5e is pretty well-known to have an HP inflation problem. It's not as bad as 4e (which gave high-level dragons over a thousand hp), but it's worse than 3.5(!!!). An AD&D ogre has 19 hp, a 3.5 ogre has 29 hp, and a 5e ogre is a bloated monstrosity with 59 hp. There's a lot of initial bloat, but it does sort of calm down by high levels, at least compared to 3.5. An AD&D 2e Ancient Red Wyrm had about 104 hp. 3.5's Very Old Red Dragon has 487 hp, and 5e's Ancient Red Dragon has 546 hp.
So what the heck happened? If you look at the raw damage of a fighter, it doesn't look that far off. A 5e fighter is attacking twice for 1d8+6 when an AD&D fighter is probably doing 3/2 attacks for 1d8+4. Eventually, AD&D fighters have a much higher chance to hit as well, so damage stats are surprisingly close.
One of Mike Mearls' guiding philosophies in 5e design is that not hitting anything for several rounds in a row is boring. At early levels in AD&D, your chance to hit was often around 25%. In 5e, your base chance to hit is typically no lower than 40%, and often much higher. In the case of the ogre, everybody's got about a 60% chance to hit.
So at low levels, the biggest difference is that the other classes do a lot more damage than they used to. An AD&D Magic-User often twiddled his thumbs until combat was over, and a Thief's big effort was to try and not die. But in 5e, every caster has cantrips that do about 1/3 to half the damage of a Fighter's attack. The Thief (now the Rogue) gets his Sneak Attack any time he attacks the Fighter's target. So when you add everything together, the party as a whole is easily doing triple or more an AD&D party's damage. So when in addition to your fighter's 1d8+5 damage, the rogue hits for 2d6+3, the cleric hits for 1d8+2, and the wizard hits for 1d10, with a 60% chance to hit for each, that 59 hp gets whittled down pretty quick.
At the high end, there is a different dynamic in play. The fact is that both AD&D and 3.5 were notoriously broken at high levels. Your wizards & clerics would pile on a few effects to turn the party into an unstoppable killing machine, your warriors looked like Christmas trees of powerful magical gear, and they would tear through the most dangerous monsters and even gods as though they were made of tissue paper. So at high levels, 5e characters have actually been nerfed quite a bit, while the monsters have been significantly beefed up. An ancient dragon is no cakewalk, even for 19th-level characters. 5e's Ancient Red Dragon has 546 hp, but he's a reasonable threat for that 19th-level party, while AD&D had virtually nothing that would challenge players of that level.
What I don't like about this system is there's a tremendous amount of bookkeeping to do in a battle. If you look at my notebook after a fight, you'll often see a dozen or more hits recorded on a monster before it goes down. It can be mentally exhausting, especially since I don't like to use digital tools in a social setting. The characters in my party are only around 6th level, and virtually every fight they have now has over 200 hp to whittle down. The warriors have two or more attacks, mage spells have a lot of dice, and the druid might summon eight animals to help him, each with its own attack. The math works out, but sometimes it all seems to get out of hand.
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