The Pitch
In standard Magic games, players take the role of planeswalkers who encounter one another and duel. The assumptions are that each competitor arrives fresh to the fight, fights for nothing, and fights to the death (0 life). That's fine for a tournament environment that has to be equivocal, but I'd rather have someone show up bloody to a fight, victory mean more than just winning, and close matches carry more weight than a one-sided beating.
Overview
Linked Campaigns use two teams of three decks (whether that's three decks with three players or three decks with one player on each team is up to you). Each Round, both teams put up a deck that will be playing while its other two decks Recover from their previous Beatings. The players for each playing deck prepare for the game, but lose life or mill cards according to their previous Beatings before they mulligan. At the end of the duel, each player totals up the beatings for their deck and the winner takes a Location card for their team's Command Zone. Location cards have points (some have abilities) and the first team to 20+ points wins.