Compromise and Conceit

Infernal adventuring…

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An Australian statistician and role-player living in Japan, interested in the theory and practice of role-playing…

  • Simpler Methods for Resolving Combat Actions in Warhammer 3

    Continuing my series of posts about the unnecessary complexity of Warhammer 3rd Edition (WFRP3) combat and skill resolution, today I want to focus on the construction of dice pools in combat. I have already shown that action cards may not provide much benefit in combat, and I have also explored an alternative method for setting skill difficulty, and today I want to explore the possibility that the combat system involves unnecessarily complex dice pools with limited value.

    The standard method for handling defense in WFRP3 is divided into two parts: action cards add 1-2 black or purple dice to the dice pool, armour adds one black die per point of defense, and the attacker can add fortune dice through the use of talents, fate points and other types of enhancement. Furthermore, the basic difficulty of all attacks is 1 challenge die, with some cards having additional challenge and/or misfortune dice. Thus a starting warrior with strength of 4, one point of training, 1 fortune die on strength and a talent that gives an additional fortune die will have a basic attacking pool of 4 blue, 1 yellow, 2 white; against a target defending (+1 misfortune) and wearing lightish armour (+2 defense) the final dice pool will be: 4 blue, 1 yellow, 2 white, 3 black, 1 purple. The number of black and white dice can get quite ridiculous at higher levels: it’s quite possible that an action card will add 2 black, the defender will chuck in 2 black from cunning points, and the attacker will then throw in 2 or 3 whites from blessings, fate points and other situational benefits.

    My question is whether all these extra white and black dice can be just cancelled out, so that the dice pool ends up with the final number of excess black/white dice. This would be particularly useful for higher levels and more complex fights, and hints at a language of skill challenges that is much simpler to express. To explore this possibility, I simulated 10,000 attacks with a basic melee weapon for a fighter of strength 3-6, and checked the average damage and success rates, using two different methods of dice pool construction. In one method, black and white dice were added to the pool and rolled together; in the other, only the net number of dice was added. For all attacks the defender was assumed to be defending actively, with 2 points of armour defense (total defense 3); the attacker had 2 fortune dice. I assumed a total soak of 0 so that I could calculate pre-soak average damage, and used a hand weapon to calculate damage. Table 1 shows the mean damage delivered and the chance of success for both methods of calculating the dice pool, for the four strength values.

    Table 1: Outcomes from two dice pool construction methods, basic Melee Attack

    Strength Success probability Mean damage
      All dice Excess dice All dice Excess dice
    3 0.51 0.52 4.50 4.50
    4 0.63 0.65 6.30 6.40
    5 0.72 0.75 8.14 8.43
    6 0.80 0.84 10.09 10.46

    It should be fairly clear that there is very little difference between the two methods, and that even at very high strengths the difference in damage is minimal (less than 0.5 wounds on average). The same differences in probability of success would also apply to probability of observing at least one boon (since boons and banes cancel on black/white dice in equal measure with success/failures).

    Repairing combat hit probabilities

    Note also the huge increase in chance of hitting as strength increases – and this is without adding additional training or reckless/conservative dice. In reality a strength 6 fighter will have additional training and fortune dice, and will be close to a 100% chance of hitting in combat against someone with a standard defense card and armour. This high probability of hitting is also independent of the target’s physical characteristics: the only way a standard PC can up their defense is to get better action cards and to buy better armour. In WFRP3 the only skill check that is largely independent of the target’s attributes is the key attacking check!

    I think this could be fixed easily by making the difficulty of hitting a target dependent on their physical attributes. We can introduce a simple language for converting difficulty into dice pools, and generate difficulties as follows:

    Target difficulty=attribute+defense-total fortune

    This can then be converted into dice pools by dividing by 2; the result is the number of challenge dice, and the remainder the number of misfortune dice. For combat, the base attribute can be agility and people can swap this for toughness or strength if they have a suitable talent and they are carrying a shield and heavy armour (toughness) or a weapon (strength).

    In combat, for a person with agility 3 this is equates to the same difficulty as would occur in the standard system when they have the dodge action card. A person with agility 1 would actually be easier to hit than in the current system, but such people basically don’t exist. A fighter with agility 4 would be as hard to hit as a fighter with advanced dodge in the current system. This would be particularly liberating for the GM, since he or she could essentially dispense with tracking aggression and cunning, as well as defense cards for everyone. Although the increasing difficulty of attacks would mean combat took more rounds, the reduction in management (of cards, recharge and dice pools) would significantly speed up each round.

    This change would also put magic and combat on a more equal footing. Many magic attacks are challenged by the target’s attribute, which means that in general their difficulty is likely to be higher than 1 challenge die. Since magic often does less damage than combat attacks, this significantly reduces its effectiveness.

    With these considerations I think I have now developed a rounded idea of how WFRP3 can be simplified into a streamlined high fantasy system. Now I simply need to put it all together in order to start using it.

  • The False Assurance of Western “Critical Thought”

    Today’s Guardian has some new notes on the ongoing scandal that is the British education system. This time it’s a new OECD report ranking countries by numeracy and literacy, and the United Kingdom has fallen near the bottom. Worse still, the study finds that on average 16-24 year old Britons perform worse on both numeracy and literacy than do 16-55 year olds – that is, educational achievement has gone backwards in recent times. The depth of failure is also astounding:

    a quarter of adults in England have maths skills no better than a 10-year-old, a conclusion that also prompted a political row in which the Conservatives attacked Labour’s record in government.

    That means an estimated 8.5 million adults are only able to manage one-step tasks in arithmetic, sorting numbers or reading graphs. The same body also concluded that one in six adults could only just decipher sentences and read a paragraph of text – the literacy level of a child in their final year of primary education.

    This is a pretty disturbing indictment of the British education system. The rankings also show it is under-performing relative to other English-speaking nations, with Australia and Canada out-performing the UK on every measure and the US close behind the UK. South Korea is top in numeracy and Japan top in literacy, which finding is particularly staggering given that literacy in Japanese requires a huge commitment of time and effort just to learn the vocabulary in comparison with English. The UK government is trying to blame Labour, pointing out that a 24 year old tested by this report would have spent their entire education under Labour, but I think that’s a little simplistic – education systems are slow to shift, and education methods, infrastructure and workforce obviously have legacy affects that would strongly influence outcomes long after the government that set them has disappeared into the trash bin of history. The Guardian is taking a more nuanced approach, attempting to understand what it is about education policy in Japan that makes Japanese students so good. It makes the good and obviously alarming point about differences in attitude towards education between the countries:

    Japanese senior high school teachers, and their pupils, are often incredulous when they learn that 16- to 18-year-olds in England can drop maths and literature and study just three A-level subjects of their choice.

