Interesting non-zero-sum games
There's an old game called Plank that might fit your description. When you play with three players, the player to the left of the winner receives a penalty of two points, and the third player receives a penalty of one point. When the players have played some multiple of three games, the player with the fewest points wins the match. This way, when a player has to choose between stopping one player or the other, he will always stop the player to his right.

I first ran across Plank in Sid Sackson's A Gamut of Games book. He added the rule for  scoring three-player games to solve the king maker problem. Plank was described in The Young Folks' Cyclopaedia of Games and Sports in 1890 (complete rules at the link). I've since seen a commercial version called Hack Attack. I really like the game as a light filler, and I made my own set out of Lego. For me it has a similar feel to Quarto.