Dice Spreads, Probability Curves, and Rules Writing

In advance of an upcoming article I am planning to write, where I review the Knights and Magick rules (by Arnold Hendrick, and published by Heritage back in 1980), I wanted to address a rather interesting dice mechanism that are employed in the rules.

That is the 1-15 dice roll.  This is accomplished by rolling two dice, and adding them together.  The first dice is a rather common d6, and the second dice is the typical d10, but numbered from 0-9 (the 0 face on the die is valued at zero, and not at ten).  Adding the two results together yields a number from 1 to 15.

Now, the asymmetry of the roll always bothered me.  How could you possible hope to have game feng shui when you are rolling two different dice, and adding them together?  It was preposterous!  Sort of like the damage rolls from early Runequest and Call of Cthulhu and other games, where you might have 1d8 for the weapon, plus 1d4 for your strength, and maybe +1 for a good quality blade - so your damage looked like 1d8 + 1d4 + 1.  Weird, but okay for roleplaying games.  Wargames were supposed to be above all that sort of thing.  Better.  More pure.

But here we were, with Mr. Hendrick's rules using a D6 and a D10 added together.

Let's look at the results.  To compare it, a brief examination of the (much more common) 2d6 bell curve.  Here we see a smooth progression from one result (for a Two), up to six results for a Seven.  And then back down again to one result for a Twelve.  There are a total of 36 possible dice pairings, so the frequency is a number of times out of 36.  This is pretty standard stuff, that is part of any study of probability, but also should be pretty intuitive to just about any long time gamer, and/or game designer.

Result	Frequency
2	1/36
3	2/36
4	3/36
5	4/36
6	5/36
7	6/36
8	5/36
9	4/36
10	3/36
11	2/36
12	1/36

But now lets take a look at the rather interesting D15 result.  Here we have 60 possible results, rather than 36.  There are more possible results (1-15, rather than 2-12).  Plotting the possible results we see:

Result	Frequency
1	1/60
2	2/60
3	3/60
4	4/60
5	5/60
6	6/60
7	6/60
8	6/60
9	6/60
10	6/60
11	5/60
12	4/60
13	3/60
14	2/60
15	1/60

Okay, so what do we see from this?  First, the numbers of all the middle results, from 6-10, have the same frequency - 6, our of 60 results, or a 10% chance of getting any of those numbers.

How does this behave in the game?  Well, we end up with many more middling values (half the possible results are from 6-10).  Plus, in a game that deals in + and - factors added to the dice, for a variety of different causes, this will tend to balance out and level the impact of the dice modifiers.  What I mean by that, is that in a 2d6 dice roll, a single +1 or -1 can have a very high percentage impact on the dice chance, especially if your base number is off the middle.  In this case, the middle (or stable region) is spread out, so that dice modifiers are more predictable in their impact (a minor change with a +1 or -1).  This also means that having additional plus or minus factors won't be an overwhelming impact, as it is in a 2d6 curve.

Once I pulled out the results, and looked at them, the asymmetry doesn't bother me AS MUCH, but it is still there.  I think I would prefer a 3d6 roll, or 5d4 roll, to balance out that middle and level it - but that is a different story.

Now, with the recently reviewed Hackbutt & Pike rules (written by Ben King, in the Tac-50 rules series), we see that the casualty table is driven by a dice difference.  Again, looking at the 2d6 probability table, but selecting results based on the difference between the two numbers (agreeing that doubles results have to be rerolled), we have the following:

Die Difference	Frequency
1	10/36
2	8/36
3	6/36
4	4/36
5	2/36

Of course, the total frequency of pairs with a difference only adds up to 30 out of 36, again because we tossed out the possible results where doubles are rolled on the dice (6 chances out of 36).

This is a very flat progression.  It is also interesting that on the Tac-50 tables, the least valued result is always a difference of 5, followed (in order) by 1,2,3,4.  A very interesting use of an interesting probability spread.

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One more thing - I have a plan to (very soon) write a review of the original Sword and the Flame rules.  In those rules, as in many others before and since, combat is resolved between two miniatures by rolling a dice on each side, and the high scorer wins the fight.  With no modifiers, and with Tie results not counting, there are 15/36 chances for each side to win, if using six sided dice.

• Side A wins - 15/36
• Side B wins - 15/36
• Neither Side (tie) - 6/36

But what happens when we change this by just a little bit?  Say, as in The Sword and the Flame, one side or the other gains a +1.  In this case, what if Side A gains a plus 1?  Very different results.

• Side A wins - 21/36
• Side B wins - 10/36
• Neither Side (tie) - 5/36

Looking at the results it shifts, from even chances, to a Two-One chance of winning - from just granting a +1 on the dice.

With a +2 on the dice, it is even more extreme.  In fact, looking that the following table, we see it is 5-1 in favor of the side with a +2 on the dice.

