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I was wondering about some of the stats that you regularly see in baseball, and what criteria must be met to achieve such a stat, primarily with the relief and closing pitchers.

I understand that a starting pitcher must go at least 5 innings and leave with his team up at the end of the inning to be considered for the win, however after that I do not quite understand the stats for the pitchers to follow.

To my best knowledge I believe that if the game ever gets tied up again or the winning team drops below the other team's score once the starter is out, they will receive a no decision for their performance, and if the team comes back and retakes the lead, then the pitcher for that inning will be eligible for the win (not sure if there is an innings pitched requirement), while the loss will be awarded to the pitcher that last gives up the lead (regardless of the amount of innings pitched).

The stats that I am wondering about include, saves, holds, and innings pitched requirements for the win in the event that a game gets tied up after the starter leaves. As far as the saves and holds, what are the differences? Is there any requirement for a relief/closing pitcher to get the win once the starter is no longer eligible for it?

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A save cannot 'change' as it is defined as finishing the game.

A hold does not change, either; if reliever A pitches a clean 8th inning, before and after his team leading 4-2, and then reliever B comes in and serves up a grand slam, reliever A still receives credit for a hold.

As far as wins, the winning pitcher is defined as the pitcher who was the pitcher of record when the winning team took the lead for the last time, subject to the restriction that the starter must have completed 5 innings to be able to be credited; and subject to the restriction that a reliever must be deemed "effective" (usually, 1 IP or < 2 ER given up). It's to some extent up to the scorer if several relievers pitch and are effective early in the game (if the starter gets hurt after 1 inning or something and a bunch of relievers each pitch one inning, the scorer has a bit of leeway as to who gets the win, though it would be unusual not to give it to the pitcher of record when the last lead was taken in my experience).

You can read the MLB Official Rules: Scorer for more details on the specifics. Holds aren't an official stat at that level, but the rules for holds are generally the same as for a save (except that the pitcher must not have been the last pitcher in the game).

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