I played a board game once and got one point of relation with some minor doofus that had been made governor. It seemed like a lot of fiddling around for just one point of relation. This was the only time that I bothered to try one of the board games and I an not at all interested in repeating the experience.
On a related note, if a lord is just -1 relation with you, he or she won't consent to playing a game. You can't get any missions from them either. One hint of negativity and you are mostly boned from improving the relationship. You could vote for them in a council vote, but that just makes you extra enemies.
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I played one time as well. Not too enthralling.
