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The Shield Master feat (PH p.170) grants the following ability:

  • If you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you can use your reaction to take no damage if you succeed on the saving throw, interposing your shield between yourself and the source of the effect.

Does the player decide to expend their reaction before they roll the saving throw, or after?

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The language is thoroughly ambiguous. It can be interpreted in either of two ways:

  • (you can use your reaction to take no damage) (if you succeed on the saving throw)

    If you succeed on the saving throw, you can use your reaction to take no damage.

  • (you can use your reaction) (to take no damage if you succeed on the saving throw)

    You can use your reaction to set up a delayed trigger to take no damage on a successful saving throw.

The main question being whether "if you succeed on the saving throw" is part of the effect, or a further condition on the reaction.

There is plenty you can infer here. For example:

  • If they intended "if you succeed on the saving throw" to be part of the condition, they could have grouped it with the other condition and come up with something much clearer: "If you [must make a Dexterity saving throw to take half damage], and you succeed on the saving throw, you can use your reaction to..."

    Since they didn't use this wording, clearly they must mean for the reaction to be used before the save is rolled...

On the other hand...

  • There are other effects which explicitly state they must be used before a result is known/rolled/etc.

    Since they didn't explicitly state that you need to use your reaction before knowing the result of the save, clearly they must mean for the reaction to be used after the save is rolled...

Ultimately, the choice is your DM's. Or Jeremy Crawford's, if he gets around to providing a Word of God answer.

(insert rant about sloppy templating)

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After: You choose to use the feat after successfully saving

Here is the order of events as per the rule you quote:

1) You find yourself in a fireball's area of effect.

2) You get to roll your Dexterity saving throw.*

3) "If you succeed" on this saving throw you "can" use your reaction to get the feat's benefit.

[The "bold quotes" here are straight from the Shield Master text you quote.]

4a) You choose not to use your feat (or you can’t because you've already taken a reaction) - you take half damage, as you suceeded on your save anyway.

4b) You invoke your feat as a reaction and emerge unscathed.

* Points 1) and 2) seem to me to be the most natural reading of "If you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage" - it is the general context in which the feat can be put to use, but it is not the trigger of the feat. "If" in this sentence could be replaced with "when", "whenever" with no change of meaning. On the other hand "If you succeed" is the trigger. In "if you succeed", "if" is a true conditional, and cannot be replaced with "when", so it represents the moment of decision to use the feat.

If you fail the save...

Conversely if you fail the save you can't use the reaction offered by the feat, so the saving throw must come first.

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Before (unfortunately). The trigger is

[being] subjected to an effect that allows you to make a Dexterity saving throw to take only half damage

And that is when the decision must be made: when the effect is hitting you. If the decision time was after rolling the save, the trigger would be "after making the roll" or similar.

But it says "if you succeed"! Doesn't that means I can choose to use it only if the roll succeeds?

There are two ifs in the description. The first is part of the trigger, the second is part of the effect.

Trigger:

If you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage…

Choice, after the trigger:

… you can use your reaction to…

Effect, after you've decided to take advantage of it:

… take no damage if you succeed on the saving throw, interposing your shield between yourself and the source of the effect.

So the order of operations is

  1. You are the subject of an effect that allows a Dex save for half damage and the ability triggers

  2. You get to choose whether to use Shield Master's optional effect

  3. You roll the Dex save and apply the effect:

    1. If you make the save, you take no damage
    2. Otherwise you take full damage

But doesn't that make it awful?

No more or less than an opportunity attack. When you get a chance to attack with a reaction, you're not guaranteed success, you're just getting an opportunity to try something. An opportunity attack uses up your reaction before the roll, but that chance of it having no benefit isn't controversial.

This use of your reaction is the same: using up your reaction creates an opportunity to get a benefit, but whether the spent reaction actually gets you a benefit depends on whether your roll afterwards succeeds.

And since it's only ⅓ of Shield Master's effects, it doesn't have to be more amazing than a regular reaction in order for the feat to measure up against the other feats — and it is a kind of reaction that is otherwise inaccessible, which itself has special value. It is not an amazing benefit, but it's not awful either, especially since it statistically improves your durability against such effects regardless of any one roll being a save or fail.

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After. As it says:

you can use your reaction [...] if you succeed on the saving throw.

So unless you succeed on the saving throw, you can't use your reaction. Since there's no way to know beforehand whether you'll succeed on your saving throw, you have to wait till you see the result to use your reaction.

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My interpretation of it is After

Shield Master

If you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you can use your reaction to take no damage if you succeed on the saving throw, interposing your shield between yourself and the source of the effect.

Emphasis added to show the pertinent wording which I would take as If you succeed then you can use your reaction to bring up the shield and completely avoid the damage.

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