"Averaging Dice" Topic

Some preferences I have with regard to wargaming design.

1) Players prefer rolling pairs of dice.
2) Players seem most comfortable with game modifiers of less than +3
3) Players prefer success/fail game mechanic resolution rather than lookup tables. (I am comfortable with "fail by X dice pips" and "succeed by Y dice pips" to drive more granular test resolution.)
4) Auto-success and/or failures for dice rolls higher than 10% (a d10 with 10's succeeding despite a target number and 1's failing) tend to make atypical units controlled more by chance than their quality.
5) Grading troops into more than 5 to 7 grades becomes cumbersome. I have adopted "3" as a typical grade with two inferior grades (1 and 2) and an increasingly logarithmically power grades of 6 to 8 as maximums seem to allow for very powerful units to exist alongside common units. This is consistent with some commercial games.
6) Ranges of dice results greater than about 6 to 8 make matching the target number exactly statistically less optimal. (I want a marginal success when you just meet your targeting number.) So, a d10 or 2d6 are just too unlikely to hit a target number exactly.
7) Players prefer to roll high to succeed.

In the past, I have contemptuously ignored averaging dice. Individually, they behave much like a d4. Too little spread for most things I want to model. In the past, I have adopted the d8, d10 and d12 to allow for a broader spread. I have always had two problems. The spread made enough uncertainty that unit quality often did not show through in the dice results. (A veteran unit on a d10 with a +1 was only 10% more likely to succeed than a regular unit.) Resolution was often easier with a lookup table than without one. Without some sort of bell curve in the dice results, tactics often resulting in throwing the dice and hoping for an unusually high result.

Recently, I looked closer at rolling two averaging dice and adding the result. (Those familiar with BattleTech might notice some similarities.)

The math looks like this…

1/36 each: a 4 or 10 (an auto-success or failure by many game conventions)

5: 4/36
6: 8/36
7: 10/36
8: 8/36
9: 4/36

Imagine unit grades of the following

+0 to +2 … rabble and non-combatants
+3 Regular Troops
+4 Veteran Troops
+5 Elite Troops
+6 Exceptional Troops

Imagine a game where the designer desires roughly one-third of firearm fire to be effective from a regular unit against an opponent not in the open.

That is a targeting number of 11+ in this hypothesis.

2d6(Avg) +3 (Regular Troops)

Say you define the following…
SNAFU: miss by 3 or more (or double-2's)
Miss: do not meet the targeting number (miss by 1 or 2)
Marginal Success: Exactly meet the targeting number
Success: Exceed the targeting number by 1 or 2.
Critical Success: Exceed the targeting number by 3+ (or double-5's)

So…
SNAFU: 14% (I would say another check where running out of ammo is the most likely result)
Miss: 50%
Marginal Success: 22% (targeted unit suppressed)
Success: 11% (target takes casualties)
Critical Success: 3% (target shredded)

Elite Troops would result this way…
SNAFU: 3% (I would say another check where running out of ammo is the most likely result)
Miss: 11%
Marginal Success: 22% (targeted unit suppressed)
Success: 50% (target takes casualties)
Critical Success: 14% (target shredded)

By the time you factor in things like multiple units creating a crossfire (+1) and aimed shots (+1) by firing every other turn, three regular squads seem about the match for one elite squad.

If someone made a d12 with the following faces: 1,2,2,3,3,3,4,4,4,5,5,6 I would adopt that instead. This would add a touch more chaos into the model and I like chaos.

I would like to solicit input on the math and game design of adding two averaging dice and comparing it to a target number. Within the context of the truism I have enumerated, is there a more elegant way to do this? Are there flaws in either my math or logic that suggest rethinking major parts of my idea?

Feedback appreciated,
Robert

I get your point – rolling 2 average dice will give a much more typical result.

I recently played a game that involved opposed d10 rolls – with small modifiers – if felt very random. Whenever you got to roll 2 or 3d10 then you tended to get a much more typical result.

Your match exactly, pass by 1 or 2 or pass by 3 – seems a bit complex on the surface. I've played a lot of Hail Caesar that uses a similar mechanism for command rolls – and it was much easier to remember, 7 or 8 for 1 move, 6 for 2 or 5 or less for 3 moves – rather than trying to remember to how many you had passed by.

Kings of War, which uses a nerve test, where a fail routs, and a pass by only 1 or 2 sets the unit wavering – expresses this as 2 target numbers (a mini-table).

"If someone made a d12 with the following faces: 1,2,2,3,3,3,4,4,4,5,5,6 I would adopt that instead."

If what you really want is custom dice, then use custom dice. Then you don't have to worry with modifiers at all.

Imagine how much faster it would be just to say: "Crap troops roll a Crap die.. and Elite troops roll an Elite die," rather than mucking about with multiple dice, averaged, plus modifiers.

