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This may be more than one answer, depending on what is being appraised. Is the following at least reasonable,or am I missing anything?

  • Arcana - magic items, spell scrolls, potions (as long as they are identified fully)
  • Nature - gems, precious metals
  • Religion, History - art (depending on the subject)
  • Investigation, Perception - everything else

I should note, I am intentionally trying to make appraisals non-trivial. This likely isn't addressed in the rules because not every DM would do this.

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    \$\begingroup\$ Note that I don't think Arcana is enough to appraise a magical item unless you somehow already know what the item does (depends how you play that), but otherwise an Identify spell would be required... \$\endgroup\$ – Alexis Wilke Feb 22 '15 at 2:55
  • \$\begingroup\$ Good point, I had thought the same @AlexisWilke \$\endgroup\$ – David Wilkins Feb 22 '15 at 2:56
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An Intelligence check is appropriate for this task. According to page 178 of the PHB, an Intelligence check can be used for a number of things, including:

  • Estimate the value of a precious item

As far as what skill can be used on this check, or whether a skill can be used at all, I leave it up to my players to suggest a skill that might apply and justify it. This puts the onus on them to decide what they want to use, and it has the additional benefit of sometimes letting them use something that wouldn't have occurred to me.

For example, a conversation might go like this:

Player: Can I tell what this portrait is worth?

Me: Give me an Intelligence check.

Player: Can I apply my proficiency with painter's supplies to that?

Me: Sure, that sounds fair.

Note that an example of a very similar conversation appears on page 5 of the PHB:

Dungeon Master (DM): OK, one at a time. Phillip, you’re looking at the gargoyles?

Phillip: Yeah. Is there any hint they might be creatures and not decorations?

DM: Make an Intelligence check.

Phillip: Does my Investigation skill apply?

DM: Sure!

Phillip (rolling a d20): Ugh. Seven.

DM: They look like decorations to you. And Amy, Riva is checking out the drawbridge?

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    \$\begingroup\$ Now, see? That is "Chris Perkins" awesome \$\endgroup\$ – David Wilkins Feb 22 '15 at 4:17

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