If in the course of an adventure you want one of your players to end up having a one on one duel versus an opponent how do you determine what level the BG should be? I ask because it says that the CR is based on the Average Party Level based on X number of characters being in the party. Or did I miss something completely?
CR for a character is supposed to be ECL-1. Unfortunately, as you may discover, the variety of builds and character types can skew it in either direction.
If you have the time, the most effective method is to simulate. Take a copy of the character and pretend fight it against your opponent during your GM prep time. Work out the average damage a round from both sides. Compare hp. Make an informed decision.
Second Goddity's suggestion to simulate.
Assigning CR for just 1 PC really depends on the character and the opponent. The calculations of CR vs. a whole party (benchmarked at 4 characters) assume a diversity of capabilities that a single character lacks (e.g., arcane casting, distance combat, close melee combat, mobility...) The fewer PCs, the less diverse the skillset, and therefore the more difficult to assign a CR.
For example, consider a ghoul, which is CR 1 under many circumstances. Vs. a party of four 1st level characters, there is a reasonable likelihood that one or more of them will have a decent Fort save (to combat paralysis), the ability to channel energy/spontaneously cast cure light wounds (to damage it without engaging or with a touch), to hit it from a distance before it closes with magic missile or an actual missile weapon, or just high initiative, a greatsword, and Power Attack...
But, even a 3rd level character, by herself, might face a "save or die" situation, especially if she doesn't have a great Fort save or way to hurt the ghoul without getting close. And without friends to drag her paralyzed body out of the fray, she's stuck for at minimum 2 rounds. More than enough time to coup de grace, with a round to spare for seasoning...
On the other hand, a 1st level elf cleric with the Travel domain could stay out of the ghoul's reach (for the most part), while readying channels for when it got w/in 30 feet. Even if hit, she'd be immune to the paralysis and so the challenge would be significantly reduced.
This is just one hastily thrown together example. It illustrates how much more variable things get when you have fewer than 4 PCs.
Another thing to think about is how much more a single bad roll can hurt when you only have 1 PC. If you've got 4, the party can bounce back from one or two or even three bad rolls - and the CR estimates take this into account. But with just 1 PC, all it takes is one bad round and there's no one to pull them out of trouble.
An average encounter is CR = APL.
An average party is 4 characters.
So, a CR 8 is an average encounter for 4 characters of 8th level.
CR goes up by 2 when you double the number encountered. It goes down by 2 for a halving.
So, for 2 characters rather than four, you reduce the CR by 2. [=CR 6 for the example.]
So, for 1 character, reduce it again. [=CR 4 for the example.]
So an average encounter for a 8th level character is CR-4.
Here is the encounter chart:
Difficulty .. Challenge Rating Equals…
Easy .. APL –1
Average .. APL
Challenging .. APL +1
Hard .. APL +2
Epic .. APL +3
So, for the above 8th level character, solo encounter, you get:
Difficulty .. Challenge Rating Equals…
Easy .. APL -4 –1 = CR 3 vs. 8th
Average .. APL -4 = CR 4 vs. 8th
Challenging .. APL -4 +1 = CR 5 vs. 8th
Hard .. APL -4 +2 = CR 6 vs. 8th
Epic .. APL -4 +3 = CR 7 vs. 8th
/cevah
But, of course, that's nonsense even though technically correct by the CR rules. A CR 4 or 5 challenge is nowhere near any kind of real threat for an 8th level character.
Ideally the challenge rating of a PC is it's level. Meaning a lvl 8 character against a CR 8 encounter has a 50/50 chance of winning. So just scale this down to the level of difficulty you are after.
That said, the mechanics of this game makes 1 vs 1 fights very hard to assign a single value to, and quite random. As others have said, you should custom built the encounter for the character: for a fighter, give him a blaster caster so that it is not save or die and so on.
Thank you for the answers. I like Goddity's idea to run some simulations.
Looked at the charts and it did not make sense to have a 12 level character fight a CR 6 foe.
Might have to kick this one back a little bit in the adventure.
If it's a 0 HD race with class levels just toss one of equal level to him. More or less if you want it to be a character more or less experienced than him.