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I'm fairly certain that you can't have Disadvantage more than once on the same roll for any reason. But it still seems possible to gain Disadvantage from multiple sources, even if every instance but the first is negated.

Can players intentionally use this to forego multiple penalties? For example, if a PC has 3 levels of exhaustion and is fighting a basilisk, can they say "I'm going to avert my eyes because I'll have Disadvantage on my attack roll anyway?" or should there be some further penalty?

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I think this question needs clarification. I believe you are asking "if a player is subject to multiple sources of disadvantage, can they choose which one? – Escoce Jan 4 at 17:05
up vote 15 down vote accepted

Multiple sources of (dis)advantage don't "stack" (PHB p. 173, Player's Basic Rules p. 57, emphasis mine):

If multiple situations affect a roll and each one grants advantage or imposes disadvantage on it, you don't roll more than one additional d20. If two favorable situations grant advantage, for example, you still roll only one additional d20.

(Dis)Advantage can't make you roll more than 1 additional d20. Having 2×disadvantage is exactly the same as having 1×disadvantage. This is true even for advantage canceling out disadvantage: 2×disadvantage + 1×advantage cancels out completely, rather than leave you with 1×disadvantage. The concept of (dis)advantage is designed to be as simple as possible, and any stacking rules would possibly negate this advantage of the system.

So, yes, that fighter could close his eyes and face no mechanical repercussions from it and still gain the benefit of avoiding the Basilisk's gaze. However, the character normally isn't aware of the rules and therefore doesn't know that. This sort of meta-gaming is often (but not necessarily) discouraged by groups. Unfortunately, it can be hard to avoid.

The key question to pose to the player here would be "If your character were not exhausted, would he avert his eyes?". "Yes, of course, that gaze is nasty!" can be a perfectly valid answer, in which case the player benefits from the simplicity of the rules. Just make sure that the player's justification for his character's closing the eyes is not just "I already have disadvantage, so I don't lose anything!"

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Following your closing question, if the character is aware of the risk of eyeballing a basilisk, wouldn't he or she close his or her eyes? – Drunk Cynic Jan 4 at 17:12
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@DrunkCynic Yes, he may. It's totally believable that he would. But the decision to do so should be made largely independently of the exhaustion. – MrLemon Jan 4 at 17:15
    
@nitsua60 done. That rule was part of the reason I emphasized "exactly", I made it explicit now. – MrLemon Jan 5 at 9:15

You either have (dis)advantage or you don't - you never have multiple (dis)advantage. (PHB p.173)

If multiple situations affect a roll and each one grants advantage or imposes disadvantage on it, you don't roll more than one additional d20. If two favorable situations grant advantage, for example, you still roll only one additional d20.

In addition, advantage cancels disadvantage. If you have 26 sources of disadvantage and 1 source of advantage than you have neither.

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