Multiplayer: Free for All
Includes any multiplayer mode which lacks teaming of any sort, where each player is against everyone else.
Alternate names: Teamless
Name variations: Free-for-all, FFA
Note that FFA is sometimes incorrectly used for any mode that does not have a specific goal (seemingly as a synonym for deathmatch or last man standing modes) rather than a mode where all players are against everyone else.
The first Multiplayer: Free for All video game was released in 1980.