If you don't finish in first place after a "free-for-all" game (Hearts, Literati, Mahjong, Word Racer, Chinese Checkers, and Sheepshead), your rung changes as though you lost the game to the winner. If you finish in first place, your rung changes as though you defeated the loser with the highest rung.
Rung computation for Word Racer, Go Fish, and Dots is similar to the free-for-all games. The only difference is that your rung is calculated against all players, not just the winner. For example, if you finish 4th out of 8, your rung will reflect losses against player #1, #2, and #3 and will reflect wins against player #5, #6, #7, and #8.