In the Assassin format, players are randomly assigned "targets" to defeat, such as via shuffle of basic land cards. Players score points for delivering the finishing blow to their assigned target as well as being the last survivor. Defeating another player grants you their "contract", and thus a new target to attack. Formats differ on what drawing your own contract means - whether it be a reshuffle (if at start of the game) or a "wild card" that allows you to gain points for any player defeated.
Horde Magic is a cooperative multiplayer variant of Magic. The Allied players face off against the Horde deck, which is automatically controlled. The Horde automatically casts a semi-random number of creatures and effects from it every turn, then attacks with everything possible. The default flavor is the Horde are mindless attacking zombies. The Horde has no life total, but damage to it reduces its library of cards. If the players can survive until the Horde runs out of cards, they win.
One variant is "Star" (also called "Pentagram", "Five-Point", "Rainbow", "Five-Player Star", and "Color War") and involves exactly five players, each playing one of the colors of Magic[->0] (Note: Artifacts are allowed, but no off color cards (including multi-color or hybrids) may be used). Players sit in color wheel order (White, Blue, Black, Red, Green) and try to defeat the two diametrically opposed "enemy" players. A player wins when both enemy colors are