Post/Author/DateTime	Post
#1 Edymnion Oct 02, 2003 23:11:24	Often the question comes up of "Which weapon is best?" Well, in another thread I showed that the critical styles were all even (aka an 18-20/x2 weapon did the exact same average damage per swing as a 20/x4 weapon, even when Bursting enchantments are taken into account). But, since the question still gets asked, I'm going to do the math here for each weapon, and then we will all have something to refer back to. Now, I won't show my math for every weapon, but here's how its done. I'll use a d6 19-20/x2 as an example here to show you how I did the math. d6 has an average value of 3.5, so the x2 crit is 3.5*2 = 7. This weapon crits on 2 numbers, so its 3.5*18 + 7*2 = 77. 77 divided by 20 to get the average damage per roll is 3.85. All numbers below are for medium creature sized weapons, but the ratios will be the same when its all scaled down to small. Any clear winner for highest damage in each catagory will be underlined. This is also organized via the 3.0 way of ordering weapons. For the 3.5 people, Light = Small, One Handed = Medium, Two Handed = Large. Simple Weapons- Melee Unarmed: 2.1 Tiny Dagger: 2.75 Dagger, Punching: 2.75 Guantlet, Spiked: 2.625 Small Mace, Light: 3.675 Sickle: 3.675 Medium Club: 3.675 Shortspear: 3.85 Mace, Heavy: 4.725 Morningstar: 4.725 Large Quarterstaff: 3.675 / 3.675 Spear: 4.95 Simple Weapons- Ranged Small Crossbow, Light: 4.95 Dart: 2.625 Sling: 2.625 Medium Crossbow, Heavy: 6.05 Javelin: 3.675 Martial Weapons- Melee Small Axe, throwing: 3.675 Hammer, light: 2.625 Handaxe: 3.85 Lance, light: 3.85 Pick, light: 2.875 Sap: 3.675 Sword, short: 3.85 Medium Battleaxe: 4.95 Flail, Light: 4.725 Lance, Heavy: 4.95 Longsword: 4.95 Pick, Heavy: 4.025 Rapier: 4.025 Scimitar: 4.025 Trident: 4.725 Warhammer: 4.95 Large Falchion: 5.75 Flail, heavy: 6.05 Glaive: 6.05 Greataxe: 7.15 Greatclub: 5.775 Greatsword: 7.7 Guisarme: 5.5 Halberd: 6.05 Longspear: 4.95 Ranseur: 5.5 Scythe: 5.75 Martial Weapons- Ranged Medium Shortbow: 3.85 Large Longbow: 4.95 Exotic Weapons- Melee Tiny Kukri: 2.875 Small Kama: 3.675 Nunchaku: 3.675 Siangham: 3.675 Medium Sword, bastard: 6.05 Waraxe, dwarven: 6.05 Hammer, gnome hooked: 3.85 / 2.875 Large Axe, orc double: 4.95 / 4.95 Chain, spiked: 5.25 Flail, dire: 4.725 / 4.725 Sword, two-bladed: 4.95 / 4.95 Urgosh, dwarven: 4.95 / 3.85 Exotic Weapons- Ranged Tiny Crossbow, Hand: 2.75 Small Whip: 1.575 Medium Crossbow, repeating: 4.95 Please note that this takes into account ONLY the pure damage of each weapon. It does not factor in reach or other special abilities. So it would be wise of you to note that while a sling and a dart have the same damage output, the sling has a much farther range. Also, no strength bonuses were taken into account, so obviously the thrown weapons will have higher damage outputs than missle weapons if you have higher strength. Well, there you go. Do with it as you will.
#2 zombiegleemax Oct 04, 2003 12:08:34	The thing that I really dislike about analysis like this is that is has to ignore the target "to-hit" number, which essentially means that it is worthless unless you hit on everything including a one. Which is never going to happen.
#3 Edymnion Oct 06, 2003 22:37:49	Originally posted by chonjurer The thing that I really dislike about analysis like this is that is has to ignore the target "to-hit" number, which essentially means that it is worthless unless you hit on everything including a one. Which is never going to happen. Yes, it does assume that every swing will hit, no other way to do it. However, all weapons have the same chance of hitting as any other (a dagger or a greatsword, they both need the same roll to hit the same AC), so the ratios are the same regardless of hit percentage.
#4 zombiegleemax Oct 07, 2003 14:52:23	The numbers need adjusting if they are all done by your sample calculation. A roll of a 1 is always a miss, thus 3.5*17 + 7*2 = 73.5. 73.5 / 20 = 3.675 average damage a round.
