It seems palace expansion disapproval reduction is broken.

Expansion disapproval is a global value. It's always applied on empire level, thus ED can't be negated in one particular city of your empire.



Once you have 2 or more cities, ED is generated by every single city you own:



ED = 10 *( Number of Cities - 1) * game difficulty modifier



EmpireExpansionDisapproval = ED generated by CityA + ED generated by City B .. + ED generated by City Z



EmpireExpansionDisapproval is applied to each city of your empire:



City Approval = (60 + Approval provided by city tiles, improvements, hero capacities - EmpireExpansionDisapproval) * Ownership





Let's say you have 3 cities: City A, City B and City C, and let's say each of the cities generates, for example, 10 ED.



EmpireExpansionDisapproval = 10 + 10 + 10 = 30. This means 30 is subtracted from City A Approval, City B Approval and City C Approval.



What happens if you have the Palace improvement in city A?



EmpireExpansionDisapproval = 0 + 10 + 10 = 20 (only 20 is subtracted from City A Approval, City B Approval and City C Approval).



The Vaulter portals in Endless Space work the same way.



update:

btw Empire Approval is calculated this way:



Empire Approval= (CityA Approval * CityA Population + CityB Approval * CityB Population ... + CityZ Approval * CityZ Population) / CityA Population + CityB Population ... + CityZ Population



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update:



Sorry, I thought the Palace is one of the Era V buildings, but it appears that Palace is a unique pre-built improvement, so I changed some of the formulae above accordingly.

All right thank you

So, just to be clear, each unique settlement causes an empire wide -10 approval penalty. The palace improvement's "0x Expansion Disapproval" aspect effectively lets you have that first city for free, from an expansion approval standpoint. I believe that the Ardent Mage faction hero skill tree also has an ability which allows an Ardent Mage governor the ability to confer a 0x expansion disapproval to a governed city.