Game Balance and Dice Mechanics
One of the fundamental factors needed in game design is balance. Of course balance is a funny term that gets interpreted differently by various people, so here I will define it as “A game is balance the same amount of resources (XP, Feats whatever) invested in different ways afford the investor roughly the same amount of impact on the story over the course of the campaign”*
The easiest way to fail at this is to allow a character to maximize one thing with no marginal cost. Consider this: you have two characters, one has even distributed their resources among all abilities, and another who has simply maximized one. The latter character is guaranteed to have a larger impact on the story over the course of the campaign.
The intent of RPG’s are to present problems for players with open-ended solutions. If one character has maximized fighting they will use fighting as a solution to everything. If one character has maximized sneaking, they will use sneaking to solve every problem. This applies to anything the player decides to have the character specialize in.
This in and of itself wouldn’t be a problem, but it becomes a problem when that character is in a group with characters that have taken a more generalist approach. In this case the maximally efficient solution to every problem for the party is to allow that character to solve all of the hard problems with the thing that character specializes in.
“Sure we could sneak into the castle and free the prisoners with lockpicks; but that might fail, and certainly I can just go in and kill everyone”
If you are doubting that this is actually a problem, consider this: Every successful RPG published has /some/ method of accounting for this. D&D classes ensure that there is a limit to specialization; GURPS uses a roll-under mechanic that means the marginal utility (see below) of increasing a skill by another point diminishes after 16; Fate required skills to be bought in a ‘pyramid’ structure; Shadowrun, World of Darkness, and One Roll Engine all use dice pool mechanics which automatically reduce the marginal utility (more on dice pools later); and the previous incarnation of Living Myth used escalating costs for stats.
To understand how to balance this, you have to understand the concept of marginal utility. Marginal utility is a concept from economics that points out ‘one more’ of something has a different value than the one before it. For example if you have no shoes, the utility of buying a pair of shoes is vastly greater than if you already have three shoes. In the application to game mechanics the concept of marginal utility could be thought of as the benefit of a +1.
The thing is that the value of a +1 is largely dependent on the sort of dice mechanic one is using. The simplest analysis though can come from investigating a linear dice mechanic. If for example we looked at d20, we might therefore assume that every +1 results in a +5% bonus, and thus the marginal utility is always the same. We could for example graph the utility as linear (The Red Line):
But, as pointed out above we would be mistaken. The Utility of maximizing a single thing is non-linear. As a single ability increases the utility of that ability becomes not only better at that one thing, but in fact becomes better at everything.
Another way to think about it is that there is a set of possible options one has to a given problem. You can imagine each solution has a particular difficulty given a particular situation. For example saving the handsome prince might have the following difficulties given various solutions: Fight 50, Sneak 20, Talk 40, Magic 30, Run 10. Someone who has invested 10 points into everything will choose run, and someone who has invested 50 points into fight will choose fight. But given a seconds situation: Fight 20, Sneak 20, Talk 20, Magic 20, Run 20, we actually have something which can be solved in many ways, but the person with 50 in fight will still fight, and the person with 10 in everything has no options at all.**
So the actual utility is more like the Green or Yellow Line.
There are three main ways of handling this problem. Each has its own trade-off. We can do structured abilities, or we can have accelerating costs, or we can use a dice pool mechanic.***
Examples: Dungeons & Dragons, Fate
With structured abilities, part of character creation and character growth is restricted in what you can and can not do in such a way that you can not take advantage of the gained marginal utility. For example, in D20 you may buy four points in a skill at first level, then only a single point per level after that. In Fate, the peak skill is capped at 5, followed by two skills at 4, three at 3, etc…
The net effect here is that a particular character is forced to have a particular sort of balance to them. On the plus side this sort of structure can reduce the anxiety caused by opportunity cost. However it does this by (often radically) limiting the players choices when crafting their character.
If one, for example, considers GURPS vs D&D, one can see that the fixed choices represented by the latter box characters into a very tight set of choices, whereas the former truly allows the creation of any sort of character that fits within the bounds of the Character Point allocation.
Increased Ability Cost
Examples: Old Living Myth, Lots of Video Games
Game systems use this surprisingly infrequently. It is an elegant solution to the problem though. The system simply makes the cost of buying the next level in an ability reflect the gain in marginal utility. Consider for example that you figure a +1 is has an N log N growth curve; by setting up the cost of a +1 to be some factor of N log N you have restructured the cost benefit ratio to be linear.
