Intuitition behind Average Absolute Odds fairness While reading a FairML paper, I saw that they use Average Absolute Odds, whose math definition will be:
$\text{FPR} = \frac{\text{FP}}{\text{FP + TN}} $
$\text{Average Absolute Odds} = |TPR_j - FPR_j| + |TPR_i - FPR_i|$
Where $j$ and $i$ are the protected group and the reference group.
What will be the social intuition behind this metricc? Why is this metric relevant for FairML?

Other references label this the Average Absolute Odds Difference [1,2] and use a different equation than the one OP suggested.
$$AAOD = \frac{1}{2}\left( |FPR_j - FPR_i| + |TPR_j - TPR_i| \right)$$
The social intuition is that an $AAOD = 0$ means the algorithm is fair because it results in the same FPR and TPR for the protected group as the reference group.  If the algorithm causes a difference in either the FPR or TPR, the $AAOD$ will increase from zero.
Also note that the difference in false positive rates has the same worth as the difference in true positive rates.  A deviation in each contributes the same to the AAOD.  This may be context specific because a false positive may be more socially costly than a true positive.
[1] https://www.mathworks.com/help/risk/explore-fairness-metrics-for-credit-scoring-model.html
[2] https://dataplatform.cloud.ibm.com/docs/content/wsj/model/wos-fairness-average-absolute-odds-difference.html
Update
Another way to look at the AAOD measure is as a single-metric approach to the Equalized Odds concept of fairness [3].  Equalized odds seeks to check the TPR and FPR separately against the reference group.  $AAOD$ is the average of those two concepts.
From the comments, for an example social context, one reference says "A negative value in EOF is due to the worse ability of a Machine Learning model to find actual recidivists for the protected group in comparison with the reference group".  EOF is Equal Opportunity Fairness as measured by the difference in the TPR between protected and reference group.  In this social case, for the $AAOD$ we are looking for differences in the rate of the algorithm to predict a recidivist given they do re-offend ($TPR_j - TPR_i$) and the difference in the rate of the algorithm to predict a recidivist given they do not re-offend ($FPR_j - FPR_i$).  Difference of both types, in either direction, are socially costly, so their absolute values are averaged together.
[3] https://arxiv.org/pdf/2001.09784.pdf