• Side A wins  - 26/36
• Side B wins - 6/36
• Neither Side (tie) - 4/36

Now, lets stretch this even further, and take a look at what happens, with no bonuses, but with one side winning Ties...  Again, lets assume that Side A has the advantage, and with no dice modifiers, will win ties.

• Side A wins - 21/36
• Side B wins - 15/36
• Neither Side (tie) - No Ties

From these results, the odds change slightly, from even chances, to 4-3 chances for the side with the advantage.  Not as extreme as a +1, but then again, conceptually, saying that one side or the other wins ties is like saying they got a +1/2.

What we see is that in the case of an opposed dice roll (found in games going back to Featherstone and Grant, of course, but still present in modern sets) getting a +1 or even the benefit of winning ties is a very large bonus.

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This idea of opposed dice rolls is common in many rulesets.  But in some others, there is the possibility of a player rolling several dice (typically more dice, for higher skilled combatants, for instance), and then selecting the highest dice, before comparing it to the opposition.  In this case, for instance, there is a much greater chance for a soldier who rolls 3d6 and selects the highest, to have a great number than his opponent, who is only rolling 2d6 to select the highest.  The component of a good number is still present, as a singleton dice can always come up as a 6, and the highest opposition roll could be any value less.  But what is the probability?

Lets construct a table, where we record the odds (and percentage chance) of getting a number, between 1 and 6, if it is the highest (or tied for the highest) out of a pool of dice.  To keep this simple, and illustrative, I am going to do it for three pools of dice - a single D6, 2d6, and 3d6.  Notice the shift in probability...

Highest Result	1d6 Prob.	2d6 Prob.	3d6 Prob.
1	1/6 (17%)	1/36 (3%)	1/216 (.5%)
2	1/6 (17%)	3/36 (8%)	7/216 (3%)
3	1/6 (17%)	5/36 (14%)	19/216 (9%)
4	1/6 (17%)	7/36 (19%)	37/216 (17%)
5	1/6 (17%)	9/36 (25%)	61/216 (28%)
6	1/6 (17%)	11/36 (31%)	91/216 (42%)

Again, there is a dramatic shift in probability, just by changing the dice rolling mechanism slightly.  By rolling 2d6, and selecting the highest number, your chance of having a 2 is halved, and your chance of having only a 1 is dropped to approximately 3% chance.

So the chances of rolling a higher number, when you can pick from a pool, are better.  That is intuitive, but looking at the table for 3d6, we see over a 40% chance that your number will be a 6.   And if you need AT LEAST a 5 (which means that the results for a 5 or a 6 will satisfy) your chance increases to 60% (42+28), and so on.

At this point, I will abandon the exercise.  It might be nice, to compare the chances of a player rolling a pool of 2d6 to beat a player rolling a pool of 3d6, but I think I would rather return to writing reviews.  And maybe lunch.

Dice games - push your luck

So, the idea with a push your luck style dice game is that with every dice roll (that you survive) you potentially reap some rewards.  As your pile of rewards gets bigger and bigger, the possibility of NOT surviving a dice roll (and losing all those goodies) becomes the intangible decision point for the player to decide about every turn.

This is true in lots of dice games - Zombie Dice from Steve Jackson Games is one of the more popular.  You get to keep brains (which score points), up until you roll three shotgun blasts - they you are out for the turn and lose all those juicy brains.  If, however, you choose to stop rolling - you get to keep whatever brains you have scored so far that turn - but you give up the possibility of rolling yet further brains (and further shotgun blasts).

Now, in comparison, take a look at Rollin' Bones - the Pirates of the Caribbean On Stranger Tides dice game.  This is an attractive enough game.  Comes in a cloth bag, with a nice color rules folio, and with bones shaped like finger bones.

From Board Game Geek -http://www.boardgamegeek.com/image/1632171 

The "bones" have four sides - 1 skull, 1 bone, 1 shovel, 1 sword.  On your turn you roll any "bones" you have accumulated (but not yet buried), including one more from the pool (the bone yard).

If you get 1 skull symbol, your turn is over, and you lose any "bones" you have accumulated, as well as those you picked this turn.  That is a 25%  chance - per dice.  So the more dice you roll, the more chance you have of losing EVERYTHING.  Why would you ever choose to roll more dice than the minimum?

I haven't figured this game out yet - the components are neat enough that a simple rules fix might make it a better game.  Still working on it....

Gaming weekend in Clemmons

The entire staff of Gaming with Chuck (except for the two Battle Cats, who stayed behind at GwC headquarters, to defend against the encroaching hordes) took a journey from Columbus GA up to Clemmons NC, to engage in a weekend of Bacchanalian Ludo-Revelry.  The games played included a variety of board and card games, as well as loads of roleplaying.

Roleplaying Games

The roleplaying consisted of two sessions of Labyrinth Lord, and one session of D&D Next.  Yours truly ran the sessions of Labyrinth Lord (on Friday evening, and also during the day on Saturday).

