### Speculative Physics

## Juggling Desks: Balance and Dice

**by Mendel Schmiedekamp**

Jan 29,2003

## Juggling Desks: Balance and Dice

There are two modes to verify a game works as it is intended: experimental and theoretical. Experimental testing primarily consists of effective playtesting. Theoretical testing, however, covers a variety of tools, from statistical verification to modeling the game play. Often the later form will be less time-consuming and resource intensive, especially in comparison with well developed playtesting methods (more on that in a future column).

### What are the odds?

Statistics are the foundation of mechanical analysis. In order to predict the way that a game will be played, the probability of the different mechanical results needs to be, at least, estimated. On the other hand, a reasonable approximation often suffices to describe a great number of features.

The key reason approximations are appropriate is the illusion of the fair die. Almost no dice are actually fair, even with reasonably well balanced dice, the method of rolling can often have some, non-trivial, effect on the results. This means that there is a blurring point where actual differences between probabilities is irrelevant.

This illusion also applies in other pseudo-random techniques. The fairness of card draws relies on the shuffling of the deck. Even a large number of shufflings often fails due to cards sticking and unbalanced shuffling technique. Drawing marbles from a bag or playing little games of rock-paper-scissors are likewise suspect.

For example, I rolled two ten-sided dice a hundred times, the resulting average value for the first die was low, at 5.45, the second was high, at 5.73. Even more worrisome, is the fact that the second die, while rolling higher, rolled a 9 more than twice as often as rolling a 10. Whatever the cause of these aberrations it is entirely likely that one set of rolls will be significantly preferred over the other. In d20 terms, the difference between these two dice is at least as significant as a +1 to your roll.

While anecdotal, these results indicate that, at least for dice, the probabilities we calculate need only be determined within two to three percent.

**Aside:** There is an important distinction between the average game, and the average over all games. In the former case there is a high probability any two dice used in competition will have a disparity. In the later case, these disparities will balance out. However, we are primarily interested in getting the mechanics to work correctly for the average game, since that will determine the reception of those mechanics.

### Core Methodologies

Extracting probabilities from mechanics takes a degree of rigorous thinking. The first step in any such situation is to break the problem down to small, easily handleable pieces. Then describe the probabilities of each piece. Lastly combine those elements to build a full picture of the mechanics.

For example, consider the Power Attack feat from d20. First we need to define the region of interest. This feat allows the attack roll to be reduced by the same amount as the damage is increased, before the roll of course. Obviously this is only possibly undesirable when the attack roll reduction affects the chance of hitting. This means the change must occur within the range of the twenty-sided roll itself.

One one hand, consider the changes that a reduction has on the chance of hitting. Clearly the damage is dependent on this change, as an attack which does not hit will fail to deal any damage. Thus we first need to find the proportional effect of the attack reduction. Lets call the range on the d20 on which the attack hits as S, and the attack reduction as N. Thus the chance of the modified attack hitting is (S-N).

On the other hand, the changes to the damage are based on the average amount of damage expected. Call this amount D. This number can range significantly, and is most easily calculated by taking the maximum and minimum damage values and averaging them. This method takes advantage of the symmetric nature of most dice rolls, especially those used for determining damage in d20. The damage of the modified attack is then, (D+N).

Since the probability of hitting determines if damage is dealt or not, the average amount of damage is proportional to the range the attack hits times the damage of the attack. So a normal attack deals average damage proportional to (S*D), while the modified attack deals average damage proportional to (S-N)*(D+N).

Expanding this later formula to determine the effect we get three terms: (S*D) + (S*N) - (N*D) - (N*N). The first of these terms is exactly the one for the default attack. So the remaining three terms gives the changes to the damage. We then note that all those terms have at least one N, so we can combine them to get N*(S - D - N). Hence it is never ideal to use any amount of power attack where the modified average damage exceeds the range of a successful attack. Strangely enough this means that weapons with smaller damage, and hence smaller values for D, are preferred for use with this feat.

In the very least this calls into question some of the flavor elements of this feat. After all, one rarely thinks of doing a power attack with a knife, but doing one with a broadsword is more likely to turn out negatively.

### Game Balancing

The simplest definition of game balance, is that each equally costly method for attempting the same effect, should have the same outcome as any other. Essentially when game resources are expended for a given effect, the results should be consistent regardless of the approach. Essentially it takes the approach and makes it a game mechanical expression of setting flavor, rather than a "meaningful" mechanic.

Some people might argue that game balance actually deals with distinct effects. But in practice there is no way to balance effects which do not on some level achieve similar goals. Until there is some unification, it is analogous to the proverbial apples versus oranges. This last level of effect gives a context to compare approaches. It may take as much abstraction as "chance to defeat an opponent through any means" or "chance to gain influence over a social institution" to reach this level, but ultimately the goals of game balance are in making these chances as irrespective of player decisions as possible, while giving the stylistic illusion of control.

