Design: Multiple Values in One Column or Many Columns

The first thing you want to do to keep your thinking straight is to choose some clear terminology. You are using the word "assessment" in two different contexts: (i) a Test and (ii) the results for a student taking the test. I understand that "test" may not be the best term, since you might be assessing based on lots of different kinds of work. The important thing is that you need to clearly differentiate between a measuring instrument and the results of an individual student against that measuring instrument.

Keeping these concepts separate will help you to keep your thinking straight when designing your database.

I'd recommend that all of your RESULTS be kept in a single table. That will make things like report cards easier to handle. Whether the result is a percentage, a letter grade or a pass/fail, you probably want to have a common representation (percentage is a good lowest common denominator) and a weight to help you aggregate marks across assessments.

UPDATED version based on feedback:

CREATE TABLE group (
    group_id INT NOT NULL AUTO_INCREMENT,
    name VARCHAR(255),
    PRIMARY KEY(team_id)
);

CREATE TABLE player (
    player_id INT NOT NULL AUTO_INCREMENT,
    last_name VARCHAR(50),
    first_name VARCHAR(50),
    group_id INT,
    player_rank INT,
    PRIMARY KEY(player_id)
);

CREATE TABLE game (
    game_id INT NOT NULL AUTO_INCREMENT,
    match_date DATE,
    overtime TINYINT(1) NOT NULL DEFAULT 0,
    PRIMARY KEY (game_id)
);

CREATE TABLE player_game (
    game_id INT NOT NULL,
    player_id INT NOT NULL,
    player_score INT,
    PRIMARY KEY (game_id, player_id)
);

...and UPDATED explanation: I think group and player are self-explanatory. The table game will have one record per match. The player_game table records the players in a match and the results; each match will have two records, one for each player.

Since you appear to be using InnoDB as the storage engine you may want to use foreign keys to enforce referential integrity, and obviously you can add other constraints and indexes as needed. If you need some functionality that this doesn't seem to cover, ask and I will either explain or add it.