I know this topic has been asked about many times before but I haven't been able to find a conclusive answer. I'm struggling to work out exactly how much XP should be awarded based on the difficulty of an encounter for non-standard party sizes, and how the difficulty level interacts with the XP gained.
Starting off with an example for a standard 4 player party, if I want to run a moderate encounter the rules say I should aim for an XP budget of 80, and that's how much XP would be awarded to each player after the encounter. It is also my understanding that if my encounter ends up between difficulty boundaries (say 100xp), I can just award that 100xp to my players. I am basing this off of XP Awards section of the rules (Core Rulebook, page 507), which uses the example:
"if the party wins a battle worth 100 XP, they each get 100 XP, even if the party's rogue..."
If I was running a moderate encounter for 3 players, my understanding is that I'd instead be aiming for an XP budget of 60, while still awarding 80xp each for the encounter. The part I am unsure about is if I run an encounter where the XP falls between difficulty boundaries (say 70), what does this mean for the XP earned by the party? Do I still award 80xp as it's closer to a moderate encounter (60) than a severe one (90)? Or do I instead calculate it as (70 * 4) / 3 and award 93xp per player?
I know I could just make sure all encounters use up the entire XP budget allocated for the number of players and this would not be a problem, but I'd prefer to have some more flexibility and know the correct way to account for the XP.
Please let me know if I've misinterpreted anything in the way this should work.
Yes, you have the mathematics correct.
The Core Rulebook's Table 10-1: Encounter Budget is designed to that GMs can adjust for party size using only addition and subtraction, but as a mathematician I prefer multiplication and division. (And exponents, too. If you are curious, I can explain how the mathematical beauty of Pathfinder's exponential leveling system based on the square root of 2 makes for the best gameplay in leveling up.)
The hidden underlying idea in the Pathfinder 2nd Edition Encounter Budget is that each player character has a combat value of 40 xp themselves. Thus, for a 4-member party, a group of enemies worth 160 xp total is an enemy force exactly as powerful as the party, so the party in theory has a 50% chance of victory. In practice, the party is more practiced at their tactics, so has a better chance of winning. For a Severe-Threat challenge, the enemy for is only 75% as strong as the party, 120 xp versus 160 xp. For a Moderate-Threat challenge, the enemy is 50% as strong as the party, 80 xp versus 160 xp.
The experience points earned by individual player characters are on a different scale four times as large. It has the 4-member party assumption built in. If a 4-member party battles 4 monsters each 2 levels below them (20 xp each) one on one, then each party member earns 4 × 20 xp = 80 xp, despite each party member having individually fought only a 20-xp opponent.
If instead, a 3-member party battles 3 monsters each 2 levels below them (20 xp each) one on one, then each party member still earns 4 × 20 xp = 80 xp, because each party member individually fought a 20-xp opponent, just like in the 4-member example.
And the mathematical conversion is Individual XP earned = (Total Enemy XP)×(4)/(Party Size). Thus, for 70 xp of enemies against a 3-member party, each player character earns (70 xp)×(4)/(3) = 93 1/3 xp. Whether to round the fraction 1/3 up or down is totally to GM discretion.
Thanks for the detailed breakdown Mathmuse, that's really useful and makes complete sense.