The shooting genre covers a few different types of games. First-person shooters, light-gun games, and arcade-style flight games all fall under this category, and 2001's releases contained strong games in all three of these subcategories. Here are the nominees for the 2001 Best Shooting Game Award:
Nominees:

Ace Combat 04: Shattered Skies

Publisher: Namco
Developer: Namco
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Ace Combat 04: Shattered Skies is a highly enjoyable, well-executed flight sim that perfectly blends the action elements found in the series' earlier games with the sort of narrative typically reserved for other genres.
Halo

Publisher: Microsoft
Developer: Bungie Studios
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Halo is a complete thrill ride from beginning to end, an experience perhaps best described as a 16-hour-plus version of the siege scene from the film Terminator 2.
Max Payne

Publisher: Rockstar Games
Developer: neo Software
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The game's incredible technical accomplishments and its original play mechanics all combine to make it one of the very finest shooters ever made.
Star Wars: Rogue Leader: Rogue Squadron II

Publisher: LucasArts
Developer: Factor 5
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Rogue Leader: Rogue Squadron II improves upon its predecessor and serves up one of the best showcases for Nintendo's new system and an equally impressive game.
Twisted Metal: Black

Publisher: SCEA
Developer: Incog Inc.
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Twisted Metal: Black is not only the best in the series, but it also manages to be the best car combat game ever created.