MySQL 8.0

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CREATE TABLE test1 (`region_id` INTEGER, `pst` INTEGER, `rank` INTEGER);

INSERT INTO test1 (`region_id`, `pst`) VALUES
  ('101', '20'),
  ('101', '19'),
  ('101', '19'),
  ('101', '15'),
  ('100', '20'),
  ('100', '13');

Records: 6  Duplicates: 0  Warnings: 0


UPDATE test1 t1
INNER JOIN (
  SELECT region_id, pst, 
         RANK() OVER (PARTITION BY region_id ORDER BY pst DESC) `rank` 
  FROM test1
) t2 ON t2.region_id = t1.region_id AND t2.pst = t1.pst
SET t1.`rank` = t2.`rank`;

Rows matched: 6  Changed: 6  Warnings: 0


SELECT * FROM test1;


region_id
pst
rank


101
20
1

101
19
2

101
19
2

101
15
4

100
20
1

100
13
2