Is Median Absolute Percentage Error useless? I'm working on a project focused on pricing houses.
Looking online I see a lot of works and companies providing the performances of their model using the median instead of the mean (see for example Zillow).
And to me this makes total sense, since its a statistic less influenced by outliers (in my scenario big and particular houses).
Also since I would like to have a greater interpretability of the performance of my model I used the Absolute Percentage Error, instead of the squared error.
In the end what I ended up using is the Median Absolute Percentage Error, and with my surprise I discovered that is not a common metrics, it doesn't have a wikipedia page, few sites provide an explanation of what it is, sklearn doesn't even have it as a metrics inside the sklearn.metrics._regression.py module.
And my question is: why?
Fixing it was a no brainer, really 2 line of code and I created my own version of it and I can easily run my model with this new metric, but I was wondering if maybe I'm missing something, like if its completely useless or maybe there are other metrics that works better.
If anyone could give me some insights that would be great!

I would be very careful about percentage errors, especially in the context of skewed distributions in the outcome (more precisely: skewed error distributions).
Look at each separate prediction. You evaluate it using an Absolute Percentage Error. This metric will prefer underpredictions, and especially so in the case of skewed error distribtutions. See What are the shortcomings of the Mean Absolute Percentage Error (MAPE)?, and note that the argument really applies to the raw APE, whether we summarize the APEs using the mean or the median.
Summarizing these separate APEs using the median instead of the mean (or using trimming, per Christian's answer) will attenuate the problem, but it won't solve it. "Optimal" predictions will still be biased low. You can simulate this by running an analysis like in the post linked to above: simulate skewed data whose distribution you know (this would stand in for the unknown error distribution in your application), and see which one-number summary will minimize the median APE.
If minimizing the APE is what you really want, and bias is not a problem for you, then by all means, go ahead. I just can't think of a business problem that would be better addressed by an APE-optimal forecast rather than an unbiased expectation forecast. As such, I would say that the "better interpretability" of APEs is a mirage.
