I see
Anysense is running a mod because "we feel threatened by your large civilization" and "a first impression is a lasting one" are not normally viewable in unmodded civ.
The total relations with Peter is -24 diplo (Furious)
Peter signs Opens Borders at Cautious (-2 diplo or higher)
Peter must have good diplo with
Anysense's vassal or vassals to be willing to sign Open Borders.
How much positive diplo is it exactly?
Let's run a test.
The player is -100 diplo (Furious) with an AI.
The AI signs open borders at Cautious.
The player has a vassal who is +10 diplo (Friendly) with the AI.
Does Furious + Friendly = Cautious?
Or is it merely (-100 + 2) / 2 = -49 diplo (Furious)?
How does the Master + Vassal diplomacy combine when dealing with another AI?
Using World Builder, make the player -10 (Furious) and the vassal +10 (Friendly) with the target civ.
The civ with Cautious as the threshold will sign Open Borders.
Change the player to -100 diplo (Furious) and keep the vassal +10 (Friendly)
The civ will still sign Open Borders, so the combined diplo is presumably still Cautious or higher.
Now keep the player at -100 (Furious) and change the vassal to +9 (Pleased)
The civ who only signs open borders at Cautious or higher
will no longer sign open borders, so the combined diplo of Master + Vassal must be below Cautious.
Master + Vassal do not combine diplomacy numbers.
They combine thresholds; Furious + Friendly = Cautious.
Now for more complex tests.
We have an AI that signs Open Borders at Annoyed.
What if the Master + Vassal is Furious + Annoyed? Can the master sign open borders?
Nope.
What if the Master + Vassal + Vassal is Furious + Annoyed + Cautious?
Yes he signs Open Borders!
Furious + Annoyed + Cautious = Annoyed.
What if the Master + Vassal + Vassal is Furious + Furious + Cautious?
Nope.
What if the Master + Vassal + Vassal is Furious + Furious + Pleased?
Yes he signs Open Borders!
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Basically, the best I can figure out is you start with the diplomacy threshold indicated by Civ Illustrated.
For the last example, an AI signs Open Borders at Cautious.
Every threshold below Cautious subtract 1 and every threshold above Cautious add 1.
If the total score of Master + Vassal + Vassal + Vassal etc. is equal to 0 or above, then the AI will sign open borders.
If it is less than 0, the AI won't.
AI signs Open Borders at Cautious.
AI is Friendly with the player.
AI is Annoyed with vassal #1 of the player.
AI is Pleased with vassal #2 of the player.
AI is Furious with vassal #3 of the player.
AI is Annoyed with vassal #4 of the player.
That would be +2-1+1-2-1 = -1, so the AI won't sign Open Borders with the player.