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Islands that are occupied by other players or computer players can be taken over by other players in two different ways.
A player has to first declare war on the owner of the island or equip and use Letter of Marquee or Black Flag items on the ships. Then the player needs to use military ships to attack the main harbor and coastal buildings of the island. After the morale of the island reaches a critical point the player is prompted with 2 options. Either gain all the shares for that island or conquer the island. If the player chooses to conquer the island the player will take over the island and claim it, with everything previously built on the island being removed. However, the harbor will keep its upgraded status.
If a player buys all the shares of an island, he/she will have the possibility to conquer the island by buying it. After claiming all 5 shares the player can select the middle of the share diagram. This is very costly as it requires 10 influence per share (in comparison to a big island which usually costs 30-40 influence to settle) and there is a 10 minutes timeout between the buying actions
Small trading outposts that are by default can't be taken over in any way.
If 2 or more players are attacking the same island together there will be no option to conquer the island. The only option is to accept shares for the island distributed based on how much each player contributed to the attack.
The 4 plateaus in The Arctic can only be taken over via shares.