#5 Feldspar Oct 08, 2003 20:30:51
#6 musrum Oct 09, 2003 2:43:20	Your analysis does break down at high DCs. I compare two weapons here: Pick, heavy 1d6 x4 Rapier 1d6 18-20/x2 DC 0 is what you calculate:
#7 dire_celestial_bonobo Oct 09, 2003 3:04:10	Nice analysis; however, looking at average damage output glosses over one point that in practice makes 18-20x2 preferable to 20x4: overkill. In general, a creature will take N non-critical hits to down. We model the creature's hit points just before it's killed by the last hit as a random number, which is reasonable, since it'll have been battered by any number of things. Let B be the base damage of the weapon (before criticals). Accordingly, if the hit that downs a creature is not a critical, on average we expect half of the damage to be wasted due to overkill (since sometimes you'll do 10 and it'll have 9, but as often you'll do 10 and it'll have 1). Things get trickier when we take criticals into account, though. Let's examine the amount of wasted damage from an 18-20x2 weapon and a 20x4 weapon against a creature that takes N non-critical hits to drop: N = 1: - 18-20x2: 85% chance to waste B/2, 15% chance to waste B + B/2 --> wastage = 1.30*B/2 - 20x4: 95% chance to waste B/2, 5% chance to waste 3B + B/2 --> wastage = 1.30*B/2 Verdict: identical (This is the case when fighting weak, low-hp opponents, like orcs - criticals make no difference.) N = 2: - 18-20x2: 85%*85% chance to take two normal hits (waste B/2), 15% chance to take one critical hit (waste B/2), 85%*15% to take a normal hit then a critical hit (waste B + B/2) --> wastage = 0.7225*B/2 + 0.15*B/2 + 0.1275*3B/2 = 1.255*B/2 - 20x4: 95%*95% chance to take two normal hits (wastage B/2), 5% chance to take one critical hit (wastage 2B + B/2), 95%*5% chance to take a normal hit then a critical hit (wastage 3B + B/2) --> wastage = 0.9025*B/2 + 0.05*5B/2 + 0.0475*7B/2 = 1.485*B/2 Hits to kill: - 18-20x2: 1.85 - 20x4: 1.95 Verdict: 20x4 wastes 20% more damage than 18-20x2 and takes 5% more hits to drop the enemy. N = 3: - 18-20x2: 85%*85%*85% chance of 3 normal hits (waste B/2), 15%*85% of critical then normal hit (waste B/2), 85%*15% of normal then critical hit (waste B/2), 85%*85%*15% of two normal then a critical (waste B + B/2), 15%*15% of two critical (waste B + B/2) --> wastage = 0.614*B/2 + 0.255*B/2 + 0.1084*3B/2 + 0.0225*3B/2 = 1.262*B/2 - 20x4: 95%*95%*95% chance of 3 normal hits (waste B/2), 95%*95%*5% of two normal then critical (waste 3B + B/2), 95%*5% of normal then critical (waste 2B + B/2), 5% of critical hit (waste B + B/2) --> wastage = 0.857*B/2 + 0.451*7B/2 + 0.0475*5B/2 + 0.05*3B/2 = 1.561*B/2 Hits to kill: - 18-20x2: 2.72 - 20x4: 2.85 Verdict: 20x4 wastes 20% more damage than 18-20x2 and takes 5% more hits to drop the enemy. N = 4: - 18-20x2: 85%*85%*85%*85% of 4 normal (waste B/2), 85%*85%*85%*15% of 3 normal then 1 critical (waste B + B/2), 85%*85%*15% of 2 normal then 1 critical (waste B/2), 85%*15%*85% of NCN (waste B/2), 85%*15%*15% of NCC (waste B + B/2), 15%*85%*85% of CNN (waste B/2), 15%*85%*15% of CNC (waste B + B/2), 15%*15% of two criticals (waste B/2) --> wastage = B/2(0.522 + 3*0.108 + 0.0225) + 3B/2(0.092 + 2*0.019) = 1.261*B/2 - 20x4: 95%*95%*95%*95% of 4 normal (waste B/2), 95%*95%*95%*5% of NNNC (waste 7B/2), 95%*95%*5% of NNC (waste 5B/2), 95%*5% of NC (waste 3B/2), 5% of single critical (waste B/2) --> wastage 1.533*B/2 Hits to kill: - 18-20x4: 3.59 - 20x4: 3.71 Verdict: 20x4 wastes 20% more damage and takes 3% more hits to drop the enemy. N >= 5: Since the enemy isn't killed in a single hit by either weapon, these will be some combination of N = 1 through N = 4. Thus, on average, a 20x4 weapon will take 4% longer to drop enemies and suffer from 20% more wastage due to overkill, or about an eighth of a hit per enemy dropped. More generally, we can define a wastage function for each critical type. Given that a monster will take N non-critical hits to drop, and assuming all damage is multiplied by criticals (i.e., not a rogue), we define W(N) for an 18-20x2 weapon as: W(0) = 1.5; W(1) = 0.5 W(k) = 0.85W(k-1) + 0.15W(k-2) In words: W(1) = 0.5 means doing one hit of damage to a monster that will drop from one hit of damage means you will, on average, waste half of the hit's damage (since sometimes you'll do 10 and he'll have 9, and other times you'll do 10 and he'll have 1). W(0) = 1.5 means doing two hits of damage to a monster that will drop from one hit of damage