This sort of balance is often quite difficult to achieve, since the probabilities are complex and involve modeling player actions as well as trying to match the results closely. Ultimately the myth of the fair die suggests that any attempt to match probabilities is going to be risky, since they will tend to be inaccurate in addition to the other problems in calculating them.

The dangers of game balance also lie in the fact that the balance cannot be too simple, or the illusion of control will be lost. There's also a concern that player choices and strategy should be able to make some limited effect on the chances of achieving an effect, ideally in a non-trivial way. From this perspective player developed strategy is also a game resource whose investment must be kept balanced.

There are several approaches to balancing the approaches seen in a game. The first is to make the system very complex so as to make it obscure if the mechanics are actually balanced or not. This allows the player to have either the illusion of balance or the illusion of control, as they wish. On the other hand, a mechanic that is this complex is usually untenable for the designer, and as a result is often not well tested. There is a strong chance that the mechanic will be shown to be deficient, or in the very least unequivocally unbalanced.

Another approach is to make each approach strong in some circumstances, and weak in others. This generates a dynamic balance which tends to cancel out any overly strong approach. A good example of this is how character classes or archetypes are commonly developed. Giving a special advantage in a portion of circumstances, and a weakness in others. These approaches are then handled by those playing the game exploiting weaknesses, and hence forcing new approaches to be used to avoid this weaknesses. This method relies strongly on the metagame, the discourse which occurs around the game, to achieve balance. When it works it actually provides both control and balance. On the other hand, metagames are poorly understood by most designers, and can cause a system like this to break down as often as it keeps it working.

Ultimately the best approach to game balance can be the easiest. To simply disregard it as a design goal.

**Game Imbalance**

Why would anyone intentionally design an unbalanced game? The reason is simple. Most settings are unbalanced. It is a rare setting that can actually have all approaches being equal. One might even argue that any setting of this sort will be necessarily less compelling and interesting than the alternatives.

On the other hand, just because a given approach is preferred does not mean there is nothing to verify. There should always be a strong sense of how much a method increases chances of success, and this chance needs verification. One nice aspect of this, is that this verification doesn't need to tangle with the illusion of the fair die. The probability difference should be large enough to be significant, even if the dice or other randomizers fail to cooperate.

Most games seem designed from the perspective that game balance is a necessity. But ultimately it is like any other design decision, purely based on what the game needs. It should never be included just out of a misguided expectation.

**Next Month:** More on Desk Juggling: Designing Players and the Statistical Metagame

### What do you think?

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### Previous columns

- Testing the Boundaries II - Putting Things Together by Mendel Schmiedekamp, 28dec05
- Testing the Boundaries I - Taking Things Apart by Mendel Schmiedekamp, 29nov05
- Need for Speed by Mendel Schmiedekamp, 21oct05
- Groups Redux by Mendel Schmiedekamp, 23sep05
- Two for One by Mendel Schmiedekamp, 19aug05
- Death of Culture by Mendel Schmiedekamp, 27jul05
- Zero Sum by Mendel Schmiedekamp, 17jun05
- Technobabble by Mendel Schmiedekamp, 17may05
- Constrained by One Another by Mendel Schmiedekamp, 22apr05
- Patterns and Play by Mendel Schmiedekamp, 18mar05
- Role, Play, and Game by Mendel Schmiedekamp, 18feb05
- Symbols and Machines: Formal Languages and RPGs by Mendel Schmiedekamp, 21jan05
- Form and Meaning: Computational Languages and RPGs by Mendel Schmiedekamp, 23dec04
- Growing Games: Natural Language and RPGs by Mendel Schmiedekamp, 19aug04
- Speaking in Tongues by Mendel Schmiedekamp, 21jul04
- Planning for Failure by Mendel Schmiedekamp, 20may04
- Between the Steps by Mendel Schmiedekamp, 23apr04
- The Road Once Traveled by Mendel Schmiedekamp, 24feb04
- Seeds of Enlightenment by Mendel Schmiedekamp, 30jan04
- Mathematics of Game Design: Probability and Statistics by Mendel Schmiedekamp, 23dec03
- Mathematics of Game Design: Game Theory and Related Concerns by Mendel Schmiedekamp, 31oct03
- Mathematics of Game Design: More Finite Structures by Mendel Schmiedekamp, 24sep03
- Mathematics of Game Design: Finite Structures by Mendel Schmiedekamp, 29aug03
- Homeworld Project 2: Modes, Grains, and Journeys by Mendel Schmiedekamp, 28may03
- The Homeworld Project by Mendel Schmiedekamp, 30apr03
- Desk Juggling: Designing Players and the Statistical Metagame by Mendel Schmiedekamp, 26mar03
- Juggling Desks: Balance and Dice by Mendel Schmiedekamp, 29jan03
- Fuzzy Alchemy by Mendel Schmiedekamp, 30dec02
- The Opposition Rests by Mendel Schmiedekamp, 27nov02
- A Brief Introduction by Mendel Schmiedekamp, 28oct02

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