ACKS: Critical Error. Also Thieves

So I got to thinking about Weapon Focus in ACKS last night, and how we perceived it as terrible.  I started doing some math, and semi-interesting results came out. 

The ACKSPC says that +1 to damage is worth about a proficiency, but Weapon Focus seems to generate much less than +1 damage per hit.  However, it turns out this is dependent on what you need to roll to hit - if you have a d6 weapon, Weapon Focus gives you >+1 damage per hit on average as long as you need an 18+ to hit.  d8 -> 17+ and d10 -> 16+.  These are, of course, the sort of to-hit numbers that PCs do their best to avoid having.  And yet, sometimes they occur...  and it's in just those sort of nasty situations that you want to put your target down as fast as possible, and extra damage is most helpful.

Thinking further on this, I begin to believe that our prior piledriver with respect to damage multipliers in ACKS is not as innocent as it initially appears.  We have been handling it as "if you have an x2 multiplier, you roll 2dn and take the total."  This generates a binomial distribution of results, with a 2 and a 12 each occurring one in 36 times, and a 7 occurring one in six times.  Basic probability stuff.  The rules as written instead say that we should be handling it as "1dn * 2".  This generates a uniform distribution, where a 2 and a 12 each occur with probability 1 in 6.

We, being human gamers, are risk averse and dislike the increased probability of rolling a 2 more than we like the increased probability of rolling a 12.  We like that when we get a multiplier, the odds of it being exactly double the average damage for one of our hits is high.  And yet...

I believe there are compelling reasons for using a uniform distribution instead of a multinomial one.  Foremost, it makes thieves and assassins a lot scarier to targets above their weight class!  In the most extreme case, you're a 13th level assassin doing a backstab polearm-charge with Weapon Focus and a natural 20 (we could stack Vorpal on there too, but that's silly - they don't make vorpal polearms!).  You're looking at an x7 multiplier on a d10.  In the multinomial case, the distribution of results is very strongly clustered around the mean of 38.5 (plus 5 damage bonus and some magic), with a standard deviation of 7.6.  Most of the time you'll do about 50 points of damage; as a result, you carry a reasonable threat of instant death to 11th-level or lower fighters.  Against an average 14th level fighter with 63 HP, your chance of an instant kill is a more-or-less negligible 0.5 percent or so, even in such a multiplier-stacked situation.  By comparison, on 1d10*7+5, your chance of an instant kill against such a 14th-level fighter is 20%.  Much scarier for that fighter.

Consider instead a 7th level assassin with x3 backstab, +3 damage bonus, and a d8 weapon in two hands.  Consider also a 10HD adult dragon with 45 HP.  With 1d8*3+3, you have a 1 in 4 chance of taking more than half of its HP in a single hit and forcing a morale roll in the opening round.  On 3d8+3, you will tend very strongly towards 16.5... which is not bad, but probably not decisive, and your odds of forcing a morale roll with 23 damage are but 6.8%.

Backstab is not supposed to be a nice reliable source of bonus damage dice in the style of 3.x sneak attack.  Backstab is supposed to be unreliable spike damage, built for hitting a high-power hard target and either generating a decisive advantage in the surprise round, or leading to an "oooh crap, that didn't work, time to start running" situation, in true thief or assassin style.

It is just in those 16+-to-hit situations against dragons and higher-level NPCs that the backstab and other damage multiplication mechanics give a chance of victory for a lucky underdog.  If you are their equal or better, and can likely win by simple application of fighters, why chance stealth and skullduggery, or leave things up to luck?  One should not.  No, these mechanics, and the classes which use them, are for those black hours when the enemy is stronger than you, the odds are already long, and you need every chance you can get.  If you fail your Move Silent roll, the morlocks will eat you...  but if you don't give it a shot, they'll eat you just as surely.  If you fail your Open Locks roll, you will starve to death in this cell, but to not attempt it has the same consequences.  If you roll a 1 on your backstab damage, the witches of the Bleak Academy will crush your puny domains beneath their beastman hordes...  but if you don't go for the backstab on the suicidal assassination mission deep behind enemy lines, there is no way you and yours will triumph.  And if lady luck smiles upon you enough times in a row, you may yet live to see another sunrise when all before seemed lost.  Thief skills are saving throws for the possible avoidance of otherwise certain doom.  Desperate times beget desperate men, and we shall call them thieves.