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Weighted dice mechanics

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releppes
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Joined: 09/17/2010

Not a game or problem. Just an interesting observation of a simple combat system.

I was tinkering around at www.anydice.com with simple weighted dice. My favorite form of weighted dice is dividing a d6 into 1:2, 1:3, and 1:6 probabilities. I like this arrangement because it's easy to represent all three schemes using one die.

Imagine a d6 die arranged as such:

1: MELEE
2: MELEE
3: MELEE
4: PIERCE
5: PIERCE
6: MAGIC

Let's also say:

MELEE = 2 points damage
PIERCE = 3 points damage
MAGIC = 6 points damage

Before the die is rolled, the attack type (MELEE, PIERCE, or MAGIC) must be declared. Any value other than the declared attack type is counted as a miss.

NOTE: All three attack types will produce an "average" score of 1.

The interesting observation is that small risk (ie: melee) stands a better chance of producing a score, but a large risk (ie: magic) can win the lottery. However, over the course of rolling many dice, the average (ie: risk vs reward) is about the same.

For example, say a monster had 12 hit points. The odds of getting a kill with 6 dice are:

MELEE = 1.56%
PIERCE = 10.01%
MAGIC = 26.32%

It's a no brainer! One would think MAGIC is the obvious choice of attack. However, there's also a much larger chance to miss with MAGIC. By the time 12 dice are rolled, the average of doing 12 points of damage is about the same (61.28%, 60.69%, 61.87%).

I found this to be an interesting, yet simple combat system where one could use one die to represent three combat moves. Each with different risk vs reward, but all three being approximately equal.

releppes
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Joined: 09/17/2010
another interesting scheme

Another interesting scoring mechanic is weighting a d6 die as follows:

1: MELEE (2 points)
2: MELEE (2 points)
3: MELEE (2 points)
4: PIERCE (3 points)
5: PIERCE (3 points)
6: (score x2)

And rolling 3 dice at a time (ie: 3d6).

As before, the attack type (MELEE or PIERCE) must be declared before rolling. Any result other than the declared attack is considered a miss (ie: 0 points). The twist is a 6 will double the score shown on the scoring dice.

Example:

Player declares MELEE and rolls (1,2,3) = 6 points
Player declares MELEE and rolls (1,1,4) = 4 points
Player declares MELEE and rolls (1,1,6) = 8 points (score of 4 doubled)
Player declares MELEE and rolls (1,6,6) = 8 points (score of 2 quadrupled)
Player declares MELEE and rolls (4,4,4) = 0 points
Player declares MELEE and rolls (4,4,6) = 0 points (score of nothing doubled)
Player declares MELEE and rolls (6,6,6) = 0 points (rare, but still nothing)

The effect of this mechanic is that 3 dice could represent two attack types where one attack will produce values (0,2,4,6,8) and the other attack will produce (0,3,6,9,12).

Like the previous mechanic, it's a risk vs reward decision for the Player. A MELEE attack will produce a lower more consistent score. A PIERCE attack can produce a better score with a greater chance to miss.

Unfortunately, I don't have any stats as I've not figured out how to represent this scheme in AnyDice yet. I'd be interested to know if MELEE vs PIERCE was equal.

innuendo
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Joined: 05/25/2010
You know, with this system,

You know, with this system, short of being at infinity rolls (which is exactly equal), magic is by far the better choice.

see: http://anydice.com/program/33c and look at the graph tab. As you can clearly see magic puts out more damage, more reliably. Sure there is the bigger chance of "whiffing" but the amount of damage put out per hit is far greater and makes up for it. This is over 100 rolls, but modify it for 10 or 5 and the pattern remains, magic is just better at doing damage.

The only time I wouldn't take magic was if what I was trying to kill had <6 health. other than that, it's strictly the better choice.

releppes
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Joined: 09/17/2010
Not so sure

I'll agree that MAGIC is better, but I completely disagree with the "clearly" statement.

The Graph (viewed on Normal) is misleading. Sure it shows MAGIC with higher percentages around the average, but those data points are very sparse. You can clearly see using your example of 100 dice that MELEE, though less, has a much higher density around the average. So yes, it's a complete risk vs reward.

If you view your Graph with the Odds set to At Least, you'll get a clearer picture. From that graph, you clearly see that the Odds all intersect at ~50%. No, I wouldn't say MAGIC is a clear winner. The steeper MELEE curve indicates that MELEE is far more consistent around the average.

However, one statement that is true is that MAGIC does appear "slightly" better. Instead of using 100 dice, use 120 (evenly divisible by 6). View the Table with the Odds set to At Least. You'll see that MELEE hits the average 53.63% of the time, PIERCE at 53.43% of the time, and MAGIC at 53.80% of the time.

