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WotC recently released the Basic Rules for the DM, and with it came a few items like the Amulet of Health:

Amulet of Health

Wondrous item, uncommon (requires attunement)

Your Constitution is 19 while you wear this amulet. The amulet has no effect on you if your Constitution is already 19 or higher.

The Basic Rules state:

When your Constitution modifier increases by 1, your hit point maximum increases by 1 for each level you have attained. For example, when Bruenor reaches 8th level as a fighter, he increases his Constitution score from 17 to 18, thus increasing his Constitution modifier from +3 to +4. His hit point maximum then increases by 8.

But it doesn't explain what happens when the Constitution modifier decreases!

My players will soon get the Amulet as a reward, and I fear that they will just pass the item around to increase everyone's max HP. how can I prevent that? Are the HP lost when a character is no longer attuned to the Amulet?

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    \$\begingroup\$ I'm suddenly imagining an old soldier with the constitution of a mountain, who has no idea why... the only constant is he takes off his amulet every night to sleep, and puts it back on the next day, like he's done every night for decades. \$\endgroup\$ – doppelgreener Aug 25 '14 at 13:56
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From PHB while describing Constitution:

...if you are 7th level and some effect lowers your Constitution score so as to reduce your Constitution modifier by 1, your hit point maximum is reduced by 7

When your CON Bonus somehow decreases by X, you lose total Max Hit Points equal to X*PG-Level.

What happens to HP when Constitution decreases? / Are the HP lost when a character is no more attuned to the Amulet ?

If your CON Bonus decreases, you lose Max Hit Point and HP over your new Max Health are lost too.

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    \$\begingroup\$ Note, it does not discuss current hit points reducing, but if your current ends up higher than your max because your max was reduced, your current goes down to your new maximum. \$\endgroup\$ – Aviose Aug 27 '14 at 16:17
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There are actually clear guidelines on this. (Thanks to GMNoob and Crico)

The next sentence in Basic is

Or if you’re 7th level and some effect lowers your Constitution score so as to reduce your Constitution modifier by 1, your hit point maximum is reduced by 7

So if you put this on, and you had A Con score of 15 and you're, say L12. Then you'd gain 24 HP. If you take it off, and you still had a Con score of 15, then you'd lose those 24HP.

Something I had missed was that putting it on, gives you all of the HP as if your CON mod had always been 19. Which is cool (and a huge boon if you boost CON as a stat increase). So taking if off removing all of it's benefits makes sense.

Some things worth talking about though:

  • Obviously this pretty much eliminates any silly business with passing the item around. There's no benefit at all to putting it on and then taking it off.
  • Attunement should always be enforced with items like this. So if a PC finds that they aren't using the bonus HP regularly and their buddy is getting beat down, and they want to pass it on, you need to make sure they spend 2 short rests doing it, and that the newly attuned character doesn't already have 3 attuned items.
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Answering the second, more specific part of your question...

(...) I fear that they will just pass the item around to increase everyone's max HP. how can I prevent that?

Attunement exists precisely to prevent this kind of abuse.

This is pre-prevented by the fact the amulet requires attunement. Changing who in the party is attuned to the item generally requires two short rests, one to voluntarily end one character's attunement, and another for another to attune.

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