I try to find the MAX value for every distinct element in a dataset (I done this using MAX in summary statistics) but I also want to have the id of the element corresponding of that MAX in my output table. Do you have any idea?
The non-python method of achieving what you're looking for would be to use a definition query to single out an institution type, like "police". The definition query would look like this (for shapefiles)...
"INSTITUTION" = 'police'
Then sort your table by "value" in descending order. The "FID" with the highest value will be at the top. You would have to perform this process for every institution type.
When using the summary statistics tool have you tried selecting IN_FID from the drop down under the Case field? Its the last option when you are using the Summary Statistics tool. This should give you the MAX_VALUE and IN_FID fields in a table.