Non Combat Resolution Games

A non combat resolution game is a kind of game that exists within the setting and is an accepted way to resolve disputes without involving direct violence.  Ideally it can be played between just two people and can be played somewhat diegetically  Here is a list of some of these games: (not all fulfill the diegetic requirement)

The Riddle Game

The advantage of the riddle game is the thematic strength and ability to be played entirely diegetically.  The disadvantage is not everyone is good at riddles and they can be pretty difficult to solve.  

Liar's Dice

This is the perfect Non Combat Resolution Game (except for on the thematic level, depending on the campaign.)

Strategy Games (Chess, Checkers, The Royal Game of Ur, Senet, Mancala, Ludus Latrunculorum)

These types of games are fun conceptually but take too long to be played like Liar's Dice.

Insult Games (Flyting, Senna, The Dozens, Rap Battle)

These types of games have the same advantages and disadvantages as the Riddle Game.

Athletic Games (Jousting, Archery, Foot Racing, Horse Racing, Chariot Racing)

These games are thematically strong depending on the campaign but don't meet the diegesis requirement.  They can be pretty quickly and easily resolved though.

Eugène Emmanuel Viollet-le-Duc