means you will waste half a hit of damage (same reason as above) plus another full hit of damage (since you needed one, not two). W(k) = 0.85W(k-1) + 0.15W(k-2) means that if you're facing a monster which will take k hits to drop and you've got an 18-20x2 weapon, you've got an 85% chance to deal one hit of damage to it (so wasted damage is whatever you waste when you keep fighting it as a monster that needs k-1 hits to drop it) and you've got a 15% chance of criticalling and dealing two hits of damage (so wasted damage is whatever you waste when you keep fighting it as a monster that needs k-2 hits to drop it). Note that you only bottom out and get a number when you get to W(0) or W(1), since that's when you finally drop the monster. Similarly, for a 20x4 weapon, we get: W(-2) = 3.5 W(-1) = 2.5 W(0) = 1.5 W(1) = 0.5 W(k) = 0.95*W(k-1) + 0.05*W(k-4) These values come from reasoning analogous to that above. Hence, for whatever number of hits the monster takes to drop before we count criticals, we can compute the expected wastage as a fraction of damage from a single non-critical hit. For reasonable numbers of hits required (up to 10), these values are:
#8 gil_gwath Oct 09, 2003 3:22:54	Originally posted by Edymnion Yes, it does assume that every swing will hit, no other way to do it. However, all weapons have the same chance of hitting as any other (a dagger or a greatsword, they both need the same roll to hit the same AC), so the ratios are the same regardless of hit percentage. The point you are missing is that if the crit threat is not the same it changes ratios. (Even if it could be visible only on limit cases.) Example (following your math): If the character needs at least a 15 to hit, assuming a successful hit we have the following average damage: Morning star (1d8 x2) = 5.25 Sword, short (1d6 19/20 x2) = 4.66 If the character needs at least a 19 (!) to hit, assuming a successful hit we have the following average damage: Morning star (1d8 x2) = 6.75 Sword, short (1d6 19/20 x2) = 7 [EDIT]: Added underline to make obvious wich weapon is better in the given case. So we demonstrated that the "best" weapon is dependent on the "to hit" number... [/EDIT] So it's easy to understand why in some cases the "to hit" number could be of interest in these calculations.
#9 harzerkatze Oct 09, 2003 10:14:20	Almost, but not exactly true. If you go up against an opponent with very high AC, the balance can change. Imagine two weapons of the same stats, only that one is 18-20/x2 and the other is 20/x4. Normally, they have the same damage ratio, just as you calculated. But imagine an opponent so heavily armed that you only hit him an a roll of 19 or 20. Now the first weapon will only work like a 19-20/x2 weapon, because the roll of 18 is a miss. The other weapon will still do 4x damage on a roll of 20. If we imagine an opponent that can be only hit with a roll of 20, the 20/x4 will do twice as much damage when you do a critical as the 18-20/x2 one. But that is just the mathematician in me speaking. Effectively, these differences are trivial. If your only hit on a rolled 20, only 0.25 percent of your attacks will be confirmed criticals. That one of 400 attacks. There are more important things to worry about then if you do double damage every 400 attacks.
#10 Edymnion Oct 09, 2003 16:10:54	Originally posted by Dalaeck The numbers need adjusting if they are all done by your sample calculation. A roll of a 1 is always a miss, thus 3.5*17 + 7*2 = 73.5. 73.5 / 20 = 3.675 average damage a round. Oops, you're right, I did indeed forget to take into account the auto-miss rule. I guess I figured it was balanced by an auto-hit 20, but that’s rarely going to be an issue, while the auto-miss is always going to be an issue. So the numbers above do indeed need the disclaimer that you are assumed to always hit with every swing.
#11 canamrock Oct 09, 2003 18:27:11	So, if the 'wastage' of the high multiplier weapons is about 2.5-5%, and the power loss to high-range weapons occurs in roughly 5% of combats (the 1:20 chance of the AC being in the range where the extra crit range is irrelevant), then it seems that, except for non-damaging components tied to the critical function, the weapons will perform the same in a overall comparison against a variety of targets. However, this also means that it MIGHT be worthwhile to keep a high-crit and high-range weapon around in those cases where either one has a tactical advantage over the other.
