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In other words, given a CR of x the official definition of CR is that four adventurers of level x will on average use up a quarter of their resources to defeat the challenge.

For parties with less than 4 characters, does this scale as one might naively expect? ie. does a CR of x mean that a party of 2 characters will instead use up a half of their resources to defeat the challenge, or that a single player who went toe to toe with the challenge would have a very even fight on their hands?

I'm looking at the d20 Modern books, but I assume the same sort of scaling applies to D&D 3.x too.

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I don't have the d20 Modern book in front of me, so I'm working out of the 3.5e DMG. I'm not sure if the numbers are exactly right, but the principle should be the same.

The short answer is, no, it doesn't scale linearly, but there is a method to the madness. To calculate the "appropriate" challenge for a smaller party, you need to break down the XP charts. The real number you need to be looking at is how much XP each individual character receives from an encounter, which you can derive from the chart. For instance, an EL 5 challenge gives a total of 1500 XP to a 5th level party, or 375 XP per person with a standard 4-person complement. You can reverse engineer this number to get the equivalent XP value for smaller or larger parties: for instance, 375 XP each for a 3-person party yields 1125 XP. You can then add the XP received from each individual monster in an encounter together, and compare that number to the 1125 XP yardstick to work out how difficult the encounter might be.

(This kind of calculation is where the CR X monster + CR X-2 monster = EL+1 encounter formula comes from. At 5th level, a CR 4 monster is worth 1000 XP and a CR 2 monster is worth 500 XP -- together, it comes out to that 1500 XP figure for an EL 5 challenge.)

Or, you could just use the d20 Encounter Calculator. It uses the same principle and does all the work for you -- calculates the EL of the encounter, estimates the challenge level for that particular party, determines an XP award for each individual character, and even calculates the appropriate treasure value. It even handles mixed-level parties.

This isn't going to give you absolutely perfect results. Especially as the party size gets very small or very large, the challenge ratings will get very wonky. A large party will have a hard time dealing with a single monster that has the right CR for them; you should probably just use more monsters of the level you'd normally use against a normal-sized party of that level. A smaller party can be tricky, since so much of d20 Modern/3.x D&D is built around small-scale skirmish combat. Three people should be fine, but you'll probably need to adjust a lot of the game's basic assumptions for a party with only one or two characters.

Ultimately, even for a party of completely standard size, the CR/EL system really does need to be treated as a guideline. In a perfect, platonic rules environment, a CR 5 challenge should use up a quarter of a 4-person, 5th level party's resources, and a 1125 XP challenge should use up quarter of a 3-person, 5th level party's resources. In practice? It depends on the exact monster, the exact make-up of the party -- both in terms of class and optimization -- and even on the whims of the dice.

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No. This is apparent when you see that a doubling of the number of creatures adds a +2 to CR instead of doubling it. CR is as much art as science and there is little linearity to it.

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I'm currently running a 3-player game, any tips on how to interpret CR's for <4 PC's? –  Matthew Scharley Jan 30 '11 at 20:12
    
"About" 1 CR, though you really have to apply your own judgement as GM. If a monster has an ability that can incapacitate a couple opponents, that's more overwhelming with fewer PCs, for example. –  mxyzplk Sep 27 '11 at 4:48
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From a 3.5 perspective, but one that applies to all "difficulty budgets" the only way to explore the specific instance of the budget is to test. One of the more effective ways of doing it is to create a danger room/holodeck type scenario. Design an encounter of very easy, easy, normal, hard, very hard, and TPK difficulty. By tracking resource expenditures through these encounters and by not worrying about a TPK ("it was all a dream/simulation") it will be possible to gauge the efficiency of the party.

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