What I find odd is that for every multiple of 6, MAGIC is always "slightly" better and PIERCE is always "slightly" worse. Using 120 dice is a little ridiculous, but you get the same conclusion with 12 dice. The values around the average are all very close. So close that I would hardly say one attack has advantage over the other. But it's odd that MAGIC is slightly higher (+0.59%), and PIERCE is slightly smaller (-0.59%). For some reason, I expected the odds at the average to be "exactly" equal.

innuendo
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Joined: 05/25/2010
Sure, when the battle goes on

Sure, when the battle goes on it's similar in results...but it seems that in any game most battles wont be 12 dice rolls.

Let's break this down into a table, vertically is the number of rolls in the combat, and the table is the % chance to deal 12 damage

Rolls....... melee........pierce..........magic
..4.............0.............1.23............13.9
..8...........14.5...........25.9...........39.5
..12.........61.28........60.69.........61.87
..16.........89.49........83.41.........77.28

(sorry this is ugly)

And if you want to compare, try doing it with n damage and k rolls. As long as n>k, magic is better. The break even point on all 3 attacks is when the number of rolls in the combat is the same as the damage required to kill the monster (this goes back to damage per roll = 1). Since I imagine most combats with a 12 health monster will last less than 12 rolls (assuming this game isn't malicious), and most battles with a 6 health monster last less than 6 rolls, and most battles with a 500 health monster last less than 500 rolls, I will be rolling magic.

You are correct though, i figured it wasn't until infinity this was the case, the it is still at a fairly high number of rolls until melee catches up.

releppes
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Joined: 09/17/2010
Using this mechanic in an RPG

So if this magic/pierce/melee scheme were to be used in an RPG, I could see a simple leveling up system.

Let a character get defined based on the attack type most used. As a character does battle, experience gets awarded based on the attack used for a kill. So if a monster was killed using a melee attack, then melee experience gets awarded. Once experience reaches a sufficient level, the character gets awarded a new skill level.

Example:

Let skill level == dice rolled.
Let monster level == experience awarded.
Award new level every 10 experience points.

Player skill stats are:

melee level 1
pierce level 1
magic level 1

melee experience 8
pierce experience 5
magic experience 3

Implying the player is still a level 1 character, but does have some experience. Being level 1, player can only roll one die for any attack. Player does battle against a level 3 monster.

Turn #1: Player rolls 1 die and does damage with a pierce attack.
Turn #2: Player rolls 1 die and does more damage with pierce attack (monster close to dieing).
Turn #3: Player rolls 1 die and kills monster with melee attack.

Player gets awarded 3 experience points for melee. Melee experience is now 11, so player's melee skill is now 2. Next melee attack, player can now roll 2 dice.

It's novel and cheap and mildly amusing.

releppes
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Joined: 09/17/2010
Still not convinced

innuendo wrote:
...And if you want to compare, try doing it with n damage and k rolls. As long as n>k, magic is better. The break even point on all 3 attacks is when the number of rolls in the combat is the same as the damage required to kill the monster (this goes back to damage per roll = 1)...

Yes, this is the conclusion I came to as well. If the number of dice rolled is less than the target damage, then you stand a better chance to get an "instant" kill using magic. However you also stand just as much of a chance to miss.

Using your example, let's say the target damage is 36. Let's also say you can roll 6 dice per turn. There's a remote chance you can get an instant kill with magic. There's no chance possible to do it with melee. It'll take at least 3 turns with remote chances to kill with melee (1.56% per turn). Three turns with magic look pretty good (26.32% per turn). That's the deception!

What you're not considering is that melee will miss only 1.56% of the time. With magic, you stand a chance to miss 33.49% of the time.

Rolling 6 dice per turn, it's likely you'll need more than three turns to do 36 points of damage. It still looks more promising that magic will do the job faster, but it's far more erratic. It's seems melee will definitely take longer, but the results should be more consistent.

That's why I found this little experiment interesting. I too look at magic as the obvious choice for attack. It would appear that rolling a 1:6 with higher payout far outweigh the odds of 1:2 with lower payout. But I'm not convinced this is true.

AnyDice is a nice tool, but I'll have to create a simulation of this to prove it to myself. Put a wizard with 36 hits points against a knight with 36 hit points and let them beat on each other over and over. Run the simulation for 1000 times and see if the wizard proves to be the consistent winner. Then see how things change if I let each player use multiple dice per turn.

innuendo
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Joined: 05/25/2010
I'll work something up real

I'll work something up real quick to give us some results...I used to be really good at java, and i'll just dust off the old skills.

releppes
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Joined: 09/17/2010
I stand corrected

innuendo wrote:
I'll work something up real quick to give us some results...I used to be really good at java, and i'll just dust off the old skills.