I am getting a special die that is numbered 7-12, no idea what I will sue it for, but it is fun.

Not a wargame, but see Formula D for a really good use of averaging dice.

Imagine how much faster it would be just to say: "Crap troops roll a Crap die.. and Elite troops roll an Elite die," rather than mucking about with multiple dice, averaged, plus modifiers.
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It's faster to say it, but it's not faster to find the correct dice during a game.
In fact it becomes a major source of frustration in my experience,

"It's faster to say it, but it's not faster to find the correct dice during a game. In fact it becomes a major source of frustration in my experience,"

Well, it would have to be restricted to just a small number of types. Three levels of quality, for instance. And then the dice would be color-coded:

Crap = Yellow (of course)

Average = White

Elite = Red

So it would be pretty easy to remember at a glance: "roll some white dice."

It's faster to say it, but it's not faster to find the correct dice during a game.
In fact it becomes a major source of frustration in my experience,

Another reason to use dice boxes. That and keeping ice off the table.

IIRC, WRG Ancients always used to give regulars average dice and barbarians D6s – I think for command rolls (pips?) (or reaction rolls) but don't remember exactly.

"It's faster to say it, but it's not faster to find the correct dice during a game."

Yes, a complete pita, especially those rules which apply modifiers by changing the special dice depending on circumstances.

Unit X always using a D8 or a D10 or whatever is less of a problem.

A game called Wembley used 6 sided dice with different pips to represent different team strengths/situations.
RED (1st Div home) 4 4 3 2 1 0
GREEN (2nd Div home) 4 3 2 2 1 0
BLUE (3rd Div home) 5 4 2 1 0 0
ORANGE (1st Div away) 4 3 3 2 1 0
YELLOW (2nd Div away) 4 3 2 1 1 0
WHITE (3rd Div away) 5 4 1 1 0 0

Every team could win a game but odds favoured the better teams.

This has a lot of benefits such as no need to add factors for unit quality or to look up results tables.
You can add rules like defenders behind cover might be able to re-roll.
The advantage from a gameplay perspective is that you get to the result immediately. We play man-to-man Dark Age skirmish games with 200 figures a side in a hour using simple dice rolls, no plusses or minuses. It let's players focus on the game not the maths. The games get very tense and exciting just like the original Wembley game did.

Another fun way to do it, is to make three different kinds of dice, by color (say, White, Yellow, Red.) All d6s, but: White dice are crappiest, Red dice are best. Yellow are just normal d6s.

Then unit qualities could be expressed as colors. For instance, each unit could be rated for Command, Move, Fight, and Shoot.

To keep it simple, unless otherwise notified, everybody rolls yellow (normal.)

But a sharpshooter unit might "Shoot Red," meaning that when it shoots, it reaches for the red dice. But they have no bayonets, so they "Fight White."

*

Lots of fun options, but the key is for the designer then to really discipline himself and not use modifiers! It's not the dice that are a problem in most game systems; it's the way the modifiers get added on later. This is tough; we all want and expect: "Plus One for Being Uphill…" or "Plus Two for Saxon Cuirassiers," or whatever.

When I do a game, I force myself to go back through the combat and shooting and movement mechanics over and over and over again, until I can comb out all but a tiny number of absolutely necessary modifiers; things that I just can't get rid of, and which can't be substituted with any other mechanic.

It is very hard to make something simple.

I have been struggling eith this idea in my game design as well (keeping it simple with very limited modifiers). These are some very interesting ideas. I like the idea of throwing a dice with the modifiers built in so to speak. Another option might be to throw extra d6 and just remove best or worst etc…, but that becomes a modifier of sorts again. I have a mechanic that I have been playtesting with different dice (d6 – d12) but it does create too much randomness I think.

Allen

Limited modifiers is a good idea for new gamers who wants to keep things simple. For some veteran players, they like complicated ideas to bring more optional strategies to the game.

I think your preferences are well-thought out, but I don't like non-standard dice. I feel in the best game designs the probabilities are easily accessible, and I find oddly numbered dice take extra effort for the player to understand the probabilities.

The average die is a randomizer with 4 possibilities, 2 of which happen twice as often as the other two. You can do that with a regular D6 very easily by assigning 4 range bands of 1, 2-3, 4-5, and 6. Example:

1 – auto failure
2-3 – miss
4-5 – hot
6 – critical

This is easy to memorize (4+ is a hit, and the max/mins are crit/fumble).

2 Average Dice give a range of 7 results with a bell curve. 2D4 does the same thing. The probabilities don't exactly match your example but the point is having a bell curve, not the exact distributions.

There's also the approach taken by the Savage Worlds games: a success is always a 4+, but the die size increases with quality/skill level. So for example poor troops use a D4, trained D6, elite D8. A 1 is a failure but note its chance decreases as troop quality increases. Also the die "explodes" (on max result roll again and add) and at 8+, 12+ etc the result is double success, triple success, etc, so the chance of a "critical" goes up with troop quality.

