How add unique key to existing table (with non uniques rows)

You can do as yAnTar advised

ALTER TABLE TABLE_NAME ADD Id INT AUTO_INCREMENT PRIMARY KEY

OR

You can add a constraint

ALTER TABLE TABLE_NAME ADD CONSTRAINT constr_ID UNIQUE (user_id, game_id, date, time)

But I think to not lose your existing data, you can add an indentity column and then make a composite key.

The proper syntax would be - ALTER TABLE Table_Name ADD UNIQUE (column_name)

Example

ALTER TABLE  0_value_addition_setup ADD UNIQUE (`value_code`)

I had to solve a similar problem. I inherited a large source table from MS Access with nearly 15000 records that did not have a primary key, which I had to normalize and make CakePHP compatible. One convention of CakePHP is that every table has a the primary key, that it is first column and that it is called 'id'. The following simple statement did the trick for me under MySQL 5.5:

ALTER TABLE `database_name`.`table_name` 
ADD COLUMN `id` INT NOT NULL AUTO_INCREMENT FIRST,
ADD PRIMARY KEY (`id`);

This added a new column 'id' of type integer in front of the existing data ("FIRST" keyword). The AUTO_INCREMENT keyword increments the ids starting with 1. Now every dataset has a unique numerical id. (Without the AUTO_INCREMENT statement all rows are populated with id = 0).