    Add me to the ranks of the incredulous. When I was finishing high school you had to do five subjects. What else would be reasonable? And to the best of my knowledge I could only drop maths in my final year, and had to do one science and one humanities amongst my five subjects. What do English students do with their time?

    This article, however, also brings up the common criticism of Japan’s education system – in fact it brings it up twice – and presents this criticism as some kind of counter-balance to the system’s strong focus on rote learning and hard work. The article states:

    Japan’s state education system is often criticised for quashing original thought among pupils in favour of rote learning, and for placing an emphasis on theory rather than practical skills …
    The stress on memorising information and passing exams, which begins in primary school and continues through to senior high, has been blamed for stifling critical, independent thought

    This is a personal bug bear of mine, and something I find really frustrating about western coverage of Japan in particular and of Asia generally, for two reasons: it exaggerates the extent to which western students learn “critical thought” and it valorizes western “critical thought” as something that somehow counter-balances ignorance, or has some kind of value separate from the basic knowledge and skills required to inform critical analysis.

    In terms of exaggeration, I remember growing up in the Australian school system, entering university, and interacting with peers during that period, and I can’t say that between us we had a shred of critical thought. We all failed essays at university and had to be taught a whole bunch of things about analysis and critical thinking skills, and university tutors in the humanities will often talk about how the students they get in first year are just repeating rote what they learnt from parents and peers. So the idea that western schools are a haven of critical thinking strikes me as a little exaggerated. Yes, high school students in the west spend more time spouting their opinions in essays than Japanese students, but so what? I’m sure that lots of British students have spent time in the library photocopying their arsehole, but that doesn’t mean they’re good at art.

    But more importantly – and the reason this annoys me – critical thinking is a complete waste of time, and can even be counter-productive, if it is alloyed with ignorance and an inability to read. Let’s review the facts about one in six adults in the UK, who could “only just read a paragraph of text.” Why don’t we slap down the IPCC summary for policy makers in front of one of these adults and ask them to critically analyse it. Are they going to produce an analysis with any critical value, no matter how well they learnt to spray their opinions at school? I don’t think so – especially if they have maths skills no better than a 10-year-old. Perhaps it might be better if these adults were first able to understand the IPCC summary, before they embarked on a critique. Indeed, it might be better if these adults refrained from criticizing things they can’t read, because if you don’t understand something it’s likely your critical thinking about it is going to be of little value. You cannot present “independent, critical thought” as a boon independent of the skills that underlie basic comprehension, because one depends on the other. This isn’t to say that both can’t be taught in school, but it’s clear that the UK and US are not doing that. If you teach “critical thought” without teaching the skills it depends on, what you are actually teaching is rhetoric: the ability to bend facts to support your pre-conceived ideological goals. That this is taught in UK schools is not a positive thing.

    Critical, independent thinking is not actually a hallmark of western culture: spouting opinions is. If we are such good critical independent thinkers, how come we got lied into a war in Iraq, participated in the massive con that was the housing bubble and the GFC, still haven’t come up with a solution to global warming, and managed to wage the biggest and most disastrous war in human history (WW2). Is it possible that what we see is a virtue is actually a flaw? Or, more likely, we aren’t doing it at all? After all, the land of limited independent thought, Japan, has a low crime rate, high employment, little inequality, and has a strong opposition to engaging in any form of war. They have an economy much larger than their population would be expected to have, exert a significant positive influence in the world, and make all the stuff you use even though they have no resources to speak of. Perhaps an education system that doesn’t focus on “independent, critical thinking” is more beneficial to society than one that does? Or perhaps the West is so full of its own opinions that it mistakes ranting for thinking?

    This article’s platitudes about critical thought might go down well with educated British readers, but to me they’re just another example of the standard rhetorical footwork employed by journalists about Japan: on the one hand, a weak and stereotypical assessment of Japanese as conformist; and on the other, a triumphalist reassurance that westerners are all free-thinking individuals. Both of these two steps in the movement are wrong, and the underlying assumptions about the value of critical thinking to a functioning society, as well as the facts about how prepared western school leavers are to engage in such thought processes are also deeply flawed. A little more nuance would be nice.

    Also of passing interest in this debate that the UK will now have with itself over its education policies is the role of inequality, and the relative benefits of development compared to birthrates in preparing for the future. How can the education levels of young adults in the UK be going backwards at the same time as average GCSE scores are going up? One answer, readily deployed by conservatives, is “grade inflation.” The other answer is inequality: that if you looked into the background of that “one in six adults” you would find they were much more likely to be poor and from certain areas. Japan, of course, has very little inequality compared to the US and the UK, and Australia and Canada are much more equal than the US and the UK. Interesting how the rankings seem to reflect the inequality within these countries. Also, if one in six of your young adults lack basic literacy and one in four of your adults lack basic numeracy, I think it’s safe to say that you have a problem with your workforce, and no industrialized, developed nation can hope to maintain its economic and cultural development with this kind of lack of investment in its workforce. Although England has a higher birthrate than Japan or South Korea, which country has the larger number of suitable new entrants to its workforce? Who is better placed to maintain a high-skilled pool of workers? The UK with something like 20% of its workers incapable of even basic office duties, or Japan and South Korea? Maintaining birth rates is not the be-all and end-all of maintaining a sustainable social order, especially if a large minority of all those born are going to grow up to be completely unable to contribute to the economy. British policy-makers need to be looking at the long-term implications of their education and industrial strategies (such as they have any) if they want to maintain anything resembling the quality of life that modern industrialized economies have come to expect.

    Perhaps they could start by reassessing what they consider to be educational priorities, and trying to look beyond party-political point-scoring. “It’s Labour’s fault” is hardly a sterling example of the “critical thought” that UK policy-makers supposedly learnt at school. But then, maybe it’s an alternative when you don’t have the skills to read the report …

  • Are Action Cards Important in Warhammer 3 Combat?

