So i want to make sure i am understanding bribes correctly.

Lets say there is a personality stake in the free zone with a main arcana (highlighted value) of 6 Military and a Ducat Value of 7.

If i play an agent from my hand with a 3 Military Value, then on my next card set down a treasure worth 4 ducats, I win the personality stake through bribery correct?

Because my 3 Military which is the stakes main value adds to the 4 ducat from the treasure to equal the stakes ducat value of 7?

Correct, if you play a relic you add that relics ducat value to your agents' combined arcana value , i.e. the relevant trait (in this example military).

Then, if this is equal to er exceeds the stake's ducat value , you claim the stake. Otherwise, the relic is discarded.

And so the only time you use your agent's ducat value is if that is also the main arcana trait (which occurs on I believe only one personality).