Number Crunching

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It’s not a big secret that Leighton and I didn’t do a lot of statistical analysis when we wrote QAGS. After all, the system was originally meant for use in pick-up games when we didn’t have time to spend hours on character creation. We never really intended for it to be a “real” system used for “serious” games (the “real” game system was Anyworlds, the overly complex system I’d written in college). Of course, once we started actually using QAGS, we quickly discovered that it worked just as well as (and in many cases, better than) the more complicated systems we were used to. When it came time to write QAGS 2E, we didn’t want to make any major changes to the system, but we did make a few changes and define the mechanics in more detail to improve the “game” aspect of things a bit. Mainly, though, we just tried to make it clear that Yum Yums were the great equalizer.
Since the math has never been in the forefront of the game’s design, sometimes things can get a little wonky in cases where the players don’t want to spend Yum Yums. Here are a few things I’ve notice or have had pointed out to me:

  1. Since the level of success is based entirely on the roll, the character’s Number has very little to do with how well he does. A character with a Number of 16 has a better chance of succeeding than a character with a Number of 10 (80% vs. 50%), but they both have the same chance of getting a crappy roll (25%, if you consider anything under 6 to be a “crappy roll”).
  2. Default rolls suck. I noticed this recently when running a Roller Girls vs. Zombies games. Since very few of the characters had fighting skills, most were making default rolls to fight, which meant that they missed the zombies a lot more often than they hit them, which isn’t really the way zombie movies generally go. Also, it’s not always clear which rolls should be default and which should be full Body, Brain, or Nerve.
  3. High Body, Brain, and Nerve Numbers are sometimes useless. I’ve noticed this in a couple of games recently where I’ve actually used the Qik Start rules to create a character. For example, in a recent Sector 13 game, I played Brock Sterling: SPACE EXPLORER!, who obviously needed high Body and Nerve Numbers, which meant his Space Explorer Job was only slightly above average. Since a lot of the rolls were based on Job, the high stats in Body and Nerve didn’t mean a whole lot mechanically.
  4. On a related note, there are times when two or more Words could be appropriate for a particular roll. For example, most people use Body when swimming, but if your character’s Job is “Combat Diver,” it makes more sense to roll Job. If the character has a Job Number of 12 and a Body of 15, his chance of success actually suffers due to his extra training, which doesn’t make any sense at all. The Second Chance rule offsets this problem a bit, but it’s kind of a clunky mechanic. Also, if the first roll succeeds, the character with the extra training doesn’t have any edge over the character making a default Body roll.

Problem 1
The first problem is due to the fact that with only one die being rolled, there’s no bell curve to skew the results towards an average roll, and since nothing gets added to the die roll, the die is the final decider of how well an action succeeds. Since we’re not going to give Happy d20 the boot, finding something to add to the roll seems like the best bet. My first thought was to do a table along the lines of the attribute chart in Basic D&D. For example, characters get a +1 if their score is 11-13 (or whatever), +2 for Numbers of 14-15, etc. That gives us an extra modifier to keep up with for every Word, which seems like a lot of trouble just to satisfy the math geeks. Then it occurred to me that my house rule for numbers above 20 could work here. Normally, if modifiers take the number a character is rolling against above 20, they only fail on a 20 and get to add anything above 20 to their Success Degree. For example, if a character has a Job of 16 and a Skill of +5 and rolls an 8, his Success Degree is 9. We can apply that to rolls against lower scores by using 10 as the threshold for the Degree Bonus rather than 20. So, if a character’s adjusted Number (after skills, modifiers, etc are added) is a 15, he gets a +5 Degree Bonus. A roll of “1” is still considered a Quirky Success.

The Math
In order to figure out how this works for us math-wise, the first thing we have to hope my Google crash refresher in probabilities turned up correct information, and that I did the math right. Next, we need to decide what Success Degrees we consider to be good, average, and bad. Given the typical human Number range of 6-16 and standard Difficulty Numbers, I’m going to say that a Success Degree of 5 or less is a poor success, 6-10 means the character did ok, 11-15 is a good job, and anything over 16 is extremely good.