    Following my analysis of success probabilities in Warhammer 3rd Edition (WFRP3) my next task is to analyze some of the major action cards, and identify whether fiddling with action cards brings any particular benefit to the game beyond different names for attacks. Before I do, I should note that there are only really a few different kinds of action cards:

    • cards which appear to do more damage (like Thunderous Blow and Troll-feller Strike), usually with extra risk
    • cards which enable the PC to use a different skill to attack with (e.g. Chink in the Armour, Nimble Strike), sometimes with less damage
    • cards which induce some kind of combat-beneficial circumstance (e.g. Cut and Run) or cause an ongoing condition (Cruel Strike)

    I think the second type of card are obviously worth having, since they enable PCs with poor combat traits to occasionally engage in melee attacks. The last kind of card may also be valuable, depending on the benefit they give the player or the condition they induce; but often the benefit is small or could be easily handled by sensible GMing (e.g. disengage for free). I think many of the effects given in these cards could probably be made available to PCs as talents with no loss of complexity or great unbalancing of the system. For example, Chink in the Armour enables a PC to use their Observation skill to attack. We could probably make this a talent available to a wizard if they want to spend the experience points on it, but it would be unlikely to unbalance the wizard class – no wizard can slug it out for more than a round in melee combat against anything nastier than an orc, and giving them the ability to use their observation skill to attack isn’t going to help if they can’t defend and don’t have armour or toughness worth speaking of.

    My question is whether the first kind of card – the ones that supposedly enable fighters to do extra damage with savage attacks – is worth using. I investigated this by simulating 10000 implementations of the Basic Melee Attack and the Thunderous Blow action cards, for fighters with strength scores ranging from 3 to 6, in both reckless and conservative stance (one deep). I chose Thunderous Blow because it has side effects (fatigue) and (at least in reckless stance) is potentially savage, enabling the fighter to double their weapon damage if they roll well.

    For all simulations in both stances for both cards I calculated the probability of successfully hitting and the average damage done (for all hits, not just successful hits). I then expressed the difference between the cards in two ways:

    • The Odds Ratio of a successful hit for the Basic Melee Attack relative to the Thunderous Blow card; that is, the odds of hitting with basic melee divided by the odds for the Thunderous Blow. This should be greater than 1, since the Thunderous Blow card is slightly more difficult
    • The difference in mean damage done between the two attacks; negative means the Basic Melee did less average damage, positive means more

    For all attacks the fighter had a great weapon (7 damage), one fortune die and one rank of training; and the enemy had defense of 2, soak of 6; and was assumed to be parrying (+1 defense). Fatigues were calculated but are not shown here.

    For all attribute scores (ranging from 3 to 6), the odds ratio of a successful hit was almost exactly 1 in reckless stance, and only slightly below 1 (usually between 0.9 and 0.95) for conservative stance. This indicates that the basic melee attack is basically just as likely to hit as the Thunderous Blow, though the Thunderous Blow supposedly does more damage. Figure 1 shows the difference in mean damage for reckless stance (black line) and conservative stance (red line). This means that in reckless stance Thunderous Blow does more damage (negative difference) on average, while in conservative stance it actually does less damage.

    BasicMelee vs ThunderousBlow

    It is clear that the difference in damage in reckless stance is not great, and the benefit of hitting slightly more often in conservative stance does not make up for its weaker damage. In reckless stance the difference in damage across attribute values is not large, and probably not worth the risk of extra fatigues that are inevitably incurred in this stance with this card.

    This analysis suggests that the fluff and crunch of having an extra combat card doesn’t deliver much benefit to the player. This card can be deployed once every three rounds for an extra 0.6 – 0.8 wounds of damage, at the risk of extra fatigue; or for less damage and the risk of delay in conservative stance. Is it worth spending an xp point on? As an alternative, this fighter could have spent that 1 xp to get this action card on either a fortune die for an attribute; an advanced parry card; an extra wound; or a talent that would deliver a constant and significant benefit in combat (talent cards can be pretty good). This card also requires you to give up a shield (it requires a two-handed weapon); it’s likely the benefits would be even smaller for similarly “reckless” and “beneficial” cards that applied to a one handed weapon.

    This result is an example of several problems that I think arise from action cards:

    • They constrain the GM’s creativity: in responding to the rich range of options provided by the dice pool system, the GM is able to come up with all sorts of interesting outcomes (these are hinted at on page 55 of the player’s handbook); however, the cards tie the outcome of dice rolls to strict effects that really in the end could just be summarized as “+1 damage” or “you get a free manoeuvre.” Thus a lot of effort goes into building dice pools for limited benefit
    • They are unbalanced and unrated: most combat action cards have no rating but, for example, the Rapid Fire card is awesomely vicious – you can kill a great many PCs with that card – while the Thunderous Blow card does an extra point or two of damage and the two weapon cards are weak. Combat action cards need to be rated like magic cards, but they aren’t; and many are just fancy names on a small amount of additional damage
    • They squeeze out talents: a PC can hold as many cards as they want, but can only slot two talents at a time. So players have to choose action cards of limited benefit, while missing out on talent cards that could really reward them

    I think then that a better solution would be to give each character class a small number of usable actions, probably support action, that are deployed more like spells; for example the thief could have “assess the situation” which is actually really effective; while the fighter could have some kind of leadership or defense card. Then all other benefits gained with increasing xp could be expressed as talents that reflect bonuses, outcomes and new success lines that the PC can deploy in normal rolls. There could then be a system in which fighters are able to take a fatigue to add a fortune die whenever they want to any attack; and similar benefits for other classes in other ways. This would make PC management simpler without significantly affecting the total level of violence that any one PC was able to direct during battle. It would also remove the complexity of recharge tokens, and make character management enormously simpler. This can all be achieved by stripping WFRP3 down to a system like the (related) Star Wars system.

    As it stands, WFRP3 has very poor management of difficulty levels and bad probability distributions, and the cards aren’t much value. I still really like the dice system, but I think the way difficulty is conceived and the probabilities of success that derive from this, as well as the action card system, could be significantly improved. From here I am going to begin developing methods to improve these aspects of the game.

     

  • The Meiji Jingu Doll Appreciation Ceremony

    And you thought an army of dolls must be sinister ...
    And you thought an army of dolls must be sinister …

    Today I visited Meiji Jingu to attend one of the stranger rituals I have seen in Japan: the Doll Appreciation Ceremony (oningyo kanshasai,お人形感謝祭). This ceremony takes place once a year under the sponsorship of the Doll Appreciation Society, and is held to venerate those dolls that are being passed out of use. The information we received when we entered the shrine told us that many Japanese people believe that dolls have souls, and that many shrines hold ceremonies to venerate or consecrate dolls when they are thrown out. The basic process is you bring your dolls (as many as you want) in a plastic bag, and get charged 3000 yen (about $US30) per bag. You are given a doll-shaped piece of paper to write a message on, and then the priests very carefully arrange the dolls within the shrine precinct. Once the viewing period is over, the good wishes you wrote on the piece of paper are incorporated into a ritual of consecration, and then the dolls pass on. It’s not clear what happens, but I’m guessing they go into landfill. But appropriately consecrated.

