Variance and Game Design

Variance: or

How Common Is Average Anyway?

Introduction

Most games incorporate an element of chance, usually with dice, though others use cards, coin flips, etc. While we are familiar with the concept of mean or average of a roll (a measure of central tendency), one aspect that often seems overlooked as designers search for a cool or easy to use mechanic is the question of variability, or how often do you get the same or similar results from the mechanic. The most commonly used measure of variability or dispersion of results around the mean is the variance. I think that it is easiest to look at an example.

Figure 1 shows the resulting distribution of four mechanics with very different probability curves but very similar average values. The possible rolls are given along the bottom while the lines show the percentage chance of rolling that number. The famous twenty sided die, 1d20, is shown in blue and every result on the die, from 1 to 20 has an equal possibility of occurring of 5%, resulting in an average roll value of 10.5. Two d10 added together are shown in red, with average total of 11. Three d6 added together are in green, average total of 10.5, while the black line shows a mechanic where the value is “Always 11”, thus the average is also 11.

It should be obvious that despite the fact that their average values are roughly the same, these different mechanics have very different probabilities of rolling at or near the average. The likelihood of rolling a 9, 10, 11, or 12 on a 1d20 is 1 in 5. For 2d10 it is more than 1 in 3, for 3d6 it is almost 1 in 2, and always for 11. The d20 is the least predictable , i.e. rolling near the average is less common than with the other mechanics. The “Always 11” mechanic has no randomness at all, it always generates an average result.

Variance and the game

Why do we care? First, the distribution of the chosen mechanic can have a dramatic effect on the style of a game. For example, let’s say that Lance Armstrong is rated as +10 bicycling skill and that we will add this to our die roll. A random fan from the audience has a mere +0 skill. If they race and the fan needs to beat Armstrong’s roll to win the chance goes up tremendously with mechanics that have a very wide distribution, i.e., a higher variance. The fan’s die roll needs to be 11 higher than Lance’s die roll, 10 to overcome the skill difference and 1 more to make the fan’s total higher.

Because none of these mechanics can roll lower than 1, the fan has to roll at least a 12 to have any chance of beating Lance. If the fan rolls a 12, a 1 in 20 chance using the 1d20 mechanic, then Lance has to roll a 1 for the fan to win, also a 1 in 20 chance using the 1d20 mechanic, for a 1 in 400 chance of the fan winning. If the fan rolls a 13, again a 1 in 20 chance, then the fan wins if Lance rolls a 1 or 2, a 1 in 10 chance, for a 1 in 200 chance of the fan winning. If we add up all the possible ways the fan can win using the 1d20 mechanic the fan has an overall 11.25% chance of winning (Table 1).

Table 1: Ways of winning for Fan:

Fan wins Lance wins or ties
Fan rolls: Lance rolls: Lance rolls:
Die result Odds Die result Odds % chance Die result Odds % chance
1 (1 in 20) Anything (20 in 20) 5%
2 (1 in 20) Anything (20 in 20) 5%
3 (1 in 20) Anything (20 in 20) 5%
4 (1 in 20) Anything (20 in 20) 5%
5 (1 in 20) Anything (20 in 20) 5%
6 (1 in 20) Anything (20 in 20) 5%
7 (1 in 20) Anything (20 in 20) 5%
8 (1 in 20) Anything (20 in 20) 5%
9 (1 in 20) Anything (20 in 20) 5%
10 (1 in 20) Anything (20 in 20) 5%
11 (1 in 20) Anything (20 in 20) 5%
12 (1 in 20) 1 (1 in 20) 0.25% 2 to 20 (19 in 20) 4.75%
13 (1 in 20) 1 to 2 (2 in 20) 0.50% 3 to 20 (18 in 20) 4.50%
14 (1 in 20) 1 to 3 (3 in 20) 0.75% 4 to 20 (17 in 20) 4.25%
15 (1 in 20) 1 to 4 (4 in 20) 1% 5 to 20 (16 in 20) 4%
16 (1 in 20) 1 to 5 (5 in 20) 1.25% 6 to 20 (15 in 20) 3.75%
17 (1 in 20) 1 to 6 (6 in 20) 1.50% 7 to 20 (14 in 20) 3.50%
18 (1 in 20) 1 to 7 (7 in 20) 1.75% 8 to 20 (13 in 20) 3.25%
19 (1 in 20) 1 to 8 (8 in 20) 2% 9 to 20 (10 in 20) 3%
20 (1 in 20) 1 to 9 (9 in 20) 2.25% 10 to 20 (11 in 20) 2.75%
Total: 11.25% 88.75%

Because the other mechanics favor average numbers, the possibility of the extremes — Lance rolling poorly while the fan rolls well — becomes lower. Using 2d10, the chance that the fan wins is 3.3%; with 3d6, it is 0.45%; and with the “Always 11” mechanic, the fan always looses.

A mechanic with higher variance allows more surprise upsets with the underdog winning. This means that players have a chance against better opponents, though worse opponents have a chance against them. Stylistically, this could work well for a more realistic World War II game as even the best soldier can go down to random gun fire. It would also work well for comedy where the random chance can fuel all sorts of jokes.

A mechanic with more closely grouped results allows more confidence in the expected results. Players can take on people they are much better than, confident that they aren’t going to get trounced due to a poor roll. Stylistically this fits things like classic samurai movies where only someone of roughly the same or better skill can possibly beat a samurai. Neither high or low variance is “better”, they just give a different flavor to the game.

