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The objectives are very simple: Kill all members of the opposing team(s), including NPCs if there are any.
In some Annihilation matches, teams have a Resurrection Shrine tended by a Priest. The Priest will resurrect all dead party members every 2 minutes. To win, the opposing team must be dead and their Priest killed as well. The Priest cannot be resurrected.
In this match type, martial classes, such as the Warrior or Ranger, often charge to kill the Priest before going after the opposing team. This is to make sure that the Priest doesn't resurrect the killed enemies. It is much easier to win if the opposing team doesn't have a living Priest anymore.
Some Annihilation matches feature Obelisks that can be used as a tactical advantage to beat the opponent team even faster.
In Obelisk Annihilation, teams often use tactics similar to flag running at the start of the match to decide who controls the obelisk, but teams rarely ever attempt to re-capture if they lose control. This is because the obelisks can often be avoided.
Of course, there is also 'normal' Annihilation. Here, you have to kill all members of the opposing team, so they are all dead together. When there is still one alive, you don't win. This is the most common PvP-match type in the Random Arenas and Codex Arena on the Battle Isles.
Mostly, the whole team goes for the monk, so he can't heal. As soon as this one is dead, people go looking for other monks or kill the weak classes who give support (Elementalist, Mesmer,...) first. Linebacking is a common tactic when caster allies are under heavy pressure.