    Were sacrifices ever cuter or more numerous?
    Were sacrifices ever cuter or more numerous?

    This is a very sweet idea for a ceremony, and very well attended by people of all ages and descriptions. The dolls were arranged carefully in the eaves of the inner wall of the Meiji Jingu main compound, and there were so many by 2pm today that they took up two whole sides of the compound (which is not small). Interestingly, the shrine workers laid them out very carefully and respectfully, with that careful attention to detail that Japanese people always apply to any task that they consider responsible for.

    The "traditional" section: new year's animals and hina matsuri dolls. Disney was on the top shelf!
    The “traditional” section: new year’s animals and hina matsuri dolls. Disney was on the top shelf!

    They also arranged the dolls in related categories as much as they could. There were whole sections full of Disney characters, a huge swathe of hinamatsuri dolls, squads of Doraemons and Winnie the Poohs, and of course lots of animals. Even toy soldiers and lizard souvenirs from museums were on display.

    The bears await their fate
    The bears await their fate

    I think one year a group of war-gamers should get together all their old and unwanted warhammer figures and bring them along. Then they could watch as the shrine workers meticulously laid out huge armies of skaven and undead, knowing that after years of war and struggle those exhausted soldiers would be vouch-safed eternal rest, protected in the bosom of Japan’s 8 million gods: surely a worthy end for every war-gamer’s most dedicated heroes!

  • Comparing Dice Pool Mechanisms in Warhammer 3

    Recently I have been examining dice pool mechanisms in Shadowrun, to compare two methods for resolving opposed skill checks. In those posts I have found that for opponents with equally matched skill the probability of success tends to nearly 50% as skill increases, and that skill checks based on target numbers lead to sudden changes in success probability due to rounding error. In this post I thought I would examine the same problem in Warhammer Third Edition (WFRP3).

    WFRP3 also uses a dice pool system, but it is much richer than other dice pools, being composed of seven different kinds of dice. It also doesn’t use the same dice for attacker and defender: the attacker adds some purple “challenge” dice to his or her dice pool, with the number dependent on the target attribute of the defender. The standard rule for determining this number in WFRP3 is:

    • Defender’s attribute is less than half the attacker’s: 0 dice
    • Defender’s attribute is less than the attacker’s: 1 dice
    • Defender’s attribute equals the attacker’s: 2 dice
    • Defender’s attribute less than twice the attacker’s: 3 dice
    • Defender’s attribute more than twice the attacker’s: 4 dice

    This leads to some obvious problems: if you have an ability score of 8 and your target has an ability score of 8, the difficulty of your attack is 2 challenge dice; but this is the same difficulty if both of you have attribute scores of 4. So as your skill increases, your chance of success against someone with your own skill level increases markedly. Also, if you have an attribute score of 2 you will face the same difficulty on your check for all opponents with a score of 4 or more. You have the same chance of success whether your opponent is just slightly above average (4) or of god-like power (10).

    I have considered two alternative ways of setting the difficulty based on the defender’s attribute: a number of challenge dice equal to half the attribute rounded down; and a similar method, but with the half value converted into black dice (so that someone with an attribute of 4 gives 2 challenge dice; while someone with an attribute of 5 gives 2 challenge and one misfortune dice). I have simulated the results of 10000 challenged skill checks – using only attribute dice – for skills from 2 to 6, against various defender attributes, using all three methods.

    Figure 1 shows the probability of success using the standard rules described above, i.e. with difficulty set by comparing attacker and defender attributes. The high probability of success regardless of defender attribute is obvious for large attribute values, and the plateau effect at higher defender attributes is also visible.

    Figure 1: Probability of success for various combinations of attributes, standard rules
    Figure 1: Probability of success for various combinations of attributes, standard rules

    For an attacker with an attribute score of 6, success is highly likely (about 80% chance!) even against targets with the very high attribute score of 8. Conversely, a wimpy attacker with an attribute score of 2 can be expected to be successful against anyone with attribute of 4 or more about 10% of the time – even if their attribute is 8. Remember, in WFRP3 a score of 8 in an attribute is almost impossible for a human, and mostly the province of giants and dragons. This means a party of 1st level mages could attack a giant and actually do physical damage against it! And this is before including stance dice, training, etc. A human with an attribute score of 6, a fortune die on that attribute, and two ranks of training could reasonably expect to hit a much more powerful opponent pretty much every time, unless that opponent burns through defense cards, cunning, etc.

    Figure 2 shows the probability of success for various combinations of attacker and defender attributes using a system in which difficulties are set at one challenge die per 2 points of attribute.

    Figure 2: Success probability for difficulty set at half target attribute
    Figure 2: Success probability for difficulty set at half target attribute

    This chart shows that probability of success declines with increasing target attribute score for all levels of the attacker’s attribute. It also doesn’t show the jagged pattern arising from rounding error that we saw for target numbers in Shadowrun or Exalted; rather, it plateaus for odd attributes. Note the generally high probability of success; a person with attribute of 6 can expect to beat someone with attribute of 8 about 80% of the time. This could be easily adjusted by making the base difficulty of all checks 1 challenge die; then all success probabilities in this chart would shift two steps to the right.

    Figure 3 shows the probability of success when we eliminate the rounding effect by turning half points of attribute into misfortune dice. Under this system, the remainder from dividing the target attribute by 2 is turned into a misfortune die. The overall pattern is similar to that of Figure 2 but we see a smoother trend with rising ability.

    Figure 3: Success probabilities without loss due to rounding
    Figure 3: Success probabilities without loss due to rounding

    This is a very smooth success curve, with somewhat high overall success probabilities and no unexpected values due to rounding error. Furthermore, the probability of success against someone of equal attribute score decreases as attributes decrease, which I guess is what one might expect as one watches increasingly amateurish people trying to thump each other; in contrast, in Shadowrun and Exalted this probability tends to 0.5 as skills increase.

    I think then that my final recommendation is to set difficulty for skill checks at 1+(defender attribute)/2, with the remainder from the division converted to misfortune dice. This will reduce the success probabilities compared to Figure 3 but retain the smoothness and other properties shown in that chart. For games where you want the PCs to have lots of success, make the base difficulty 0; for really challenging, gritty games make it 2.

    By setting difficulty in this way and using challenge dice that are different to the attack dice, the WFRP3 system is able to generate a sophisticated and realistic set of probability results. Unfortunately, the method for setting difficulty provided in the original rules doesn’t take advantage of these properties at all, and should be revised.

  • Why do so Few Women do Science?

    The New York Times has an interesting and thoughtful article asking why so few women do science, a topic somewhat related to questions sometimes asked on this blog about women and role-playing, and dear to my heart since I graduated in physics and now live in Asia, where science is cool. Why do the English-speaking countries have a problem with women doing science?

