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In baseball, I'm wondering how home teams are awarded winning runs. If the visiting team in, say, Inning 10, plates 5 runs, then the home team has the bottom of that inning to match or exceed this number.

But what if this scenario occurs: game is tied in the 10th with the home team up to bat. 1 runner is on base, and the batter launches a home run over the fence. The game is clearly over via a "walk-off" at this point, but since the home team doesn't "need" 2 runs to win (remember, it's tied), are they only awarded the 1 run from the on-base runner? Or are they given a 2-run win? If it's the former, is the batter then only awarded a single RBI for his stats, thus "robbing" him of the extra RBI (in this case, himself)?

TL;DR: can the home team in an extra-inning baseball game ever win said game by more than 1 run?

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According to Baseball Almanac prior to 1920 this was not possible. In fact, if a runner was on base, the batter would not be given credit for a home run.

However, this is now explicitly allowed in the rules.

While 7.01(g)(3) states that

... the game ends immediately when the winning run is scored.

it also lists the exception for home runs:

EXCEPTION: If the last batter in a game hits a home run out of the playing field, the batter-runner and all runners on base are permitted to score...

So with runners on base, a home team can win in the bottom of the ninth or later by up to 4 runs.

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Yes, the home team certainly can win by more than one run.

For an example, see Royals vs Tigers on 2019-05-05: the score was tied at 2 after nine innings, the Royals didn't score in the top of the tenth, but then in the bottom of the 9th Brandon Dixon hit a three-run home to win the game 5-2.

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