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I have a table which has 3 fields, I want rank column based on user_id and game_id.

Here is SQL Fiddle :
http://sqlfiddle.com/#!9/883e9d/1

the table already I have :

 user_id | game_id |   game_detial_sum  |

 --------|---------|--------------------|

 6       | 10      |  1000              |   

 6       | 11      |  260               |

 7       | 10      |  1200              |

 7       | 11      |  500               |

 7       | 12      |  360               |

 7       | 13      |  50                | 

expected output :

user_id  | game_id |   game_detial_sum  |  user_game_rank  |

 --------|---------|--------------------|------------------|

 6       | 10      |  1000              |   1              |

 6       | 11      |  260               |   2              |

 7       | 10      |  1200              |   1              |

 7       | 11      |  500               |   2              |

 7       | 12      |  360               |   3              |

 7       | 13      |  50                |   4              |

My efforts so far :

SET @s := 0; 

SELECT user_id,game_id,game_detail, 

       CASE WHEN user_id = user_id THEN (@s:=@s+1) 

            ELSE @s = 0 

       END As user_game_rank 

FROM game_logs

Edit: (From OP Comments): Ordering is based on the descending order of game_detail

order of game_detail

In a Derived Table (subquery inside the FROM clause), we order our data such that all the rows having same user_id values come together, with further sorting between them based on game_detail in Descending order.

Now, we use this result-set and use conditional CASE..WHEN expressions to evaluate the row numbering. It will be like a Looping technique (which we use in application code, eg: PHP). We would store the previous row values in the User-defined variables, and then check the current row's value(s) against the previous row. Eventually, we will assign row number accordingly.

SET @r := 0, @u := 0; 

SELECT

  @r := CASE WHEN @u = dt.user_id THEN @r + 1

             ELSE 1

        END AS user_game_rank, 

  @u := dt.user_id AS user_id, 

  dt.game_detail, 

  dt.game_id 



FROM 

( SELECT user_id, game_id, game_detail

  FROM game_logs 

  ORDER BY user_id, game_detail DESC 

) AS dt 

Result

| user_game_rank | user_id | game_detail | game_id |

| -------------- | ------- | ----------- | ------- |

| 1              | 6       | 260         | 11      |

| 2              | 6       | 100         | 10      |

| 1              | 7       | 1200        | 10      |

| 2              | 7       | 500         | 11      |

| 3              | 7       | 260         | 12      |

| 4              | 7       | 50          | 13      |

View on DB Fiddle

An interesting note from MySQL Docs, which I discovered recently:

Previous releases of MySQL made it possible to assign a value to a
user variable in statements other than SET. This functionality is
supported in MySQL 8.0 for backward compatibility but is subject to
removal in a future release of MySQL.

Also, thanks to a fellow SO member, came across this blog by MySQL Team: https://mysqlserverteam.com/row-numbering-ranking-how-to-use-less-user-variables-in-mysql-queries/

General observation is that using ORDER BY with evaluation of the user variables in the same query block, does not ensure that the values will be correct always. As, MySQL optimizer may come into place and change our presumed order of evaluation.

Best approach to this problem would be to upgrade to MySQL 8+ and utilize the Row_Number() functionality:

Schema (MySQL v8.0)

SELECT user_id, 

       game_id, 

       game_detail, 

       ROW_NUMBER() OVER (PARTITION BY user_id 

                          ORDER BY game_detail DESC) AS user_game_rank 

FROM game_logs 

ORDER BY user_id, user_game_rank;

Result

| user_id | game_id | game_detail | user_game_rank |

| ------- | ------- | ----------- | -------------- |

| 6       | 11      | 260         | 1              |

| 6       | 10      | 100         | 2              |

| 7       | 10      | 1200        | 1              |

| 7       | 11      | 500         | 2              |

| 7       | 12      | 260         | 3              |

| 7       | 13      | 50          | 4              |

View on DB Fiddle

SELECT user_id, game_id, game_detail, 

       CASE WHEN user_id = @lastUserId 

            THEN @rank := @rank + 1 

            ELSE @rank := 1 

       END As user_game_rank,

       @lastUserId := user_id

FROM game_logs

cross join (select @rank := 0, @lastUserId := 0) r

order by user_id, game_detail desc

SQLFiddle Demo

The best solution in MySQL, prior to version 8.0 is the following:

select gl.*, 

       (@rn := if(user_id = @lastUserId, @rn + 1,

                  if(user_id := @lastUserId, 1, 1)

                 )

        ) as user_game_rank,

       @lastUserId := user_id

from (select gl.*

      from game_logs gl

      order by gl.user_id, gl.game_detail desc

     ) gl cross join

     (select @rn := 0, @lastUserId := 0) params;

The ordering is done in a subquery. This is required starting around MySQL 5.7. The variable assignments are all in one expression, so different order of evaluation of expressions doesn't matter (and MySQL doesn't guarantee the order of evaluation of expressions).