    The article has attracted 671 comments, which shows that the topic is of interest to a lot of people, and the author herself gives a strong example of why any form of barriers to participation in science are wrong. She studied physics, so in preparing the article she returns to her old notebooks, and she writes

    The deeper I now tunnel into my four-inch-thick freshman physics textbook, the more equations I find festooned with comet-like exclamation points and theorems whose beauty I noted with exploding novas of hot-pink asterisks. The markings in the book return me to a time when, sitting in my cramped dorm room, I suddenly grasped some principle that governs the way objects interact, whether here on earth or light years distant, and I marveled that such vastness and complexity could be reducible to the equation I had highlighted in my book. Could anything have been more thrilling than comprehending an entirely new way of seeing, a reality more real than the real itself?

    As someone who didn’t have what it takes to continue in physics, but really enjoyed my third year of study and really loved the topic, I can only say that it’s wrong! wrong! wrong! to construct any barriers that would prevent someone capable of exploring that world from so doing. And the article identifies a huge range of barriers that still exist to women trying to enter science. Despite these barriers, the statistics that the author quotes are reassuring for those of us who graduated from physics in the ’90s:

    Only one-fifth of physics Ph.D.’s in this country are awarded to women, and only about half of those women are American; of all the physics professors in the United States, only 14 percent are women. The numbers of black and Hispanic scientists are even lower; in a typical year, 13 African-Americans and 20 Latinos of either sex receive Ph.D.’s in physics.

    I think I also read somewhere once that there is a Native American professor of physics (I could be wrong, this is a very vague memory). In comparison: when I was studying physics there were no women in my year, and none had preceded me. In the year after me was a single woman, and we young idiot men as we were had already decided to interpret her tiger-skin mini-skirts and low-cut blouses as proof that she was “taking the easy way” and trying to impress the profs with her body[1]. How much has the field improved in the intervening 20 years!

    The author also points out that there is a basic problem in the interpretation of femininity and its acceptance in English-speaking academia. She cites a scientist who worked in Europe, who states that women from France and Italy

    dress very well, what Americans would call revealing. You’ll see a Frenchwoman in a short skirt and fishnets; that’s normal for them. The men in those countries seem able to keep someone’s sexual identity separate from her scientific identity. American men can’t seem to appreciate a woman as a woman and as a scientist; it’s one or the other.

    This is also my experience in Japan. In Japan it is acceptable for women to be professional or experienced and feminine; it is not a case of either/or, and people are simply impressed that a woman is feminine and skilled – or even they take it for granted that a woman with technical skills will also be well dressed, elegant, womanly, etc. There isn’t the same sense that being feminine is a sign that one is unserious. While in the west femininity is seen as a kind of performance that young women do to pull a mate, and therefore somehow false or deceptive (though expected), in Japan it is just seen as a part of being a woman, not an accoutrement of femininity so much as a part of its essence. There is no expectation that women will walk away from their femininity in order to be taken seriously as scientists. And women’s place in Japan is judged on the basis of their position more than their sex. The way I have come to think of this does not reflect positively on the west: Japan has sexism, but the west has misogyny. There is a deep-seated fear and hatred of women in western culture, while in the east there is a strict set of roles. And in amongst those roles, women are allowed to be scientists. Or at least, that is my impression. This western fear and hatred of women is declining, of course, as we grow up and reject a fundamentally misogynist religious history, but it is still there. The article describes a much more subtle and weaker form of sexism though, that pervades the sciences and makes the task of women just that little bit harder than that of men; and making science just a bit harder means making it inaccessible to mortals, because doing science is difficult at the best of times. You don’t need people denying you lab space, salary and funding, especially on top of the inevitable requirement that young scientists move through several countries as part of the process of building their career. But that is what happens: straight out old-fashioned discrimination.

    There are also subtler cultural factors at work: lack of encouragement, and the continual claim that women are not as smart or as talented as men. The writer of the article experienced both of these directly and speaks to other women who had the same problems. It’s a fascinating insight into how a million tiny cuts can drive a person away from a goal, and how those million tiny cuts can be strongly gendered. You may think you’re the first person in history to make an unsavoury joke about women in your engineering course; but to the woman you are talking to, it’s just another day on the frontlines. This kind of stuff adds up, and then women get to the decision point where they are looking at years of hard work, low pay and really, really difficult problems, and with that background of discrimination and discouragement they just think, “fuck it!”

    That’s why there aren’t many women in science. It’s a fascinating article, and well worth reading for people outside science too. It really describes openly the subtle ways in which gender bias works to alienate women from a field. And this is obviously relevant to role-playing – a hobby where in the west there are very few women, but in the east there are many more, and for many of the same underlying reasons.

    And obviously, excluding women from role-playing is a vastly more important issue than exclusion from science. Read the whole thing!

    —

    1: incidentally, I dropped out of physics ’cause I didn’t have what it takes[2], but she stuck around for a PhD. Probably now she’s working in the City, snorting cocaine off the bottoms of Abercrombie & Fitch models, and here I am living in a 2-room apartment in Tokyo on a completely moderate wage. Who was the loser in that story?

    2: My friend got a PhD in Canberra. He dropped out for IT. He told me: “I realized I don’t have what it takes to be a physicist when this Russian physicist visited to do a 3 month placement. He had no funding. He had no money. He slept on the floor in his office and ate rice for 3 months. I can’t do that for any reason. I will never compete with people like that. I’m outta here!”

  • The US Government Shutdown, in Human Terms

    I have a friend in Sydney, Australia who has things a little tough. She has a decent professional job – though its a job in a woman’s career, so it doesn’t pay as well as professional jobs should – and she’s a good worker. She has been working ever since I met her without a break, and keeps the same job for years at a time, so no problems with her work life. Unfortunately she’s a single mother, not because she’s one of those dirty sluts who pop out sprogs by the month to get on welfare, but because after the birth of her child her husband turned into a weirdo Men’s Rights Activist and became insufferable, so they divorced. They have a custody arrangement (one week each) so she doesn’t fall afoul of any of the Men’s Right’s Movement demands for Good Women, but that’s not enough for him: when her child is at her ex’s house he denigrates her verbally, and he refuses to pay for any kind of extra-curricula or developmental activity, so if her child wants piano or ballet or rugby lessons, she has to fork it out herself. Nonetheless, her child is well-adjusted and she’s a good parent.

