From The TimeSplitters Wiki

Bag Tag is an Arcade mode that features in all three games in the series. This mode is available to play on every map and features just one bag, usually in a central position on the map.

Aim

The aim of Bag Tag is to be the person who holds onto the bag for the longest amount of time. The person holding the bag immediately becomes a target for all other players and being killed results in the bag being dropped. Whoever gets to the bag first becomes the carrier, as well as the new target for everyone else.

The game is set a time limit and whichever player has the longest time holding the bag after the time is up, wins the game. In the first TimeSplitters game, you could set teams so that the carrier accumulates time for their team instead of just themself.

Advice

In order to get the most out of this game, try playing on a small or medium map so that the carrier gets caught more and there is a much closer match. For an added challenge, you can set the game so that the bag carrier is unable to shoot.