A common way of doing this is by making the cost of a +1 equal to the next value of the stat. For example going from a 19 to a 20 would cost 20 points, and going from a 3 to a 4 would cost 4 points. Thus the cost for an arbitrary value is equal to the sum of all the values before.
I have play tested this extensively, and it also has a magnificent ability to automatically balance play to a certain extent. Consider for example two abilities X and Y, one of which is twice as good as the other; however the designer failed to take that into account. In that case we might say the utility of Y is equal to twice that of X. If we buy one point in each, then the person who bought Y is indeed better off by 1. However the person that buys three points in Y (at a cost of 6) would then buy two points in X (at a cost of 3) because getting the equivalent from another point in Y costs 4 points.
There is one major problem though. You can’t let abilities stack, or must do so in strictly controlled manners. Consider for example you wanted to put an advantage in your game called “Weapon Master” or something. If you make this grant a +1 bonus to a characters “melee” skill, then 100% of Character will by this as soon as the cost of Weapon Master is less than the cost of buying the next point in Melee. If for example Weapon Master costs 15 points, and we’re using the costs as described above, then everyone with a melee skill of 14, will always buy Weapon Master next instead of buying the 15th point in Melee. This may seem like it’s not a problem but it limits the sorts of advantages that may occur in a game to dealing very strictly with one thing; balancing no longer relates to how to balance individual things, but instead related to how to keep combinations of things from being overpowered – which is a much more complicated problem.
To make this problem clear, in the Old Living Myth system player had to buy their character bonuses mechanically. They had to say “Well my character is like this, and so that should increase this and also this.” There was no way to specify bonuses narratively the way you see in GURPS advantages, Fate Stunts, or D20 Feats.
Dice Pools are something I discussed before. In retrospect I was too quick to dismiss them. They have one fantastic property – they automatically scale because each additional dice in a pool automatically (based upon the statistical effect of the dice mechanic) produces a diminishing return relative to the previous dice.
This solves the problem all by itself. If you have several different sources that give you +1 dice, that’s fine because each additional die gives a diminishing return and the things giving that bonus can all be given a flat price.
But there is one big problem. It doesn’t scale. At all.
Many games consider ‘doubling’ to be the criteria for scale. If for example we look at D20, the DC’s range from 0 to 40 on a d20. Fate has 1 to 8 difficulties on dice that range from -4 to 4. Personally I’ve always found doubling to be unsatisfactory, preferring three to five laps to be a better representation of the changes from apprentice to godlike.
How many dice does it take for a character to have a 95% confidence of hitting the same number as a character with 1 dice on an exploding dice pool?**** It takes 45 dice. The 5% point on Exploding D10’s is 16, in order to have 95% confidence of hitting 15 you need 45 dice. 45 Dice in turn gives you a 5% chance of hitting a 33 – in order to have a 95% chance of hitting a 33 you’d need 2600 dice! Hitting 2600’s 5% would require 150000 dice!
Obviously, there is an inherent scaling problem here. If rolling 150000 dice were physically feasible I might say “Screw it, let’s do that!”. But it’s not. 45 Dice isn’t feasible. 10 Dice is probably the best you’re going to get, and where does that leave you in terms of the equivalent d20? At the 5% interval, it’s pretty much the same as a +6. You’re basically reduced to a game that scaled down to e6 levels.
Honestly I don’t like any of the choices very much. Three properties I’d like to have in a game system are: Freeform Character Creation, Narratively Relevant Rules, and Scaling. None of the three options presented here allow that without some sort of hack after the fact. In fact no system that I know of handles all three of these, and many only do one of them. Hacks are in place to address these deficits in many games, but they are always at best clumsy add ons.
I have already done a lot of analysis of dice pool mechanics and will explore that in a follow up post with possible solutions to the scaling issue; but ideally I would imagine myself a mechanic that both has diminishing returns built in but a wider variance so that scaling could be more readily achieved.
* This definition comes from Greg Chattin-McNichols
** This analysis suggests that the marginal utility of a +1 in a linear mechanic is n * log(n)
*** Gurps handles this in a fourth way. The Roll Under mechanic means that the marginal utility changes depending where you are in the curve. I didn’t want to go into this, but it is an example of another solution.
**** Assuming d10’s here, but the numbers aren’t very different with any number of dice faces.