    Unfortunately she has a minor mental disability which, although it doesn’t stop her working and raising children, means that she isn’t so good with money and she’s had a long history of financial troubles. It also means that she has a “pre-existing condition,” and anyone who has lived in Sydney knows that rents are punishing and being dodgy with money is not an easy trait to live with. She’s lucky because she has a good job, but for every person like my friend you can bet there’s another similar person whose job is not so great, who has serious financial troubles and is, as they say, a single pay-cheque away from disaster.

    I think I know what this situation is like, though I can’t imagine the additional stress that gets piled on when you have a child, and I can imagine that my friend comes home from work sometimes, sits down and lets out one of those slow breaths, the one’s where you’re mentally thinking “Fuuuuuuck” as you wonder at what you can do and worry about what will happen to you if you don’t do it.

    Fortunately, however, my friend lives in Australia, so she is guaranteed health care. She knows that no matter how badly things go, even if she isn’t working (which she is), neither she nor her child are going to lose their health. Which means that if it becomes her goal to shift down from her professional job to a manual labourer, cleaner or bar worker – she will still have guaranteed healthcare. Those weekly worries where she sits down and thinks about what she has to juggle don’t extend to her or her child’s health.

    Not so in the USA. The same woman in the USA – changed jobs as an adult, pre-existing condition, child with same pre-existing condition – is likely unable to get health insurance even if she can afford it. The same woman in the USA will come home and she won’t just think “can I afford anything nice for my child this weekend or next,” but will also think “I hope I don’t get seriously sick before my child becomes an adult,” because if she does she will be facing ruin, and her child’s future will take a massive nose dive. Even though she can afford health insurance in any other country in the world, in the USA she will be denied it, or her entire income will be blown on it. And she’s not alone, nor is her case limited to single mothers who had the importunacy to refuse to tolerate Men’s Rights Movement husbands – there are between 10 and 40 million Americans who can’t get health insurance, and for a sizable proportion of them the problem is either that they have a pre-existing condition, or that as sole business operators or independent contractors they don’t have group purchasing power, and simply can’t afford individual insurance.

    But not anymore. On Monday Obamacare started, and those millions of people have access to the health insurance exchanges. Insurance companies can no longer refuse them insurance, but have to offer them a basic plan, and the government will subsidize some plans. Medicaid has been expanded to cover the working poor. The primary beneficiaries of Obamacare will be the working poor, the lower middle class, and those with pre-existing conditions. The estimate for the first year is that 7 million people will gain access to health insurance, and the total number of people expected to gain access over the long term is 28 million. This isn’t a flight of fancy either – the Health Insurance Exchanges have been overwhelmed by the unexpected number of customers, just as happened to the NHS when it first opened.

    This scares the Republicans. The next presidential election is in two years and they desperately need to win it, but they have a problem: they are implacably opposed to Obamacare. The election is in two years, and the prediction is that in one year 7 million people will take it up. This means that by the time of the election 7 million people will be benefiting from a Democratic policy that the Republicans will be campaigning to abolish. Judging by the scramble to the exchanges, many of those people will have been receiving their insurance for more than a year. For those people, that Friday night collapse onto the couch and “oh, what am I gonna do!?” will no longer include worries about healthcare. If they have two years to experience this level of relaxation and then, at the next election, the GOP and its Tea Party mates rock up claiming a virtue of abolishing the law, what are those 7 million people going to think? Will some of them perhaps think one option is voting?

    Furthermore, the biggest beneficiaries of Obamacare are going to be working and lower-middle class white males with families. These are the stalwarts of the Tea Party’s campaign, and in the long term they are going to be looking at convincing up to 40 million people that gaining access to health insurance – including subsidies for the working poor – is a bad idea. What are their chances?

    This is why they have to throw down now. This is why their specific condition was that Obamacare be delayed a year. They need those 7 million people to be naive, fresh to Obamacare, not yet settled in their new comfort zone, so that they can go to the election with a slogan that appeals to their base and doesn’t simultaneously alienate – or worse still, activate – 7 million early adopters. With a one year delay they have a chance; if Obamacare is enacted now they lose. And if they lose, they lose the following election too, because whoever wins the White House (Hilary Clinton?) is going to be able to say to more than 7 million new Democrat voters “do you trust these people? Last election they said they would remove your health insurance. Do you trust them this election?”

    That’s why the GOP is willing to shut down the government, because in two years time they risk irrelevance. They have to destroy the tea party and accept universal health coverage, or they have to fight. And if they choose to fight there is going to be no room for compromise. Will they go so far as to force a default? Do they have any political reason not to, if they are facing a sea change at the next election? I guess not …

  • Fixed Targets and Dice Pools in Opposed Skill Checks

    In comments to my post on balance in Shadowrun’s opposed skill checks, Paul asked me whether the distribution of success probabilities for opposed skill checks with equal numbers of dice is equal to the success probability you get from simply fixing an expected target number for your opponent. In practice what this means is that if the target number for success is, say, 5 or 6 on a d6 (probability 1/3) and both you and your opponent have, say, 6 dice, then you set an expected number of successes for your opponent as 6*1/3=2, and then try and roll over this expected target. Apparently Exalted 2e moved from challenged dice pools to using this process, fixing the target number to be half the opponent’s dice pool, and then having the attacker roll above it.

    My guess in response was that this would be equivalent at larger dice pools. Turns out I was partially right and partially wrong. I ran a simulation in R, for dice pools from size 1 to 100, and set the target number of successes to be 1/3*(opponent’s dice pool), rounded down. So for a dice pool of 12, attacker rolls 12d6 and counts the number of successes (5 or 6s); they need to get over 12/3=4 to win. For 11 dice, the target is 11/3 rounded down, or 3. Figure 1 shows the results for opposed dice pools (black line) and the expected target number approach (red line).

    Figure 1: Success probability with and without opposed dice pools
    Figure 1: Success probability with and without opposed dice pools

    Note two interesting properties of this graph:

    • The probability of success for the expected target approach bounces around a lot, going from above 0.5 to below 0.5 in little jagged steps. This is because of the rounding problem in setting expected targets. This means that even at large dice pools (100! imagine that!) you can still get large variations in success probability depending on whether your dice pool is a multiple of 3 or not
    • The limiting value for opposed dice pools is not 0.5 as I thought, but actually closer to 0.47. I think this is because of the discrete nature of the probability distribution – there is a non-vanishing probability that both sides will roll the same number, whereas if the two dice pools were normally distributed this chance would be zero – someone always wins, and there is a 50% chance it will be you. In this case the normal approximation to the binomial distribution contains a small error even at dice pools of size 100 or more

    The rounding problem is interesting because it is quite punishing at small dice pools. For example, if you have a dice pool of size 4 and your opponent also has size 4, then their expected target is rounded down to 1, which is actually the precise expected target for a dice pool of 3; you have actually gained a +1 to your dice pool through rounding error, and if your dice pools are both size 5 then this bonus increases to +2. We could use the opposite approach of rounding up, so then you would get a -1 or a -2 on your dice pool compared to your opponent. Rounding off smooths this problem a bit – in this case a target dice pool of size 4 gets an expected target of 1 (equivalent to 3d6); that of 5 gets an expected target of 2 (equivalent of 6d6). So your dice pool benefits or suffers. From the chart we can see that this effect is noticeable even at dice pools of 100d6 (which is why I extended it that far).

