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I think the key here is when you wrote "middle innings". A 1-0 game in the middle innings is far from decided, and taking the out and having the runner score to tie is not the end of the world. Generally, when you have the chance to shorten the game, do it.
However, if the runner was the winning run, I'd let the ball drop. With almost no chance to throw him out you've effectively decided the game by catching that foul ball. If the next pitch results in a base hit, well, the game would have been over anyway. If the next hitter gets out you're a genius, at least for one more batter.
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