Under the standard QAGS Success Degree rules:

  • A character on the lowest end of the scale (Number of 6) has a 25% chance of doing poorly, a 5% chance of doing ok, and a 70% chance of failure. The best he can hope for is to do a passable job (and that only happens rarely), but then again he’s only a little stronger, smarter, or more socially adept than a grapefruit.
  • A character who is below average (Number 9) has a 25% chance of failure, a 20% chance of doing an acceptable job, and a 55% chance of failure.
  • A character with an average Number (11) has a 25% chance of doing poorly, a 25% chance of doing an average job, a 5% chance of doing a good job, and a 45% chance of failure.
  • An above average character with a Number of 14 has a 25% chance of doing poorly, a 25% chance of doing ok, a 25% chance of doing a good job, and a 25% chance of failure.
  • The pinnacle of human perfection with a 16 has a 25% chance of doing poorly, ok, or well, a 5% chance of doing extremely well, and a 20% chance of failure.
  • The world’s greatest with a 16 Number and a +5 Skill Bonus gets a +1 to his Success Degree and only fails on a 20. That means he has a 20% chance of doing poorly, a 25% chance of doing an average job, a 25% chance of doing well, a 25% chance of doing extremely well, and a 5% chance of failure.

With the Degree Bonus added in:

  • Everyone’s chance of failure stays the same.
  • The grapefruit and the below average character are still just as crappy as before.
  • The average character with a Number of 11 gets a +1 Degree Bonus. That means he does poorly 20% of the time, does an ok job 25% of the time, and does well 10% of the time.
  • The above average character (Number 14) gets a +4 Degree Bonus, so his chance of getting  a poor success is only 5%. He has a 25% chance of an ok success, a 25% chance of doing well, and a 15% chance of doing extremely well.
  • Batman gets to add +6 to his degree. He never does a bad job when he succeeds (though a Quirky Success can cause problems), does an average job 20% of the time, a good job 25% of the time, and an extremely good job 30% of the time.
  • The god-like character with the 16 Number and +5 Skill Bonus gets to add 11 to his Degree, which means he always does at least a good job (20%) but usually (60% of the time) does extremely well.

Problems 2-4
The obvious solution to the Roller Girls problem is to just let characters use Body for fighting, but that doesn’t work because combat was the main reason we instituted the idea of Default Rolls to begin with. If characters are allowed to roll Body for combat, a florist with a high Body Number has a good chance of taking an average Kung Fu Master in a fight. Still, it seems like most characters should have better than a 25-30% chance of successfully hitting a zombie with a hockey stick, so some kind of tweak would be nice. At some point when we were tossing around giant monster rules, somebody (Josh, I think) came up with the idea of letting big critters roll extra dice and pick the best to give them an edge. I think that’ll work here, too and solve the last two issues on the list above as well. Here’s the revised way of dealing with things:

  • The traditional Default Roll (1/2 Body, Brain, or Nerve) is only used when succeeding without any training is highly improbable, even for heroic characters–piloting a space shuttle, performing field surgery, that kind of thing.
  • For anything that an average person could reasonably be expected to do adequately without training, use full Body, Brain, or Nerve. The exact limits of “could reasonably expected to do” will vary from ficton to ficton. In a zombie movie, it would include using basic weapons and firearms. For a super-spy game, it would include a lot more (including the aforementioned space shuttle piloting).
  • For actions that would (under the new rules) allow a full Body, Brain, or Nerve roll, a character who has an appropriate Job or Gimmick rolls an extra d20 for each additional Word that applies and takes the best result. Rolls that are higher than the character’s Job Number but less than Body, Brain, or Nerve (whichever is appropriate) succeed, but do not get a Degree Bonus. If the character ends up using his Gimmick Number (either because his Job doesn’t apply or because none of the rolls was less than or equal to his Job Number), he only gets a Degree Bonus if the Gimmick is specific to the action being attempted. For example, a character with a Gimmick of “Crack Shot” would get a Degree Bonus on firearms roll, but a character with the “Lucky” Gimmick would not. Modifiers, if any, apply to whichever Number ends up getting used, so a character with a Body of 14 and a Job Number of 12 with a -1 penalty will need to roll 11 or less to get a Degree Bonus (of +1), but will still succeed (without a degree bonus) on a roll of 11, 12, or 13.
  • Characters who are rolling Body, Brain, or Nerve (or who choose to use their Body, Brain, or Nerve Number for a multi-die roll) do not get a Degree Bonus. They may, however, add any appropriate Skill Bonus to the Degree.
  • If multiple dice are rolled, a natural “20” on any die means that the action fails, but it isn’t considered a Bad Break unless all of the dice come up 20. This actually means that the character’s chance of failure increases with each die rolled (From 5% to 9.75% for two dice and 14.2625% for 3 dice). However, except at the extreme ends of the spectrum (1, 2, and 18 or greater), the chance of success increases by more.
  • If the die that the character decides to use would normally be a Lucky Break, it still counts.
  • If any die comes up a “1,” the success is Quirky, but the player still uses the best roll for determining Success Degree.
  • These rules take the place of Second Chance rolls, so if you use this scheme,  “Second Chance” is just a show that used to be on Fox.

