Auto-updating rankings in SQL

Trying to learn how to make a query that auto-updates rankings in a table. In my situation, calculating ranks on the fly is not feasible. I have 60+ separately-ranked events and they all have to be served at the same time.

Is it possible to make a query that does this without rank(), as my version does not have this.

Current Ranking Database              Ranking database after adding new time    +-------+-------+------+                     +-------+-------+------+ | Name  | Score | Rank |                     | Name  | Score | Rank | +-------+-------+------+                     +-------+-------+------+ | Adam  | 98    | 1    |                     | Adam  | 98    | 1    | +-------+-------+------+    Entry being      +-------+-------+------+ | Betty | 87    | 2    |    added            | Emma  | 98    | 1    | +-------+-------+------+    +-----------+    +-------+-------+------+ | Carl  | 87    | 2    |    | Emma | 98 |    | Betty | 87    | 3    | +-------+-------+------+    +------+----+    +-------+-------+------+ | Danny | 80    | 4    |                     | Carl  | 87    | 3    | +-------+-------+------+                     +-------+-------+------+ | Emma  | 76    | 5    |                     | Danny | 80    | 5    | +-------+-------+------+                     +-------+-------+------+