Just confirmed a few more things re: Economic Warfare in my current game, and thought I would share...
18 hours in I took out Beryl's main starter island for a little over 8 mil. I got 17 influence back, after spending 50 total influence in shares. Now, there is an influence cost to settling a new island, so that is in there, but I doubt it comes out to 33 influence.
In addition to that there is the rather substantial income hit from losing five shares worth of income from a two-star AI's main island.
However, it dropped... over 250 jewelry, over 250 heavy weapons, over 200 dynamite, over 200 eyeglasses, and over 250 watches. My Engineers and Investors are swimming in gravy now. 
If one takes over an island this way... have cargo ships at the trading post waiting to pick that stuff up. It is flotsam and de-spawns rather quickly.
I also found out that having war declared on me is NOT an automatic event. I had trade rights, non-aggression pact, and over 80 reputation with Beryl at the time. Still took a huge reputation hit, and she quickly canceled the trade rights, but not the non-aggression pact(yet). I plan on bribing, flattering and questing my way back into her good graces.
Another tip is to have a squadron of warships next to that AI's LEAST defended island in the Old World when you buy the main island out. If they declare war(like Carl did when I took his island), that squadron eats another island on the cheap... and causes an easy ceasefire.
Economic Warfare cons - Expensive in both invested coins, income and influence lost forever. You should be greedy... not needy to implement this strategy.
Pros - Cripples that AI forever, both economically and militarily. Gives you an Old World starter island with all that land and some oil. This means more for players of sandbox mode and without DLC-land like Trelawny. Real estate rules... location, location, location.
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