#12 zombiegleemax Oct 10, 2003 4:25:46	Re: preferable crit ranges There is also 'death from massive damage' to consider and 'coup de grace's. Obviously if you are doing x4 damage on a crit then 50 points of damage is a lot more easily reached. The advantage of the higher multiplier for a coup de grace is also obvious. Keening
#13 dire_celestial_bonobo Oct 15, 2003 14:41:23	So, if the 'wastage' of the high multiplier weapons is about 2.5-5%, and the power loss to high-range weapons occurs in roughly 5% of combats (the 1:20 chance of the AC being in the range where the extra crit range is irrelevant), then it seems that, except for non-damaging components tied to the critical function, the weapons will perform the same in a overall comparison against a variety of targets. However, ACs are not distributed randomly. For weapon-based characters, ACs of monsters whose CRs are appropriate tend strongly to cluster, usually in the "hit reasonably often" range. A monster whose AC was so high that it wasted any of all but the broadest threat ranges is very likely to be so powerful as to be inappropriate for the party to be fighting. For example, a 10th-level fighter will probably expect to have about +20 to hit (+10 BAB, +5 Str, +3 magic, +1 WF, +1 misc), meaning an AC of about 32 would be necessary before even a secondary attack with a Keen longsword would lose any threat range. Quickly scanning the MM, there's only a handful of monsters with ACs that high, almost all in the CR 16-20 range. The only one a 10th-level party could reasonably face is the CR 13 Ice Devil, whose AC is exactly 32. Doubtless, NPCs could be made with AC 32+ (+3 mithril full plate, +3 Dex, +3 large shield, +3 misc is possible for a 10th-level character, for example), but they will also be very rare (most sample NPCs in the DMG never reach AC 32, and none before 18th level). More threat range will be wasted as more attacks are gained (for example, a 17th-level fighter would waste part of a 17+ threat range on his 4th attack against AC 35), but the effects are also very diluted - wasting 1 point of x2 threat range lowers the expected damage from an attack by 5%, but since that attack is necessarily very unlikely to hit, that 5% of a 15% chance to hit will be almost irrelevant compared to the other, more likely attacks in the sequence (< 0.1% difference). Typically, threat range wastage will rarely occur, and is exceedingly unlikely to make even a 1% difference. However, this also means that it MIGHT be worthwhile to keep a high-crit and high-range weapon around in those cases where either one has a tactical advantage over the other. No - splitting your weapon-enchantment money and feats (WF/WS) between two weapons is a bad idea. Even +1 to hit is worth about +10% damage, far outweighing the benefits of a slightly more suited weapon. Obviously if you are doing x4 damage on a crit then 50 points of damage is a lot more easily reached. True. However, note that most creatures capable of taking 50 points of damage and not dying anyway are likely to have a high enough Fort save that DC 15 isn't much of an issue. Assume that every 20x4 critical will require a Fort save and no 18-20x2 criticals will require a Fort save. A creature requires H hits to kill and has a chance of failing the Fort save of C. Accordingly, the expected decrease in the number of hits to kill the creature for the 20x4 weapon will be: (H-4)*0.05*C + (H-5)*0.05*C*(1-0.05*C) + (H-6)*0.06*C*(1-0.05*C)^2 + ... ...down until H-N = 0, which is when the critical's damage alone will kill the monster. About a best-case scenario would be C = 0.5 (Fort +4) and H = 10 (which is the monster having 150+ hp to start). In this (unrealistically good) case, the massive damage rule means we expect the monster to drop approximately 0.5 hits earlier, or in 9.5 hits instead of 10, or a 5% difference. The worst-cast scenario is when the monster will drop in 4 hits or less, meaning the critical damage will kill it without the massive damage rule. This worst-case scenario is also a quite common scenario. In a non-contrived good-case scenario, we might get a creature with 100hp and Fort +9 being attacked by someone whose average damage is 15-20 (H=6), which would give us about 3/80 = 0.04 hits saved. On average, then, we might expect a 20x4 weapon to kill opponents in about 0.01-0.02 fewer hits when we take into account the massive damage rule. Note also that wider x2 threat ranges will progressively gain more of this bonus as base damage rises. (Aside: do creatures that are not subject to critical hits - like elementals - still need to make Fort saves against massive damage?) The advantage of the higher multiplier for a coup de grace is also obvious. If Coups de Grace weren't so rare, carrying around a x4 weapon for them might be worthwhile. For a weapon-based character, though, it's unlikely to be worth it. A character's average damage will tend to increase by about 2 points every 3 or so levels (taking into account Str, magic, sneak, WS, flaming, etc...), most of which will double on a crit. For most creatures, 10 + 10-20+lvl is likely to be a tough Fort save to make. For example, a 10th-level longsword fighter is likely to do about 15 damage on average, making for a DC 40 Fort save on a CdG, even without Power Attack, and very few creatures a 10th-level party should fight can make that save on anything but a 20.