Using Math for Fun!

Mathematical!

Buckle your seatbelts, because you’re about to enjoy reading an article about mathematics. Yes. It is possible, in-fact sometimes I struggle to understand why people are so math adverse … then I have to remind myself that the majority of people actually play games to escape the daily grind. That might be their job, their personal issues and struggles or just enjoy the immersive environment an alternate universe provides. In the same vein that people read books, play video games or watch movies, the appeal of an alternate reality is a major drawcard. Mechanics and code that enable that transition are key to creating a truly immersive environment.

 

A significant factor that can enable this immersive environment is when technical and mechanical calculations are integrated and operate seamlessly in the ‘background’ of a game, allowing the user a more immersive ‘frontend’ experience. Most computer games are a pretty good example of this – imagine how bad World of Warcraft would be if you had to sit there calculating the value of twenty odd modifiers and variables on your sword before you swung it … every second… for hours on end. Or if in Pokemon you had to do the following calculation every time you tried to catch a wild Pokemon:

 

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Oh, and for what it’s worth, one of the latter generations capture formulae:

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Source: Bulbapedia

A core reason World of Warcraft peaked at over ten million subscribers (now only approximately five million) is because these mechanics are all powered by powerful technology in the background – your experience as a user may never involve a true comprehension or appreciation of how CPU’s have enabled you to fly through a dungeon in a matter of hours let alone a new generation of gaming. The eloquence of the ‘back end’ just chugging away, processing all these numbers and calculations has enabled a multi-billion dollar industry… and don’t even get me started on the fact that the mobile gaming industry is predicted to crack 100-Billion dollars in 2016.

Using Calculations to Increase Fun!

So, why are games like Guild Ball, Warmachine and Hordes, Bucket Roller Kings Warhammer and Dungeons and Dragons which use more ‘traditional’ mechanics so appealing and more broadly, how have ‘dice’ (or Random Number Generation [RNG]) survived the test of time? Well, in a nutshell, the answer is pretty simple but has a lot of complexity when you look beyond the surface:

 

“When calculations are simple enough to understand, incorporation as a front-end element of user experience adds value as a complementary dimension of gameplay that enhances user experience.”

No, I totally did not just come up with that then.

 

Good tabletop games take advantage of this; it’s one of the major factors contributing to the current golden age of board and tabletop games – good designers producing great games that are more easily accessible than ever before. How many times have you had an epic play out over two hours for it all to come down to a final roll? You need that boosted nine on three dice; you need to score a critical to take out that final boss or heck you need to pull that ace on river to win the final all-in pot. When those mechanics are disguised as elements and features of the game, especially when you’re there to celebrate or commiserate with your opponent right in front of you (as opposed to a faceless internet opponent), it’s emotionally draining and it’s fucking exciting. Automation and complexity of calculations are not necessarily the foundations of a strong game.

Here’s the next step – by exposing the underlying mechanics that make the gears and cogs of a game turn, it provides users with a concrete source of data to start delving into a deeper understanding of how they can improve that specific element of their game. Whilst the math behind a game certainly isn’t the be all and end all to strategic improvement, board games can be a lot more obviously intuitive as opposed to video games because the calculations are happening right in front of you as opposed ‘behind the scenes’. Example: In most first person shooters, you’ll generally reduce a players’ health to zero more efficiently if you’re able to shoot them in the head as opposed to the arm or leg. This might be not as immediately intuitive as you’d think. If you were able to see the calculations going on behind the scene, trends would indicate your ammunition is more effective when targeting various areas of the body; in fact you could probably come up with a relatively effective heat map based on a few trial and errors. For visual or numerical based learners, seeing a heatmap or the number crunching can be a more efficient learning tool as opposed to trial and error. I acknowledge that some people are actually quite happy not knowing – gaming is a pure escape for them, and if this feels like laborious work they shouldn’t force themselves to invest time into a deeper understanding but with a game of this nature there will always be competitive players looking for any sort of edge they can get. When the casual gamer meets the competitive gamer, expectations can clash.

 

Personally, due to my mathematical and problem solving background, I’ve been attracted to the industry partially due to the game mechanics being represented literally right in front of me: Dice. Any game that is serious about having a competitive player base must get the balance between skill and randomness and/or luck correct. There’s a whole conversation we could have along the lines of ‘what is the right amount of Luck/RNG (variance) and how best to incorporate it into a game”.  It’s super interesting – Chess has arguably zero variance (other than who goes first) and is considered one of, if not the most competitive game in existence. Other games like Monopoly, Snakes and Ladders and even Scissors Paper Rock have incredibly high variance that make taking the game as a serious competition practically impossible. The line in the sand exists where variance has a non-overwhelming impact on the game that still forces players to make tactical decisions based on a possible limited variance. Additionally, there are games with no luck, but due to the scope and depth of the game, this limits its replayability (e.g. Noughts and Crosses). The right amount of variance can increase the lifespan of a game exponentially – that is part of the reason some of us (*cough* Colin Hill *cough* epic Morvahna *cough*) are happy to play the same list over and over again for years on end – no two games are ever the same! It’s exciting and, refreshing and fun.

 

Games like Guild Ball and Warmachine hit the sweet spot. Let’s begin to compare the core mechanics of what I consider games with strong foundation mechanics.