    We can see more accurately what the true probability distributions would be like if we consider only dice pools that are multiples of 3 – that is dice pools of 3, 6, 12 etc. – because in this case there is no rounding error. This result is shown in figure 2, again with the opposed dice pool shown in black and the expected target number in red.

    Figure 2: Results of dice pools with no rounding effect
    Figure 2: Results of dice pools with no rounding effect

    Interestingly,with no rounding the expected target number method produces a slightly lower probability of success than the opposed dice pool method. This is because it restricts the range of extreme success available to the player – e.g. a player with a 6d6 pool can’t get success on a roll of 1 or 2 successes, even though this will (occasionally) happen.

    I guess this means that the expected target number system is slightly broken, because rounding is very important at the scale of the dice pools that most people use. In the case of Shadowrun, for the first three dice pools (1d6 to 3d6) against a target with the same size dice pools, the probabilities of success are (respectively) 0.34, 0.56 and 0.26. So dice pools of 1 and 2 benefit hugely compared to dice pools of size 3. The same effect will exist in Exalted. What an expected target number system gains in simplicity, it loses in fairness (at least for small dice pools).

    These kinds of considerations show that developing an effective system that is fun to use, simple and fair in all situations is fiendishly difficult. Next I am going to try and look at the WFRP 3 system to see if their methods based on opposed dice types are more robust to these kinds of concerns.

    —

    Update: Since Paul mentioned it in comments, Figure 3 shows an approximate example for Exalted 2e. It uses a target probability of 0.4 (7 or better on d10) but does not use exploding dice. The effect is still there but some of the jags are not as clear. Again, red line is the expected target number method, black line is the opposed check (so red=2e, black=1e?)

    Figure 3: Target number vs. opposed check for Exalted dice pools
    Figure 3: Target number vs. opposed check for Exalted dice pools

    Update 2: apparently I got the dice pools wrong for Exalted, so I’ve updated Figure 3 using the correct numbers – a target probability of 0.5 and two successes on a roll of 10.

  • I Will Only Use Statistical Techniques Designed by Men…

    … Because they are so much more Dudalicious. In honour of the David Gilmour (not the guitarist!) school of teaching, from now on I will only use statistical techniques designed by men. Sure, I could use Generalized Linear Latent and Mixed Models (GLLAMM), but just listen to the name of the damned thing. It’s like the Jane Austen of stats, and unsurprisingly it was developed by a woman (Sophia Rabe-Hasketh). Hardin and Hilbe just had a much more indefinably cool … manliness … about them, so I think for clustered binomial or count data I’ll just wing it with Generalized Estimating Equations. Luckily I don’t do much in the way of RCTs, because the classic text on experimental design by Cochran and Cox is half-authored by a woman – I can’t tell which bit she wrote so I’ll just have to dump the lot to be sure. This could be a bit tricky, because that stuff is pretty fundamental to how we think about efficiency in experimental design. No problem really, though, I’ll just make sure I apply for bigger grants and recruit more subjects. Typical of a woman to write a book about how to be thrifty with sample sizes really, isn’t it? Real men just recruit more subjects.

    David Gilmour also doesn’t like Chinese authors, so if I’m going to follow his approach I’ll probably have to drop any adjustment for probability sampling, since a lot of the development work for those methods was conducted by Indians after independence. That shouldn’t be too bad because there are still some low-grade journals that let you publish without adjusting for your sampling process. Of course, to be sure I think I should develop a few stock phrases to deploy in explanation of why I’m avoiding certain methods:

    Although region-level variables were available, they were not incorporated in this analysis because the methods required were developed by a woman

    or

    To avoid feminization of statistics, the clustering effects of school and classroom were not adjusted for in this analysis

    and maybe

    Probability weights were not incorporated into the analysis, because that method was developed by Indians

    I’m sure the peer reviewers will appreciate that, but just to be sure I’ll be sure to specify in all submissions that I not be reviewed by women. That should cover it.

    Now, some of you might suggest that I should just relax and use all the techniques available to me, or at least not go through the canon with a fine-toothed comb checking the gender of every contributor – I mean, couldn’t I just drop the techniques only if I find out that they were written by a woman, without active screening? A kind of passive case-finding approach, if you will (but can I employ case-finding – it may have been invented by a woman. I should check that!) But this is not how the David Gilmour school works. You have to assess your authors first and foremost on their cool manliness:

    Chekhov was the coolest guy in literature. I really think so. There’s a few volumes of his there, what a great looking guy. He is the coolest guy in literature; everyone who ever met Chekhov somehow felt that they should jack their behaviour up to a higher degree.

    And really, when you look at the kinds of canon that are taught in English at high schools and first year uni courses, it is quite often the case that they are all (or almost all) male. Every statistician knows that those kinds of imbalances in a sample don’t happen by accident – that’s a deliberate selection bias. If it’s good enough for dudely English teachers it’s good enough for me, so I think from now on I should screen out any beastly feminized stats. Sure, you can’t get into any half-decent journals if you can’t use GLLAMM and good experimental design, but I say hell to that. It’s time to fight back! Men-only stats for the win!

    —

    In case anyone thinks I’m being serious[1], there’s been something of a storm of controversy about this David Gilmour chap, and I think you can see how stupid his approach is if you imagine trying it in a technical field. Stats being part of maths, it has its fair share of chick lit, but it is still male dominated; nonetheless, if you screen out the main work done by women, you suddenly lose a huge range of tools and techniques that are essential to the modern statistician. Surely the same applies in English literature, but moreso given the huge role women played in the development of the novel. Check this Crooked Timber thread for more entertaining take-downs of this position (with some prime grade Troll Meat thrown in the mix). It really is outstanding on so many levels that a literature teacher would judge who to teach in such a juvenile Boys Own Manual way; that they would take their responsibilities so lightly as to think that their sole task was to teach students their own opinion rather than … something useful … and that they would not try to hide it behind some more mealy-mouthed apologia. I mean really, there are a lot of very good female writers in the last two centuries and yet people like this David Gilmour chap manage to construct a syllabus without a single woman in it. Usually their argument would be along the lines of “I judge on merit” but you do have to wonder, don’t you? And then along comes a naif like Gilmour and makes it completely clear how these canons are really constructed – the women are screened out from the get go.