Here are a couple of examples to illustrate how the new default/dice pool system works:

  • A character with no medical training trying to perform field surgery rolls 1/2 Brain. So if his Brain is 12, he needs to roll a 6 or less to succeed.
  • A character with no combat training who wants to hit a zombie with a baseball bat rolls Body and does not get a Degree Bonus.
  • A character with a non-combat Job but a Skill of +2 in “Fencing” who wants to poke somebody with a rapier rolls Body +2 and gets a +2 degree bonus if the roll succeeds. If her body is 13, she’ll succeed on a 15 or less. So if she rolls a 6, her Success Degree is 8.
  • A character with the “Martial Artist” Job at 13 and a Body of 15 would roll two dice and take the best roll when attacking. If he rolls 13 or less, he gets a +3 Degree Bonus. If he rolls a 14 or 15, he still succeeds, but does not get the degree bonus since he’s relying on instinct (Body) rather than training.
  • If the same Kung Fu master has a -3 penalty to his roll and a +1 “Kick Things” skill, he needs to roll 11 (13 + 1 -3) or less to get a Degree Bonus of +1 (+3 +1 -3). On a roll of 12 or 13 (15 +1 -3), he’ll still kick whatever he’s trying to kick, but won’t get a Degree Bonus.
  • A character with the “Cop” Job at 14, “Crack Shot” Gimmick at 13, and a Body of 11 would roll 3 dice and pick the highest roll that’s under 14. Since his Body is lower than his Job, he’ll always get his Degree Bonus on a successful roll.

Since Gimmicks that could improve a character’s chances are getting added as an extra die, it would be nice to do the same kind of thing with Weaknesses. This is possible, but a little trickier. Whenever a Weakness might hinder a character’s ability to do something, he designates an additional die as a Weakness die when he makes his roll. If the Weakness die is less than or equal to the character’s Weakness Number, he subtracts the roll from his Success Degree. So, if a character has a Body of 13, the “Rassler” Job of 12, and a “Clumsy” Weakness at 11 wants to Rassle an opponent, he would roll three dice, designating one of them as the Weakness Die. If he rolled a 5 on the Weakness Die and 15 and 11 on the other two, his Success Degree would be 8 (11 +2 Degree Bonus -5 Weakness).

The Math
According to my crash course in statistics, to find out the chance of rolling a particular number on either of multiple dice, you work out the chance of not getting that result on each die (expressed as a fraction), multiply them together, and subtract from 1. I’m going to go with that.

  • A character with a Body, Brain, or Nerve Number of 12 trying to do something really hard still has a 30% chance (6 or less) of succeeding. The best he can hope for is a success on the low end of ok (6).
  • A character with a Body of 11 trying to whack a zombie with a baseball bat has a 55% chance of hitting. There’s a 25% chance that this will be a poor success, 25% chance it’ll be ok, and 5% chance it’ll be a good hit.
  • If the same character has a skill in “Baseball” at +2, his chance of success goes up to 65% and the resulting hit will be poor 15% of the time, ok 25% of the time, and good 30% of the time.
  • A character with a Body of 12 and a “Zombie Fighter” Job of 11 rolls 2 dice and will successfully whack the zombie 84% of the time, which gives him a definite edge over an untrained person with an equal Body Number (who has a 60% chance to hit). His chance of rolling a poor success is 36%. His chance or rolling an ok success is 43.75%. His chance of getting a good success is 27.75%.
  • The Cop with Body 11, “Cop” at 14, and “Crack Shot” at 13 would roll 3 dice and succeed 97.3% of the time. His chance of doing poorly is 14.2625%. His chance of an ok or good roll is 57.8125%. His chance of doing very well is 38.5875%.
  • Just to see what happens on the ludicrous end of the scale, let’s see what happens with Green Arrow. We’ll give him a Body of 15, a “Robin Hood-Themed Super-Hero” Job at 16, the “World’s Greatest Archer” Gimmick at 16, and an Archery Skill of +5. According to the standard math, he’d succeed 99.97875% of the time. However, he’s got a 14.2625% chance that one of the dice will come up a 20, so his actual chance of success is more like 85%–about the same as a person rolling against a 17 (which is still above the normal human maximum). To balance it out, though, his chance of a Bad Break (all 3 dice coming up “20”) is only .000125%. With his incredible degree bonus of 11, so he’s got a 48.8% chance of a good shot and a 98.4375% chance of a very good shot. And that’s why they call him the World’s Greatest Archer.

