How should multiclass classifier performance be measured when one type of error is preferred over another? Sorry if this question has been asked before--I am having trouble searching this topic since I'm not sure of my wording.

Say you have a classification problem where there are more than two labels which are discrete but roughly correspond to an increase in some quality--call these labels A, B, and C. Also say in this problem it would be preferrable to over-estimate that quality, rather than to underestimate. Is there a type of metric that captures this skew and penalizes a predicted A on an actual B more than it penalizes a predicted C on an actual B? Or is this preference better handled in a different part of data science methodology?
Define a scoring table like this (You will need to tweak this table to satisfy your particular use case. I am only using it as an example).

Pred   | True Label
Label  | A    B   C  
-------+--------------
A      |  0  -1  -2
B      | -3   0  -1
C      | -4  -3   0


Notice this scoring table has the property of favours over-estimate than under-estimates. 

Multiply (element-wise) the scoring table with the number of predictions fall into each of the 9 possible scenarios then sum the scores together will give you a metric with the desired properties.