Thursday, January 5, 2012

Why probability is important


It's a dirty word, to a great many people.  It has connotations of being a discipline that only 'geeks' and dweebs should be interested in, and of being an arcane practice that few people can understand well.  It underpins much of our modern life, and has enabled a great revolution in human development, but remains seen as something not quite approachable.  However we may feel about it, each and every time that we unfold our favorite gaming system to play a game and throw dice we're at the mercy of the gods of math.  Specifically, we're dealing with the disciplines of statistics and probability with each chuck of those cubes we love to hate so much.

Now, I'm not here to talk about why each of us should learn basic probability - it simply isn't something that everybody needs to know to make it through their day.  However, its my opinion that every wargame designer should be at least acclimated to the ideas of probability, if not outright able to articulate every detail about the math backing their gaming system.  That's a pretty bold claim, especially coming from a guy with zero game design credentials to his name; why precisely does a designer need to understand these interactions?

Here's the deal.  As I see it, rules sell games to a very limited minority of the gaming populace.  Instead, miniatures sell games.  People buy games because they like the miniatures, or they like the background material - in short, the aesthetics of a game are what makes sales.  But the rules of a game are what give it life, and make you want to use those models over and over again.  They are what drive communities to get together and play a fun game.  And at the very core of that idea - of a fun game - is the unalienable fact that probabilities control the interaction between the players and the ideas your game expresses.

At the most basic level, that's what rules are - they are expressions of the designer's intent, filtered through the players, about what makes for a 'fun' time.  Taking my favorite system as an example again (Heavy Gear Blitz!), the game tries to portray semi-modern combined arms warfare - with the addition of giant robots.  At some point, there's going to be an intersection between an infantryman (or squad of infantry) and a Gear - and how that situation resolves is as much about what the designer writes the rules to be, as the probabilities those rules work out to be.  A designer that's ignorant of probabilities just says something like 'a machinegun should kill infantry most of the time', fudges a number that works with the dice system they are given, and 'playtests' until that 'looks' right.

Unfortunately - playtesting isn't enough.  It's important, most certainly - and I'll talk about it soon enough. But playtesting is subject to many vagaries; different people can remember the same situation slightly different, dice can be badly weighted and few iterations of a playtest can give skewed results.  All of these things are just part and parcel of dealing with people, but they are a poor standard to gauge an interaction over time.  Because if the play testers happen to be lucky, or have a bias towards thinking infantry are too weak, or any number of other factions - the designer gets the idea that the rule isn't solid, and needs toned down.  Repeat ad infinitum and soon enough a rule goes out the door that appears solid on paper; but once it hits the wilds of the gaming tables it's obvious that it's a bad rule.

Enter probability.  By knowing how your system works mathematically, you can match your assumptions to the game mechanics more precisely.  If your expectation is that 'a machinegun kills infantry most of the time' works out to be 60% in your mind, working the probabilities for several scenarios will tell you if that 60% mark is reached or not.  If a playtester comes back saying 'infantry die all the time' you know their report is bogus, or biased - over some aggregate amount of tests, infantry only die 60% of the time, not 100% of the time.  As a designer, this strips away some of the ability for an individual playtester's bias to shine through, and puts that knowledge firmly in your own control.  If every playtester comes back and says 'infantry are dying all the time', maybe your 60% mark is too high.  If only a small group say that, then maybe they have a bias - and you should discount their remarks.  Or maybe there's an interaction you're missing that you need to account for.  But either way you have more context than the guy who flung a dart and hoped for the best.

Therein lies the difference between a designer that understands how his system works mechanically, and one that does not.  The latter designer is a slave to perception, bias and luck; the former still has to contend with bias, but has checks against it, and luck is out the door.  In addition the designer who understands his mechanics can establish 'principles' of the game as a sort of test to see if new units or rules are skewing the design too far from his or her idea.  He can do design independently of playtesting, which allows for more rapid production of material and a more unified finished product.  After all, the success or failure of the line should rely on the people responsible for creating it - not the fans who want their own particular bias to be enshrined as law.

Taking an example from Heavy Gear; in a standard roll of 2D6+0 versus 2D6+0, there's a 40% chance of 'success' for the attacker.  Somewhere down the line, the designers said you need 2 points of success to do damage with a typical weapon.  So the actual chance of doing some sort of total damage is in the 22% range, not 40% - in short, less than 1/4 of the shots in the above situation will result in something a player would consider a 'success' (i.e. damage boxes on a target).  Changing the roll to 2D6+1 vs. 2D6+0 makes it a 60% chance of a 'success'; but still only a 40% chance of a perceivable 'success'.  Does that match the expectation of the designer for the game?  Hard to tell.  But I suspect that somewhere a designer didn't consider exactly how often they wanted a player to have 'success'; instead they just kept massaging numbers until it 'looked right'.  But they are groping in the dark in this case - instead of knowing firmly "I want the success rate to be 60%".

