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But for the battle with the outcome 3:0, when they deal 3 hits, do they get 1 improvement, or 3 improvements?

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from what I understand now, just the one! ;)

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I think so also, but wanted to be 100% sure.

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A 3-0 would only get 1 improvement. So long as one die is 1 or greater, you get 1 improvement.