It's appears you were right innuendo!

I ran some (sloppy) simulations using Perl, and here's the gist:

$VAR1 = '5.09689031096893';
$VAR2 = \bless( {
'wins' => 3164,
'average' => '28.1008899110089',
'skill' => 'pierce',
'name' => 'Archer',
'dice' => 6,
'score' => 27
}, 'Player' );
$VAR3 = \bless( {
'wins' => 2588,
'average' => '28.0843915608439',
'skill' => 'melee',
'name' => 'Knight',
'dice' => 6,
'score' => 18
}, 'Player' );
$VAR4 = \bless( {
'wins' => 4249,
'average' => '28.7149285071493',
'skill' => 'magic',
'name' => 'Wizzard',
'dice' => 6,
'score' => 36
}, 'Player' );

The results needs some explaining. I put three players in a race to get 36 points rolling 6 dice at a time. I then ran the game 10,000 times. The first $VAR1 output is the average number of turns it takes before someone wins the game. As you can see, it takes about 5 turns for someone to reach 36 points.

Each player scores on average about 28 points per game. That seems even enough, but the revelation is the number of wins each player tallies. The wizard definitely scores more wins than the archer and knight. You were correct!

I ran the simulation varying the number of dice thrown per turn, but that doesn't change the outcome much. If I raise the target score to 1000 points (instead of 36), the results are much closer, but the wizard generally wins more games.

There's variance each time I run the simulation, but in general, the wizard always wins. So I stand corrected on my initial belief that all three attack types are roughly equal. It's hard for me to conceptually accept why this is so based on the probabilities, but the simulation speaks the evidence.

BTW: I did randomize the player order each game as to not play favorites.

innuendo
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Joined: 05/25/2010
Well I can admit when I'm

Well I can admit when I'm wrong...I made a program that I can customize the starting health and number of battles and these are some sample results.

Mages attack only magic
Knights attack only piercing
Warriors attack only melee

Each round of battle both parties attack. In events where both are left with 0 or less health, a draw is recorded.

Mage vs. Warrior
Over 100000 battles with 32 starting health:
14042/83824/2134
Percent Mage Wins: 14.042%.
Percent Warrior Wins: 83.824%.
Percent Draw: 2.134%

Mage vs. Knight
Over 100000 battles with 32 starting health:
43399/53964/2637
Percent Mage Wins: 43.399%.
Percent Knight Wins: 53.964%.
Percent Draw: 2.637%

Warrior vs. Knight
Over 100000 battles with 32 starting health:
87854/9841/2305
Percent Warrior Wins: 87.854%.
Percent Knight Wins: 9.841%.
Percent Draw: 2.305%

I'm honestly not sure I understand the results, but I'm 99.99% confident in the tool preforming the operations right. It seems like consistency pays?

innuendo
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Joined: 05/25/2010
Well ignore my last post

Alright, Figured out my errors, Please ignore my last post, it was woefully wrong. My results match up exactly with yours. The Mage is a clear favorite... (I also changed my terms to yours, archer makes sense).

Mage=magic; Archer=pierce; Knight=melee;

Knight vs. Mage
Over 100000 battles with 30 starting health:
44130/52815/3055
Percent Knight Wins: 44.13%
Percent Mage Wins: 52.815%
Percent Draw: 3.055%

Archer vs. Mage
Over 100000 battles with 30 starting health:
45345/51760/2895
Percent Archer Wins: 45.345%
Percent Mage Wins: 51.76%
Percent Draw: 2.895%

Archer vs. Knight
Over 100000 battles with 30 starting health:
49767/45912/4321
Percent Archer Wins: 49.767%
Percent Knight Wins: 45.912%
Percent Draw: 4.321%

I'm working a system where if all three fight each other who wins, just because I'm curious.

Again, really sorry about the above posts...gotta double check your work right :D

releppes
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Joined: 09/17/2010
Good stuff

Looks like we took similar approaches but arrived at the same conclusion. In my scenario, I made it a race between all three players. The first to 36 points of damage was the winner of that game.

I still don't understand the results. I don't understand why the wizard is favored to win. The average probability of each attack are identical, yet it's clear that over the course of several attacks that the higher risk pays off. It's like the motto is high risk pays off in the long run. I just don't understand why.

innuendo
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Joined: 05/25/2010
I figured out why and it

I figured out why and it makes sense. A lot of it has to do with the starting health. If it is a multiple of 6 it helps the mage (obviously). Try running your scenario with 30 as the goal and then run it was 32. The Knight gains some ground in the 31 scenario since that extra point of damage may as well be 6 to the mage.