After all, that's what a designer is paid to do - they are there to build a mental scaffolding for the rest of us to participate in.  They are there to chart a course through the conflicting ideas in all the different wargames out there, and hopefully navigate through the shoals correctly.  A large part of that relies on business sense, a good sense of aesthetics and knowledge of their market.  But it also relies on understanding just what their rules are doing - beyond the filter of the biggest fans.  And that requires knowing probability.


  1. I know plenty of games designers. And I've met a fair few in my time. Every single one of them could be fairly described as a math geek. Most wargames start out as a mathematical probability engine at the start of development. Having play tested many games systems though over the years I think you might have slightly over stated the importance of probability. I did a series of articles on game balance that I think show there are other things at 'play' in wargames. Where I do agree with you 100% though is my confusion at times with the mathematical logistic certain games churn out. I find myself scratching my head and saying 'is that really what they meant to happen?' Most of the time I think not! So that then begs the question, why didn't they see it? I think that leads back to your first article... lol.

    I've put you on my blog roster. Hopefully that will give you a hits bump. :)

  2. @FG After a review of what I wrote, I don't think I expressed exactly what I was trying to convey. The core of any minis game is it's vision I think, which is the combination of it's miniatures aesthetic and what type of combat the rules are trying to express. You need both of them to be really strong to have a good game.

    But what I've seen break down (in my limited experience) is when a gamedev tries to map an idea that's based on the softer elements of design (knowing what your players want, what type of game you're trying to create, etc) to rules, and fails horribly, and still doesn't get why the mapping failed. You need to understand how you get from a principle that's based on those softer elements (I want a LMG to be very deadly) to the actual probabilities of how it works out. You can fumble around and playtest until you get to what you think is correct; or you can recognize from the start that playtesting should be out there to confirm that the ideas your rules are expressing are fun ideas, not necessarily that you got your math correct.

    I most certainly agree that math is only a tool that's used to confirm your suspicions - you can't build a game out of it alone. But my experience has been with people who say "I want X to do Y" and when someone shows them probabilities that say "X only does Y 15% of the time, is that what you meant?" they say "well, the math doesn't reflect the full picture".

    So yeah, probability is only one (small) piece of the puzzle. And the designer proper doesn't necessarily need to be the math geek - just have someone who can confirm their instincts about the game. I just get frustrated when people try to act like their idea will work even when the math won't support it.

    I'll try to find the articles you reference on your blog. I'm curious to know your thoughts, since you have much broader knowledge of the industry than I do. My only knowledge really is about DP9, and not any of the other big players in the market.

    Thanks for the link on your blog!

  3. @IceRaptor, I think I get what you're saying now. And I agree. 40k is the best example of what I call 'paralysis by probability'. There we have a game where actually given that when shooting or trying to hit something in close combat it's rare to get a better than 50% or at the very highest 66% chance of hitting, to follow that up with then having similar or worse chances or wounding and to then give them an armour save too... well I'm amazed that anything ever happens in a game.

    I once sat and explained to a couple of youngsters why their two 'awesome' close combat units were basically doing nothing to each other. The probability meant that at best they were looking to kill an opponent each every 2 turns, This would then effect their opponents ability to do damage back. They were kids and after a while they realised it wasn't them but the game... the store manager didn't thank me. But I could see the frustration on their faces.

    All too often in this industry games designers get married to ideas or even the conventions of a venerable system. At some point as a designer you have to step back and look at the game as a system and analyse whether the output is what is intended. If it's not then the part of the system that isn't working needs to be changed. I don't believe there are any sacred cows in game design.

  4. "By knowing how your system works mathematically, you can match your assumptions to the game mechanics more precisely." Very good logic! I will be following.

  5. @BragonDorn - thanks! I hope to keep you entertained.

  6. Great post and speaks to something I think most gamer assume have been thought through by the time of purchase.

    In my opinion the reason most games use the D6 mechanic is because a) other games do b) availability of dice vs being more adventurous and building a game around a D10 or D20 mechanic.

  7. @Michael Gellar - I would tend to agree; D6 is certainly far more common than other dice types. The biggest implication for using a D6 though is that your possible variation in dice is limited, unless you do a throw + add type pool. How much randomness you want in a game is a think a central design question - so the choice of how you layout your dice (including the number of sides) is a big deal.