If you are looking to hack an existing system, or considering writing an RPG or board game from scratch that will use dice, considering the dice system is a key element of this process. The system can evoke a feel of power from the players, and your thresholds will help emulate or detract from the genre you system attempts to interpret.
Below, I will present the three of the most important systems in RPGs (the 2d6, the dice pool and the roll and keep/diminishing returns) in graphical format and proceed to discuss how design around them works.
But why not the d20 system?
This is something I considered myself when writing the blog post. Its arguably the most ubiquitous of the dice systems. It is also, however, very mathematically easy when it comes to the die maths. Imbalance in d20 systems largely creeps in through static modifiers, rather than dice mechanics being poorly designed, which I might talk about later.
The 2d6
It's simple, nice, clean. The dice system you can take home to your mother (Although she mightn't want to know all about last night's Apocalypse World or MonsterHearts session). The 2d6, or indeed any 2dx system runs off a very nice gaussian distribution.
Now, many years ago, this child prodigy called Gauss lived a frantic life packed with Maths and physics. He proved Euclid wrong, scared the life out of the men of his day in doing so, and promptly found the normal distribution to provoke them further.
Below is a graph of one. It shows chances of rolling any particular number on 2d6:
And this graph shows the chances of rolling this number or higher on 2d6:
As you approach the central number here, you see, you also approach the point of the balancing roll, and the point of most commonly occuring roll. On 2d6 in particular, it is a 7. On 2d10, it is an 11. If you take the maximum number of one of the dice and add 1, you have the rolling average. Design around it. The Apocalypse World system is the most obvious example of this. Players without a bonus will hit that 7+ 50% of the time. Players with positive stats will hit it substantially more often. This facilitates the 'being awesome, having consequences, +3 is amazing' low power curve of AW really really well, and helps emulate the Post-Apocalyptic genre themes, in which people who aren't much better are still well outside a statistical standard norm.
The dicepool
The dicepool works, for the most part, by rewarding players with additional dice for skills or traits, and you'll find it in stuff like White Wolf games, but it is pretty common elsewhere (Indeed, Ubiquity is a dicepool system, rather ironically.) The rewards of investment in the dicepool system are linear, which is why many of them try for a heirarchy to build the pool (Generally of a broad trait and a more specific skill that is linked to this trait). This linear can make it pretty easy to design around, but also has the potential to make the game pretty swingy. Dicepools, for the most part, work in two ways: Success/failure based (White Wolf, Fudge) or numerical (Over the Edge), and knowing how their graphs work can really help design for them. Below, the graph assumes you are using a 50% chance of dice succeeding, 50% chance of failure, in which the dice does nothing. The graph could be easily modified for numerical systems, or for systems with a greater/lesser chance of success or failure. It would just change the x and y axis, the curve remains the same.
Roll and keep
This variant on the dicepool system has fascinated me for a long time. Roll and Keep is a mechanic utilised most famously by the Legend of Five Rings RPG, and produces very odd curves. A number of dice are rolled, and then the highest dice, or the highest and lowest, or the two lowest, depending on the system, are 'kept', meaning that these are often totalled and compared against a target number. It rewards investment less the more investment is introduced, as you will see below. This means that designing a game using the system hits a hurdle.
Do you reward player specialisation by other additional bonuses(Such as L5R's additional abilities for high levels of skill, and a higher chance of 'exploding' dice, the ability to effectively add an extra kept dice for every ten you roll that you can keep) for having more rolled dice than kept? Or do you reward player diversification by keeping the curve the same?
If you choose to do both things in the same game, it can be very difficult to make all players in a party feel worthwhile and competent. (If, for example, one player is good at 5 things, they may rarely get the chance to shine if the party also contains four individuals who are great at 1 thing). Also worth noting on the graph, though it may seem obvious, is that of course having more kept dice than you roll is statistically useless.
This graph assumes you are using roll and keep with a simple ten sided die, with no chance of explosions. The curve is the same no matter the die size, but the target numbers on the y axis will be affected. The introduction of exploding dice would cause the curve to trend slightly more towards a linear function, like the dicepool.
Granularity
The other important thing to consider when thinking about dice systems and dice averages is a system's granularity. The dice size changes the 'steps' between numbers. For example, the difference between 5 and 6 on a 2d6 system(2.78%) is very different from the difference between 5 and 6 on a 2d10 system(0.01%), despite the overall curve of the plot being the same. In that case, this is a question of how you want those playing your games to set thresholds for success, and how that is important in your game. A percentile system like, say, Dark Heresy, expects players to succeed roughly around a third of the time at its lowest level of power, and this is reflected simply in the system. Balancing around a dice mechanic that isn't percentile based is harder, but consider 'how much do I want players to succeed against appropriate problems' and the answer will start to show itself using the graphical curves. Plot and plan your design around the curve.
But surely I want to roll 7 or more, or 15 or more, or 5 successes or more?
Well, yes. But the graphs above show average rolls. You really want to be balancing around average rolls, and anything higher doesn't really matter. Systems like apocalypse world, again, balance around a +3 modifier for a person who is very good at their stat. And where do the critical successes come in? 10. 10 is 7+3. Any more is really just gravy and kudos.
So after boring you with graphs and probabilities for a while, next time I'll be doing some nice fluffy setting work! YAY! Hope you all stay tuned~
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