    —

    fn1: I really hope not, but this is the internet.

  • On Balance in Shadowrun’s Opposed Skill Checks

    Shadowrun uses a skill check system based on dice pools and opposed checks. The basic mechanism for opposed checks is quite simple: each party constructs a pool of d6s based on their combined attribute and skill score, and success occurs on a 5 or 6. The person who rolls more successes wins, and the number of successes decides their degree of success.

    When I saw this system I thought that there must be a way to recalculate it as a single dice roll. A dice pool of this kind is essentially binomial distributed, and the sum of binomial distributions is binomial, so I thought that the difference of binomial distributions would also be binomial distributed and it would be fairly easy to obtain analytically a formula for a new dice roll based on the probability of success (1/3) and the number of dice in each pool. In fact the difference of two binomial distributions is not binomial (see my appendix below) and the dice pool mechanism is quite complicated. In the case of dice pools of equal size it creates a symmetric, non-binomial distribution that tends towards normality as the size of the dice pools increases; for uneven numbers of dice it creates an appropriately skewed distribution that has no easy calculation formula. In fact, it is fairly easy to show that for equal numbers of dice in the conflicting pools, the probability of success tends towards 50% as the size of the dice pools increases.

    To show this, I wrote a simple program in R that calculates the probability of success for opposed dice pools ranging in size from 1 die in each pool to 30 in each pool. I ran the simulation for 10000 rolls for each dice pool, and calculated the probability of success for each roll. In all cases the dice pool of the opponents are of equal size and the success probability is 1/3, as in the standard rules. Figure 1 shows that as the number of dice increases the chance of success tends towards 0.5. That is, a PC with skill and attribute of 10 each, and modifiers of 10, when doing an opposed check against an exactly equally matched PC, will be successful 50% of the time; whereas the same situation for characters with just an attribute and skill score of 1 will show a vastly reduced chance of success.

    Figure 1: Probability of success in opposed checks for equal dice pool sizes
    Figure 1: Probability of success in opposed checks for equal dice pool sizes

    I’m not sure whether I like this outcome or not. Superficially, given low-skill characters are more likely to fail generally, it makes sense that they should be more likely to fail against an opponent of equal skill. But then, it seems reasonable to suppose that the chance of success when opposed by someone with the same skill as oneself should be constant. Which assumption is better? In WFRP3, difficulty of the check is set by the opponent’s skill but is not random, and usually involves competing against dice with a higher chance of generating failure than one’s own dice have of generating success. Is this a better model? Other dice pool systems probably use a fixed target number – is this better? Maybe a fixed target number can be manipulated to generate a fixed failure rate (if it is based on the contrast of the PC skill and the NPC skill). But then again, this opens the possibility that PCs can do better in opposed than unopposed checks. For example, in Shadowrun, when doing an unopposed check the maximum probability of success for a PC with attribute 1 and skill 0 is 1/3. Presumably when they oppose someone with attribute 1 and skill 0 their chance of success should be less than 1/3? If one accepts this proposition, then Shadowrun is perfectly balanced, and the only question is how long it takes to get to 50% success. This pace can be changed by using different success targets and dice sizes: for example, a success threshold of 7 on d10 slightly reduces the chance of success for any given dice pool.

    Note that by the Strong Law of Large Numbers, it is impossible to change the limiting probability for opposed dice pool checks, no matter the threshold probability or the die size. This is because as the dice pool grows in size each dice pool becomes increasingly close to normally distributed; but when subtracting one normal distribution from exactly the same normal distribution there is, of course, a 50% chance of getting a positive number. So as the distributions get more normal, so too does the average chance of success tend to 50%. Increasing the dice size and reducing the success threshold will delay the onset of this 50%, but Figure 1 shows that for most PCs and most campaigns, d6 will suffice.

    Given these results, I think that the Shadowrun dice pool system is pretty close to perfect; and there is no easy way to modify it or any similar dice system to get more nuanced results. I will shortly be examining WFRP 3 dice systems to see if they produce more subtle outcomes. Stay tuned!

    —

    Appendix: Proving that the difference of two Shadowrun dice pools is not binomial.

    When both the PC and their opponent have a total skill of one, the opposed check becomes a challenge of 1d6 vs. 1d6. In this case there are three outcomes: -1 success (opponent wins and PC loses); 0 success (both win or both lose); +1 success (PC wins and opponent loses). For a single success probability of 1/3 the probability of each event can be easily calculated without special mathematics as 2/9, 5/9 and 2/9 respectively. This means that the probability of -1 and +1 are equal. If this distribution is binomial, then it can only occur from a binomial distribution with 2 trials and a probability of p, since this is the only binomial distribution that allows three distinct outcomes. Thus if we calculate the probability of 0 successes or 2 successes under such a distribution and set it equal to the extreme probabilities obtained for the 1 vs. 1 shadowrun check, we can see the conditions under which they are equal. Under a binomial distribution with probability p and 2 trials, the probability of 0 successes is (1-p)^2; the probability of 2 successes is p^2. Comparing with the 1 vs. 1 Shadowrun check, we see that these two probabilities must be equal (as they are in the Shadowrun check). That is, p^2=(1-p)^2. This is only possible if p=1/2. But in the Shadowrun check p=1/3. Thus, by contradiction, the Shadowrun check cannot be binomial. If any one check is not binomial then it follows that we cannot expect a general rule in which checks are binomial. Thus, through contradiction, Shadowrun opposed dice pools are not binomial and no formula can be deduced which will enable calculation of binomial probabilities in Shadowrun.

    For general opposed dice pools, the probability distribution is obtained by calculating the cross-correlation of the two binomial probability densities. An equivalent calculation for the Poisson distribution is shown in Wikipedia (the Skellam distribution) and is obviously nasty – it involves Bessel functions, which is an immediate “do not enter” sign. The equivalent calculation for the binomial distribution involves a calculation of products of binomial coefficients, and my combinatorial kung fu is not up to it, but I think at least for opposed checks with equal numbers of dice it can be solved analytically, though not in a way that is useful for gamers. I think such a solution is available in a textbook by Ashkey (?) but I don’t have the book or the will to read it. So more complicated solutions to the problem will be found numerically or not at all. I may revisit this problem in order to compare Shadowrun with WFRP 3. But for now, I’m shying away from it for obvious reasons!