How to obtain conditional use accuracy equality with communities with different real positive rates?

The short version is that I would like to know what the confusion matrices (numbers of true positives, false positives, true negatives, and false negatives) should be to achieve conditional use accuracy equality among two communities, one with 40% real positives and one with 60% real positives.

Here is the long version...

I am trying to understand the difference fairness metrics described in Understanding Fairness. It contains an interactive pair of pie charts representing two communities' confusion matrices. Blue portions represent real positives (RP); striped portions represent predicted positives. To the right of the charts are different fairness metrics and how well the proportions in the pie charts satisfy them.

Here is the original configuration:

The small red and blue circles are handles for adjusting the sizes of regions.

As shown in green, these fairness criteria are achieved:

• Group fairness
• Equalized Odds
• Overall Accuracy Equality

I have been unable to come up with adjustments that satisfy Conditional Use Accuracy Equality. As described in the document, the two communities should have the same:

• positive predictive value (PPV) or precision, i.e. TP / Predicted Positive, and
• negative predictive value (NPV), i.e. TN / Predicted Negatives

Can Conditional Use Accuracy Equality be obtained in this scenario?

• Can you please include some more background in the post? We prefer posts to be self-contained, and here is difficult to understand without reading the length linked document! But some ideas: Statisticians are heavily criticising accuracy-based measures, because they might lead to misleading conclusoins in classification, and hat will probably carry over to this setting. In its place we prefer proper scoring rules, see for instance stats.stackexchange.com/questions/312780/…. You could benefit from trying to .... Commented Nov 29, 2021 at 17:40
• ... translate those ideas to your setting! The basic idea is that if you change to a fairer setting, it should be impossible to get a worse score. An exercise for you is to create examples where the accuracy-based measures do not comply with that requirement! Commented Nov 29, 2021 at 17:41
• @kjetilbhalvorsen Thanks so much for the feedback! I added the self-contained version at the beginning of the post. Commented Nov 30, 2021 at 0:47
• @kjetilbhalvorsen Thanks too for letting me know about proper scoring rules, although the answer to the current question is still important to me (and my students). Commented Nov 30, 2021 at 1:14

$$\begin{array} {|c|c|}\hline {\rm Cats} & {\rm predicted \ positive} & {\rm predicted \ negative} \\ \hline {\rm real \ positive} & 7 & 33 \\ \hline {\rm real \ negative} & 3 & 57 \\ \hline \end{array}$$
$$\begin{array} {|c|c|}\hline {\rm Dogs} & {\rm predicted \ positive} & {\rm predicted \ negative} \\ \hline {\rm real \ positive} & 49 & 11 \\ \hline {\rm real \ negative} & 21 & 19 \\ \hline \end{array}$$