From The TimeSplitters Wiki

In this mode, the characters must kill each other to earn points. 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 playable in all levels except some Sample Maps in the original TimeSplitters for the Playstation 2 and if it is not selected or not created in the MapMaker mode.