Another way to go is a spinner. Your results table is a circle with different size pie wedges. This makes the distribution very visual and easy for the player to grasp.

Thanks for the feedback.

I have designed games with variable dice similar to both Sam and Don's suggestions. I like it fine with only 3 grades of troops, such as how Don explains Savage Worlds. The math gets more tricky with 4 or more grades of troops. Exploding dice solve part of the mathematical problem, but the odds appear awkward for at least one step up in grade when 4 troop grades exist.

Let's say you have d4, d6, d8 and d10 troops under and exploding dice model. If the success target is 4+, the odds are 25%, 50%, 63%, and 70%. From a design standpoint, the lowest grade troop is half as effective and the Second Tier troop. This is too great a difference IMO. To address that, up to target number to 6+. The odds are 13%, 17%, 38%, and 50%. In the second model, the big jump is 2nd and 3rd tier troops. You can fix it in either with a custom dice inserted somewhere, but I am trying to avoid special dice.* As my local FGS sells averaging dice inexpensively, I don't characterize them differently from polynomial dice. They just aren't D&D dice.

* In a 4 Troop grade model, I would be most satisfied with a Target number of 5+, 25%, 33%, 50%, 60%. But it is necessary for the percentages to map well to what you are trying to model. If you want garbage troops to hit less than 25% of the time, you are out of luck. I don't wish to contemplate the math behind a 5-grade variable dice magnitude model trying to use standard polynomial dice. Exploding dice can introduce clumsy mechanics if you wish to have some sort of "roll-great-and-cause-a-critical" result.

The point about being stingy with modifiers is well taken. My design preference #2 comes from that truism. I most admire the elegant game design of Richard Borg.

While I did not explicitly identify such a preference, I like the way a d6 and a d8 roll. In addition to the expense of a custom d12, I don't particularly like the tactile way the d12 rolls. It also makes a dice box necessary. I love the math of the d12, but it lacks curb appeal.

I own a set of Ubiquity Dice. These are d8 dice colored red, blue and white. They number from 0 to 3. I don't recall which color is the strong dice and which is the weak dice. They are much like the Wembly dice thehawk shared.

I disagree with Don's preference that the best games are when "the probabilities are easily accessible." I would counter that the best games are when the probabilities support intuitive tactics based on historical sound tactics. I am not advocating complexity, in fact I forcefully advocate for simple mechanics, but rather it is not necessary for even the experienced, savvy player to fully understand the odds before they roll. This is demanding for the game designer. The result must win the trust of the savvy player.

I have seen many clever, inexperienced players oppose more experienced players. The tactics employed by the inexperienced player would have been just as sound as the more experienced player in RL. However the mathematical understanding of the more experienced player leads to a lop-sided game. The inexperienced player gets frustrated. Over time, this problem can worsen with tactical games as the gap between experienced and inexperienced players grows. Ideally, I would think an inexperienced player using common sense and RL tactics should have at least 35% chance of defeating a very experienced opponent. Keeping the math deep and the mechanics simple can promote longevity in a game.

Thanks again for humoring me,
Robert

Good well-thought out comments.

I'm no expert on the Savage Worlds game mechanics, having only played one game and otherwise just observed their mechanics from afar. But I believe the D4 represents "untrained" and the D6 "trained". This probably justifies a skill jump from 25% to 50%. The rest of the jumps are more gradual as you note (D8/63%, D10/70%, D12/75%).

Also the number of successes is tied in with the effect. For example -- I am guessing here -- in a modern warfare game each fire team rolls a die, a single success suppresses the target, and additional successes cause the casualties. This makes an untrained fire team rather unimpressive (D4 giving 25% suppress, about 6% 1 kill, and 1.5% 2 kills) whereas an expert fire team is much more effective (D8 giving 63% suppress, 13% 1 kill, 8% 2 kills, about 1.5% 3 kills).

Morale checks to throw off suppress would use the same system, with untrained troops throwing off suppress slowly (D4 giving only 25% chance each turn) and staying pinned far longer than veterans (D8 giving 63% to throw off suppress).

. . . best games are when the probabilities support intuitive tactics based on historical sound tactics . . . it is not necessary for even the experienced, savvy player to fully understand the odds before they roll . . .

I don't disagree with this per se but . . . IMO the best way to illustrate best tactics to a new player is to compare the success probabilities of 2 different proposed actions. This puts him on the fastest path to familiarity because he can teach himself by continuing to compare probabilities -- that is, if the probabilities are accessible.

We all know cases where bad game design is exploited through this same approach, and we sometimes call that rules lawyering. Personally I feel the fault there lies with the game designer, who made the game probabilities hard enough to discover that only certain players were willing to go to the effort.