From The TimeSplitters Wiki

In a Deathmatch, the premise is simple. If you kill one person, you acquire one point. When the battle is over, the person with the most points wins the match. This mode has appeared in all the TimeSplitters games, and is considered by many the game's primary mode. Deathmatch is always playable in all levels, with the notable exception of some Sample Maps in the original TimeSplitters for the Playstation 2 and if the mode is not selected or not created to be compatible with a map in Mapmaker mode.