Dan Proctor's wonderful clone game, that copies early D&D so very well

The Friday night session followed a character building session for folks who didn't have characters.  That included two GwC staff members (Anita and Heidi) as well as.  It was to be the first RPG session ever for Heidi, although she has grown up playing all sorts of games, and hearing her parents and all the friends of the family talk about the RPG aspect of the hobby.  She chose to create, and play, a 1st level Gnome Thief.  The session involved a bridge crossing (with a goblin ambush), some outdoor encounters, and was going to culminate in a short dungeon adventure, but the evening ended before the gaming did.  The action was loosely based in the Gran March of the World of Greyhawk.

Section of the phenomenal Darlene map for World of Greyhawk

Heidi's character, the Gnome Thief (named Trixie) was not the only newly created character for the evening,  the other players all had new 1st level characters - including Anita rolling up a 1st level female Dwarf fighter named Badb Bigaxe.  Carol had a half elf Ranger called Dil (her name was much longer, and extremely elven, but Dil is a nice nickname).  Marcy played an Elven Druid called Glengaraeth.  Rachel also played a Druid called Aerilyn.  John played a Magic User.

The Saturday day session, featured an installment in the ongoing Greyhawk campaign (detailed, moderately, over at the Sword and Potion blog) that I run.  This featured four of the five regular weekly players being present face to face, as well as one player being present via Google Hangouts (he couldn't travel to Clemmons).  In addition to the Wednesday Night All Stars, we had some additional players.  Heidi and Anita had their characters, Trixie the Gnome, and Badb the Dwarf, upped to 2nd level (for survivability reasons) and they joined the fray.  Also, Dil and Aerilyn joined the group, allowing Carol and Rachel to join the game (they were also bumped to 2nd Level).  John and Marcy are regular players on Wednesday nights, so they switched over to their regular characters (John plays a halfling thief named Flinders, and Marcy plays an Elven cleric named Rhysgil).

Saturday's adventure was a continuation (and part sidebar) of this much larger group, along the main campaign quest, to investigate a curious dungeon located inside the Rushmoors.  The Rushmoors is a swamp that lies over what was once the heart of the Occluded Empire of Vecna.  One of the most noteworthy locations in the Rushmoors is the Black Tower (or Rotted Tower), which was once the capitol of the Empire.  The characters in this campaign, however, are seeking a curious spot inside the fetid and evil swamp, where a curious dragon ship has crashed into the heart of a huge and ancient cypress tree.  This ship has opened up a portal to another world, and is called "The Ship between the Worlds" - also the name of the adventure path the players are following.  This sidebar adventure is inside the Rushmoors, but the dungeon is one where a curious little tribe of fishmen (not too different from Kuo Toa) have an underground temple.  The players invaded, began investigating, and disrupted the nest of evil.  The adventure wasn't finished, but it was a fun time for a long Saturday afternoon of gaming.

A different version of Greyhawk, this time showing the Rushmoors, lower left.

Sunday's gaming was run by John, and was a short adventure using the latest playtest package of D&D Next.  I made up a character - hailing from the city of Dyvers, at the northern edge of the Gnarley Forest, named "Robin of the Gnarley Wood".  The character has a heavy Errol Flynn influence and was quite fun to play as a swashbuckling bravo!

Robin of the Gnarley Wood

There were all new characters around - Steve with a wizard, Rachel with a Druid, Marcy with a Cleric, Heidi repeating her Gnome Thief, Rowdy with a Monk, and Carol repeating her Elven Ranger.  It was fun to play, but it still doesn't feel like AD&D (or D&D) to me, yet.

Board Games

A wonderful weekend of gaming, that also included some board and card games.  We played a game of Eight Minute Empire.  This is turning out to be quite a fun little game, I can't wait for the release of the successor (which I backed) to come out - Eight Minute Empire Legends from kickstarter.

We also played a game of When Zombies Attack. This is a new dice game based on the players trying to survive a zombie attack.  Slightly different than the (superior) Zombie Dice game from Steve Jackson Games, where the players are themselves Zombies, trying to collect Brains.  This game has the dice representing Zombies, and the players trying (through dice rolls) to get rid of the Zombies.  If a player survives a whole round being zombie (dice) free, then they are the winner.  In one or two respects it is superior to Zombie Dice, but in general, I think it will not surpass the original.  The biggest advantage, is not having to keep score.  But, I have heard that it is easy to keep score in Zombie Dice, if one uses little plastic brains or skulls (or similar) instead of writing the score on paper.  Must try.

Finally the group played a round of The Resistance.  This was a fun game as a filler between the long Labyrinth Lord session on Saturday, and the evening activity (which was a ctered BBQ dinner and a group viewing of "The Avengers" at Steve's wonderful home theater at his house in Greensboro).  We had a great time playing Resistance, even though it was a new game to most of the players.  Everyone fell into it right away, with lots of accusations and paranoia out of the gate.  It was a seven player game, and the Government Spies won!!  Treachery!