Different levels of variability

Real life is also full of things with very different levels of scatter and variability. Drawing for high card is completely random and no amount of skill will help you unless you cheat. A bicycle race is much less variable, however — much more predictable. Lance Armstrong will certainly have some day-to-day variation but barring a major catastrophe will always win except against other high-end professionals. This can cause problems when your mechanic has much, much higher variability for certain tasks than the real world.

Games generally do not address of the question of variance directly. Though sometimes they do: one old miniatures game used d6 for resolution but veteran troops counted 1s as 2s and 6s as 5s. So novice troops could roll the extremes, 1s and 6s, but veteran troops could not, making the performance of veterans more consistent. However, all games have to deal with this in some way.

The old school method was to simply not bother rolling for things that everyone assumed should work. People always rolled for direct conflicts but not necessarily for driving to the mountain chateau, or even trying to get into the chateau. Even during a fight, some game masters might make players roll to leap the railing while others might not. This handled most things reasonably well, but there was always a question of where to draw that line. Once you decided to roll for something, it went from 100% chance of success to a significantly lower chance. Also, the person’s skill didn’t come into play unless you rolled. Either no one had to roll or the very skilled and the unskilled had to roll.

A modern modification to this technique is the idea of “Take 10” or “Take the Average”. This allows the randomness to be taken out of certain tasks by adding a static number, the equivalent of our “Always 11” example mechanic. However, it also incorporates the character’s skill. If the 10 or average value that is added to the skill is sufficient for the task then no roll is necessary, otherwise you roll. This allows highly skilled characters to avoid certain failures from poor luck but leaves less skill characters with an opportunity to succeed.

Of course, these techniques still give only two types of variance: none at all, or whatever the ordinary mechanic produces. Now, some games do use multiple different mechanics depending on what they are trying to resolve. Old school DnD used 3d6 for generating characteristics, 1d20 for armed combat, and percentile dice (1 – 100) for unarmed combat. However, the different mechanics seem the result of other design decisions rather than an attempt to model different amounts of randomness. Also, some games have mechanics such as using different dice to represent different skill levels, so variability can change with skill level. Again, this seems to be an unintended consequence of the mechanics rather than a deliberate change of variability. Trying to add multiple levels of randomness may simply not be worth it in a game design. Still it is something that can at least be considered.

Other ways to change consistency

So far the examples given have been about changing the amount of variance by changing the die mechanic, but this is not the only way to alter the distribution of results. One can also adjust the values of the skill added to the die roll. The examples had no skill = 0 and high skill = 10. If this were changed to good skill = 50 and high skill = 100 then the choice of dice mechanic for generating numbers 1 to 20 would mean very little. The size of the skill would swamp out the value of the die roll. No matter which mechanic was employed or what the outcome of the die rolls were, a highly skilled character would always beat a moderately skilled one. Likewise, if skill values were very small, for example, a high skill = 3, then the die roll becomes incredibly important with high chances of upsets using all but the “Always 11” mechanic.

Requiring that the player make multiple rolls to succeed can also change the variance. If in our cycling example the fan has to beat Lance four times in a row to win, then the chance of success when rolling 1d20 drops from about 1 in 9 to less than 1 in 6000. After all, multiple rolls will tend to converge on the average. One of the interesting upshots of this and the way most games are put together is that non-combat actions are often resolved with a single roll, while combats are often resolved with a series of rolls. The end result is, barring instant death rolls and the like, that the outcome of the wild, woolly, and chaotic combat is more foreseeable than what should be more reliable and predictable tasks, like picking a lock or repairing a car. As a result there are many systems where an increase in combat skill makes success much more likely than the same increase in a non-combat skill, despite the fact that they often cost the same in terms of character advancement. Mechanics like “Take 10” can help to alleviate these problems somewhat, but designers really don’t seem to think these things through.

Beyond dice

This discussion has been presented in terms of dice rolling mechanics; however, the same ideas apply to any sort of randomizer. For example, a big problem with many card-based systems is their over-reliance on suits. A poor example was a game where cards of the proper suit, say spades for combat, counted at face value with Aces being 14 giving a number from 2 to 14. Cards of any other suit counted as 1. These systems usually give people a hand of cards, but if you are doing the same type of action several times in row then the appropriate suit gets used up quickly and it then basically comes down to drawing cards for the value. This gives a number from 1 to 14 with an average value of 2.75 and a 3 in 4 chance of getting the minimum value of 1. The result is that continuing tasks in such games become incredibly unreliable. Such a set up can be worked around, the GM just has to make sure that the same type of actions don’t occur several times in a row. But the mechanic really forces the GM to compensate for the mechanic.

Conclusion

Obviously, there are many other considerations in the choice of mechanic, such as ease of use, but giving some thought to how much variability a designer wants and how much is produced by the mechanics could make games flow more smoothly.

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7 Responses

  1. Thank you for the article! As you know from exchange we’ve had before, this poor understanding of the effect of variance is one of my pet peeves with game designers (and even gamers in general).

    Several systems out there have poor or inconsistent use of variance, especially the ones that use several different die sizes in skill or challenge resolution, for example, the original Deadlands system and even to a much lesser extent, its heir Savage Worlds.

    However, I think one of the worse offenders out there is/was the Earthdawn system. It’s a level-based system where your abilities move up in step increments where, for each step, the average roll goes up by exactly one point. (See the table here.)

    However, the variance changes wildly with every increment, so that when trying to hit a certain target number, your likelihood of success may actually decrease when you go up by one or more levels. It was a great source of frustration for me when I played this game, despite the really flavourful setting.

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