Multiplayer: Last Man Standing
Includes a multiplayer mode where the winner is the player who is the last one alive or the team with the only surviving players. There is no respawning during a match, or it is severely limited per player or per team.
Alternate names: Elimination, Last Team Standing
Name variations: LTS, LMS, Last One Standing, Last Person Standing, Battle Royale
The first video game about Multiplayer: Last Man Standing was released in 1995.
In shooters this is quite straightforward with each player controlling single character who after death loses that round.
In strategy games, this may mean that the all player's units and facilities need to be destroyed, though this may actually only require destroying their capacity to produce more units and buildings or simply killing the army commander.
NOT to be confused with some variations of deathmatch which count deaths to determine victory. Though some games may offer a limited number of lives for each player so the first loss is not the final.