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The Founder

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The aforementioned empire has been sending survey ships into my territory. I largely ignored them because I have been busy with building up my colonies and stations. However, just now they sent in a fleet. I sent my fleet to the same place, and they left.
The admiral AI will attack anything marked as Hostile it can defeat. So they sometimes go around, attacking Spaceborne aliens. Might also have been their or your Pirate event.

At this point, I can't just do nothing. I already decided to close my borders. Should I also declare them my rival? I don't see a reason why not at this point because dimplomacy will obviously get me nowhere, but I just want to see what you all think.

I now am also recently in contact with a second empire that is much more compatible with mine. I immediately got us both into a NAP and have a trade deal going with them. If I were to go to war, how likely am I to get them to fight with me?
Rivalry will at first give you influence. It can be a major source. They will rival you back.
It also has ramifications for Diplomay: "Mutual Rival" and "Rival to Allies" are relevant Diplomacy modifiers.

How likely to get them into War? Just look. Declaring war does nothing until you confirm the Wargoals and you can always invite them.
They will need contact with them and see if they dislike them as well. You can not control either of those.
If you told us the Ethics and AI Personality of the 2rd empire would could help you more.

Thus far we have:
You are stock UNS

They are Authoritarian/Xenophobe/Materialist wich I asume would give them a agressive AI personality. Propably Hegemonic Imperialists.
 

CocoCincinnati

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You can close borders to them if you don't want their ships running around your empire. Their fleets won't actually do anything while inside your borders unless you are at war so you don't have to go intercept them while at peace. Closing borders will worsen relations but it sounds like those are already bad. As many have already mentioned, declaring them a rival will significantly worsen relations and close your borders to them to boot while giving you some influence as well. If they're authoritarian xenophobes and you're egalitarian xenophiles and you have border friction on top of that, I don't see any chance for friendship in your future, the relationship is likely to continue to deteriorate. Either ignore them or work to get a superior fleet and defend yourself by attacking them first. ;)
 

General Retreat

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One thing that can be really useful for judging galactic diplomacy at a glance is map modes. At the very bottom right of the screen you will find:
  • 'diplomatic' - shows any arrangements the selected empire has, results are colour coded. Very useful for scoping out federations or where other parts of a defence pact are located in relation to your target.
  • 'Attitude' - really not very useful. Situationally useful for working out who will be up for a trade deal.
  • 'Opinion' - extremely useful. Empires turn shades of green to red to show how friendly/hostile they are towards the target. Great for seeing how potential allies feel about a mutual enemy. Can really assist with the diplomatic game.
 

TheBromgrev

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(BTW they are Xenophile and I believe militaristic, though I'm not sure because I'm off the computer atm)

They wouldn't happen to the Blorg, would they? Just asking because that prescripted nation has a unique AI behavior.
 

The Founder

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They wouldn't happen to the Blorg, would they? Just asking because that prescripted nation has a unique AI behavior.
It is a AI personality like every other.
Granted the requirements for it a so specific, a empire has to be designed to get it.