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Message: <blockquote data-quote="comrade raoul" data-source="post: 2312540" data-attributes="member: 554">Two separate issues have come up. FreeTheSlaves raised the point that Sean's argument is unconvincing, since it relies on an unrepresentative example. S'mon suggested that high-threat weapons are better than high-multiplier weapons, and a bunch of people have said that this turns out to be false if you do the math. Since it's easier to make an abstract argument about why Sean is right once you see the math, let's do the basic math on critical hits now.I'll be using two related notions that are misleadingly, but similarly, named. The average damage of an attack is the average damage it does if it hits. The average damage of a greatsword, with a +3 bonus from Strength and nothing else, is 10. (The average result of 2d6 is 7, and that gets a +3 damage bonus.) The expected damage of an attack is its average damage taking into account the possibility of a miss or critical hit. Not counting for critical hits, the expected damage of an attack with that greatsword if I only hit half the time is 5.Ignoring special factors like concealment, the expected damage for any attack is equal to the base expected damage for the attack, plus any expected damage from critical hits.The base expected damage for an attack is really simple. It's just:(chance to hit) * (average damage) We saw that from the greatsword above. My chance to hit was 0.5; my average damage was 10, so my expected damage is 5.How much of a statistical difference do my crits make? I have to threaten, confirm the threat, and then do extra damage equal to my extra damage times my multiplier - 1. (With an x2 multiplier, I just do my average damage over again, so I end up doing twice my average damage.) This is:(chance to confirm) * (chance to threaten) * (multiplier - 1) * (average damage)So the total expected damage we do with an attack that has a possibility of a critical hit is equal to: ([chance to hit) * [average damage]) + ([chance to confirm] * [chance to threaten] * [multiplier - 1] * [average damage])Now, notice that my chance to confirm is exactly equal to my chance to hit. This means that we can combine each of these, using middle-school algebra, to get a total, simplified expected damage of an attack equal to:(chance to hit) * ([1 + {chance to threaten} * {multiplier - 1}] * [average damage])We can notice two things here. First, the effect of critical hits is (mostly!) independent of my chance to hit, even though I have to confirm my threats. A weapon's crit effectively increases its average damage by multiplying it by a constant factor equal to (1 + [my chance to threaten] * [my multiplier - 1]).Once you see this equation, you can see how high-threat and high-multiplier weapons balance out. A 19-20/x2 longsword has a 0.10 chance to threaten (it threatens on two results out of 20) and a multiplier of x2: it multiplies damage by 1.10. An x3 battleaxe has a half the threat range but double the multiplier -- a 0.05 chance to threaten and a multiplier of x3, so it, too, multiplies damage by 1.10. If we modifiy these weapons with keen and Improved Critical effects, we find that our 15-20/x2 longsword has a 0.30 chance to threaten and the same multiplier: it multiplies damage by 1.30. The 18-20/x3 battleaxe still has half the threat range but double the multiplier -- a chance to threaten of 0.15 and a multiplier of x3, so it also multiplies damage by 1.30.The moral is that what matters is the ratio between threat range and multipliers: as long as an axe's threat range is half that of a sword but its multiplier is double, they do the same amount of extra damage from criticals. The only way to preseve this is to increase threat ranges by an amount proportional to the original threat range: the doubling rules of 3.0e. Increasing them by constants skews the balance.Now, are there other reasons why high-threat weapons are better, or vice versa? Well, sometimes the extra damage of a high-multiplier weapon is wasted. But then, sometimes my attack roll is good enough to threaten but not good enough to hit! This is more common with high-threat range weapons (especially when you get into the territory of 15-20 or 12-20), and does much to balance them out. It's really a largely subjective preference.</blockquote>

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