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The Game Rating provides a numerical 1-100 score that measures how effectively the player performed his assigned role in a game.
Take careful note of that last sentence. Game Rating is not intended as a raw measure of the player's performance in a game relative to the other players. It's an evaluation of how well he met the expectations his team's coaches and management have for him.
Offensive Game Rating
An indicator of how well the player performed defensively during the game, particularly in relation to his assigned offensive and power play roles. Affected mainly by successes or failures while the team is in the offensive zone or carrying the puck up-ice, e.g. scoring a goal will bring most players the largest possible offensive score increase (except for players with playmaker-type roles, who will get slightly more from getting an assist), while giving away the puck will mean a large decrease.
Defensive Game Rating
An indicator of how well the player performed defensively during the game, particularly in relation to his assigned defensive and penalty killing roles. Affected mainly by successes or failures while the team is in its defensive zone or defending a rush, e.g. being on the ice when the opposing team scores a goal will bring most players the largest possible defensive score decrease, while getting a takeaway will mean a large increase.
Overall Game Rating
A general indicator of how the player performed his assigned roles during a game. It's derived in large part from the Offensive and Defensive GR, but the proportions depend on the kind of roles assigned to the player - if they're very defense-oriented, his Defensive GR will contribute more than the overall score than the Offensive GR will. Additionally, some "neutral" events (fights, penalties, some events in the neutral zone, etc.) will adjust this score directly - an Enforcer, for example, may get a decent overall GR despite bad Offensive and Defensive scores if he wins a couple of fights.
Goaltenders only get an overall GR, since they don't have tactical roles; theirs is the only GR that's a direct indicator of overall performance, rather than performance within an assigned role.
As a technical note, GR is calculated on the fly during games, not at the end of them. It'll always be accurate up-to-the-second when you're watching a game. This lets us consider a bunch of statistical events in the GR calculation that would be very expensive in memory and performance to keep separate track of.