From The TimeSplitters Wiki

Elimination is similar to Deathmatch, in that there are no teams and all players fighting against each other, however the aim of the game differs slightly. This mode is only available in TimeSplitters 2 and Timesplitters: Future Perfect.

Aim

The aim in an Elimination match is to be the last player left alive at the end, since each player does not have an unlimited amount of respans. Once a player has died the set number of times, you are unable to respawn and participate further in the match. The amount of kills each player has is irrelevant, since the player with the most kills could still lose all their lives before the end of the match. If a time limit is set and the match ends before there is just one player left, the player(s) with the most lives left is/are declared the winner(s).