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Question: How do the filled circles increase the odds of the favour being granted if it's based on a 1d6 roll?

(BTW I'd suggest "agree" instead of "obey".)

it increases by +1 for each fulled dot. So the odds of rolling the first is 1/6, after the first success is 2/6, then 3/6 and so on.

It gets easier to achieve the more you succeed!

So are you saying you first need to roll a 1, then, once you do, you need to roll a 1 or 2? Continuing, once you roll a 1 or 2, you need to roll a 1 or a 2 or a 3? (And so on?)

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yup! It’s meant to be like if you were training a dog. The better your relationship with commands, the more  trust you build up.

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Ok, cool. Got it. I understood the intention, just not how the mechanic worked with a die.