice within the bounds as fair as she can.
So, can this actually be used more deeply as the central philosophy of a board game? I think I've seen this in a few games so far:
Dragon Valley asks you to draw a hand of beneficial cards, then divide them up into two separate offers for your opponent. Your opponent will accept one of the two offers, leaving the remainder for you to keep. So, you're compelled to try to keep the offers as balanced as possible, so you both at least get something you both want.
Tokaido features a linear track on which players move forward as many spaces as they like on their turn. The twist: Whoever is in the lead cannot move any further. So, sure, you can jump ahead to get first dibs on something up the road, but you're gonna be stuck there for a long time while everyone else calmly gathers all stuff you skipped over.
Coup lets players take whatever action they like in the game. However, another player may ask you to prove you have a card that allows you to take that action. If you don't, then you take a very severe penalty which eventually leads to your elimination. So, you don't want to be too greedy and you don't want to be dishonest... Until you do.
Settlers of Catan, Bohnanza, Pit and any other active trading game where players look to make mutually beneficial exchanges. I think this is sort of a tangential example though.
And actually, I think that's it. I'd be curious to see another game use this core mechanic more directly. Would it lead to a lot of kingmaking? In a multiplayer game, would each action be subject to committee approval? That might get a bit mad, but I'd still be curious!