Let’s see what this does to fight sequences:

  • Round 1: Roller Girl Rainbow Blight has a Body of 12 and is fighting  a zombie who also has a Body of 12. However, since trying to eat people is sort of what zombies do, the GM rules that he gets to roll an additional die for his “Zombie” Job of 11. If Rainbow rolls a 2 on her attack roll, the zombie has a 79.75% of beating her. If she rolls a 6, the zombie will get the better of her 57.75% of the time. If she rolls a 12, the best the zombie can doe is tie (roll of 11 or 12), which will happen 19% of the time.
  • Round 2: Switchblade Suzie is also fighting zombies. She’s also got a Body of 12, but she’s got a “Bar Room Brawl” Skill at +2. If she rolls a 2, the zombie will only have a 69.75% chance of getting the better of her. The zombie has a 36% of winning if Suzie rolls a 6, and if she rolls a 12-14, she’s going to do damage to the zombie no matter what he rolls (since she gets to add her Skill Bonus to the Degree).
  • Round 3: Moving away from the Roller Girls, lets see what happens when Oliver Queen takes a shot at a random bad guy with no appropriate Job and a Body of 14. If the thug rolls 11 or less, Ollie’s going to hit him unless a 20 comes up. If the guy rolls a 14, Ollie’s still got a 98.4375% chance of winning, but there is that 14% and change chance that one of Ollie’s dice will come up a 20. So if the mook makes his best possible dodge roll, there’s still basically an 85% chance that Ollie will hit him. Again, “World’s Greatest Archer.”
  • The math for working with multiple people rolling multiple dice is way over my head, so I’ll leave that to somebody with a scientific calculator. Same goes for the Weakness Die idea.

As usual with rules ideas in this column, none of this has been playtested. Will it work? Who knows? The Degree Bonus seems like it will make trained characters slightly less erratic in their performance, but could potentially pump their final results up too high. If that happens, you could raise the threshold for a Degree Bonus from 10 to 15, but that would require actual subtraction instead of just taking the ones digit of the number you’re rolling against. The modified default rules give untrained characters a fighting chance, but make sure that trained characters still have an advantage due to the extra die (or dice). The biggest potential flaw with the dice pool is that characters who get to roll multiple dice have a much better chance of succeeding than under the standard system. Even if both Numbers are 11, the character will succeed just under 80% of the time (90% of the time with 3 dice at 11). For simple rolls that actually makes sense–an average person with training should usually succeed at tasks that aren’t especially difficult (ie, no DN). Even a modest DN of 5 knocks the chance of success down quite a bit:  36% for characters rolling 2 dice, 58%(ish) for characters rolling 3 dice. In the standard system, the chance of beating a DN 5 with a Number of 11 was 30%.

The other consideration of this set-up is whether it will make skills too valuable. One of my early ideas for counteracting the lack of a bell curve was to let players add their Skill Bonus to their success degree. Since this would only help characters with Skills, this would inevitably lead to the kind of skill laundry lists that I’d just as soon avoid. It also didn’t help characters making Job rolls unless they duplicated job skills with Skill Bonuses. The system above still makes skills a little more useful when character’s are making Body, Brain, or Nerve rolls, since it lets them add the Skill Bonus to their Success Degree. However, I think that’s ok; People who actually know something should on average be better at it than the untrained.

For Job-related skills, we need to consider the costs. The costs for Skill Bonuses at character creation are 4 for +1, 5 for +2, 6 for +3, 7 for +4, and 8 for +5. Since each point of Job costs 2 Yum Yums at character creation, a character could increase his Degree Bonus for all Job rolls (by raising his Job Number) for less than it would cost him to get a +1 or +2 Skill. For a +3 Skill, the cost would be the same as raising the Job by 3, so it makes more sense to put the points in Job. For +4 and +5, the character saves 1 and 3 Yum Yums respectively by raising the Skill Bonus instead of the Job Number. Given the small difference in cost, it seems to me the player would be better off spending the extra Yum Yum (or 3) to increase his Job Number (and therefore Degree Bonus) across the board unless he specifically wants the character to be an expert in a particular aspect of his Job. Overall, I don’t think letting the Skill Bonus serve double duty by increasing both chance and quality of success increases their importance so much that players will be spending all their Yum Yums on Skills.

I think I’m going to try these rules out in my All-Stars game. Give them a try yourself and let me know how things work out in the forums. You can also let me know where the math is screwed up (and I’m sure it is–I suck at math) if you’re into that kind of thing.

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