 

The Bell Curve

Warmachine

Warmachine has an incredibly strong core attack mechanic that is the foundation of all RNG in the game. Your models Attack Value plus 2D6 must equal or exceed your opponents defence. As opposed to a number of D6s or a D20, this is not a linear scale but rather a bell curve (also known as a ‘normal distribution’). The bell curve has been proven as a statistical model that most accurately represents common distributions in nature or population (e.g. the height of all humans or the weight of all lions). This opposes the single D6 system where the average is 3.5 on a single die and thus the probability of rolling a 1 is the same as the probability of rolling at 6 at any single moment. By incorporating multiple instances of this (2D6 or more) results are standardised of a standard population immediately. 1D6 to 2D6 and you get a bell curve instead of a uniform distribution – brilliant.

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Source: Doomstead Diner

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Source: Stack Exchange

These graph shows what percentage of a population will lie in each segment of a bell curve. 68.2% of the population lie within a standard deviation of the average roll (7 on 2d6). You can see that unlike a single D6, the chances of rolling a high result are not the same as rolling an average result. I.e. two thirds of the time you will roll a sum between 5 and 9. That is immediately vastly more accurate than a single D6.

 

The second graph is quite good. It illustrates exactly where each dice roll will occur. The only way you can roll a 2 on two dice is by rolling a 1 (1/6) on one dice and a 1 (1/6) on the other dice. Chances of that = 1/6 x 1/6 = 1/36 = 2.7%. Similarly, your chances of rolling a 7 or more are equal to 6/36 + 5/36 + 4/36 + 3/36 + 2/36 + 1/36 = 21/36 or 58%.

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Source: PP Forums

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Source: Transitive Gaming

Where it gets really interesting is how boosting (rolling an extra dice) changes the math, dramatically. Whilst neither are the best graph, they illustrate that when you roll an extra die, you’re in a completely different ball park of expected results. Whilst they do have a small amount of overlap, the different between a 2d6, 3d6 and 4d6 roll have completely separate standardised curves. Essentially, you are more easily able to predict what is a ‘reasonable’ outcome for the number of dice you’re rolling. Warmachine has a significant number of models with the ability to boost at the cost of resources that can make dice have a minimal impact on the game. In instances where you cannot boost, a higher volume of attacks gives you a population of rolls that are much more likely to fit into a standard bell curve. Where this system can fall down is that quite often over the course of a game, there may only be a small number of meaningful attacks that determine the outcome of the game and due to the small sample size do not accurately represent the bell curve model. I’m surprised that more systems haven’t tried to mimic this design.

 

Guild Ball

Dice in Guild Ball are an interesting creature and completely different to Warmachine. Picking up a bucket of die for your charging TAC 8 model can feel very Warhammery / gambly (technical phrase) as each success is measured on a single die’s result, not the sum of the die. This means that a linear model measures success – “how many of the dice rolled met a certain target number” seems like a poor mechanic on the surface however it has a number of factors that turn it from completely random into a much more predictable result.

 

Firstly, volume of dice. A model that is very combat oriented (let’s say Boar) has a high TAC Value, thus it has a higher volume of dice rolled to represent the strength of his attack – this volume is incredibly important. Secondly, this is modified by other members of the combat – got buddies fighting alongside you? Have some extra dice. You’re going up 1v3? Lose 2 dice. Charging? Gain 4 dice. Now that we’ve got an appropriate volume of dice to represent the force of your attack, we can take into account the agility of your opponent. Big, hulking slow ogre? Each dice only need a 2 to hit. Quick and nimble elf/goblin? You’ll likely be hunting for 5s or 6s. Due to the volume of dice being rolled to represent a single attack and noting that a model can make multiple attacks, each attack has actually become a very small bell curve of expected values. A combination of all these factors plus other external factors like spells that effect your attacking efficiency and you’ve got a unique combat system, however I haven’t even mentioned the best part.

 

The playbook is what takes this combat system to the next level. Whilst all your initial modifiers can be calculated to determine how many dice you’ll be rolling in any given scenario, you can only approximately predict what your outcome will look like in the form of a bell curve. The playbook takes this unique bell curve for each scenario and adds a layer of safety to it. Again let’s use boar as an example: Your hulking beatstick of a character has rolled significantly under average? In another game system, this might equate to a miserable failure, but even a sub-par roll for Boar translates into damage that represents this character still connecting his axe, just maybe not at full efficiency. Your attacks are successful in the majority of situations; the question to what degree was it successful? Was it less successful than you thought so you have to make some decisions on the fly about what kind of attack you’ll make to deal with this character, whether it is a reposition, damage or a tackle or a denial effect? Was it more successful than you anticipated, meaning you’ll get to allocate some extra resources elsewhere or was it (in most circumstances) exactly what you were hoping for… or did you just not care and want to smash? The Playbook mitigates the majority of truly horrible ‘dice fails’ – the real strength and genius of the system. It’s one thing for me to explain the theory behind it, but watching players from all backgrounds of all skills see the strength of this system translate onto the table is a very rewarding experience.

 

What both systems have in common is a mitigation of randomness through sound mathematical and statistical probability to the point where luck is simply a feature of the game that keeps it fresh, exciting and unique. Luck is a secondary element of the game that is harnessed rather to keep it interesting and provide neigh infinite combinations and permutations of different scenarios that make no two games the same. It is not the unbridled dictator that allows the players little to no say in how their character(s) navigate their way from the beginning to the end of the game. Again, when used efficiently, the right amount of variance adds to the lifespan of a game, rather than detracts from it, and that is brilliant.

If you got this far, congratulations and I apologise for splitting this up into segments… however I’ve only ended up skimming the surface of the math/statistics side of the equation and written more instead about a mathematical and philosophical approach. As such, more articles! I plan on going into the math of Guild Ball over the next few articles, including why people try to ‘break’ games with RNG.

 

Feel free to bug me over on Twitter @warcast.
Moops out!