That said, the reason is the the mage get's a benifit for finishing quickly. In the battle if it is tied at 6 health, all it takes is one hit by the mage and it's over, even though the knight is going to get to 0 at the same rate, once it's over it's over and the percentages go out the window. So even if over 6 rolls it's the same, if the game ends in 2 because the wizard hits a home run, it's over, even if the wizard would have missed every one of the last 4 rolls he already won.

That's why anydice was misleading, it was always giving the Knight his remaining rolls, when really once the wizard one hits him, he doesn't get the rest of the dice to "catch up" the wizards big hit potential.

That said, some really interesting things happen when you start having 3 and 2v1 fights (something I configured my app to run). For instance, since Mage vs Knight is so clearly biased towards the Mage I assumed a Mage vs Knight vs Knight fight, were each fighter attacks the other two each round until they are dead would have the mage winning by a huge amount as well but the knights actually fare better than the Mage here..

This was weird to me until I realized what is happening. Having two knights on the mage effectively doubles his per round damage compared to only have one knight against him. Whereas from the knights point of view, having the extra knight attack him is negligible compared to the mage in the first fight (a 33.33% increase from the first scenario).

I actually really like this system of combat. there are lots of nuances here and although strait up magic is more powerful. seeing how easy it is to combat by teaming up or applying small bonuses to the damage makes it a great system overall.

For instance, adding +1 damage to each attacker makes the knight 50% more effective and makes the mage only 16.67% better. So you can help close this gap with clever in game bonuses and the like.

If there are any scenarios you want to see the stats on let me know, i can test them quickly.

releppes
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Joined: 09/17/2010
Yes, an interesting mechanic

innuendo wrote:
...I actually really like this system of combat. there are lots of nuances here and although strait up magic is more powerful. seeing how easy it is to combat by teaming up or applying small bonuses to the damage makes it a great system overall...

I think I can understand why things are not equal. For one thing, even though the average per throw are the same, the probabilities are not. Instead of asking what's the average probability is per throw, ask what the probability is to get a specific number. What's the probability for each player to do 6 points of damage. For the Mage, it's a 1/6 chance. For a Knight, it's 1/8. For the Archer, it's 1/9. That would seem to explain things, however the Archer generally outperforms the Knight. Even when I think I'm close to an understanding, there's always more to it. In any case, I can understand that the system is not equal. Why? That I still don't know.

I like your investigation into multiple battles. It's very interesting. Not an outcome I would have expected.

I like the novel simplistic approach to this mechanic, but I'm not sure of the fun factor in game play. It's worth making up a RPG to test the waters.

Another system with a similar approach is to put damage into the number of dice thrown instead of the dice.

http://anydice.com/program/345

Here I'm saying a Knight will throw 2 dice per turn for a damage of (1..2). An Archer will throw 3 dice per turn for damage (0..3) and a Mage will through 6 dice per turn for damage (1..6). In such a system, the Knight would have the overall advantage, however it's not as biased. Plus it normalizes the result better. The only thing I don't like is the multiple dice per turn.

innuendo
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You have to stop looking at

You have to stop looking at average damage per turn, it's irrelevant in an actual battle. This is why the mage always wins, and the archer is 2nd best. Instead of looking at average damage, Start with 6 health each and look at the % chance to end the fight round by round.

----------- Mage vs Knight
Round 1: 16.67% -- 0%
Round 2: 16.67% -- 0%
Round 3: 16.67% -- 12.5%
Round 4: 16.67% -- 31.25%
Round 5: 16.67% -- 50%

Sure in the long run the odds of the knight winning go way up because he's more consistant, the mage is basically rolling for the home run every round. But if the mage hits in the first 2 rounds, it's over. Even if the knight in the long run is more consistent; he has 0% chance of winning until the third round, and even in the third round, the mage is still more likely to have rolled a 6 by now than the knight is to have rolled a hit three times.

Does this help explain it?

::EDIT:: those numbers above are actually a little low for the mage, who has a better chance round by round of course, but the point stands, the mage has the ability to win the game in the first round rounds before the other fighter even has a chance to catch up, this is the mage's advantage.

releppes
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Joined: 09/17/2010
I get it now

Thanks for the better explanation. I get it now!

Here's an updated table showing the probability of doing 6 points of damage.

Turn Mage Archer Knight
----- ------ ------- ---------
1 ----- 16.67 -- 0 ------- 0
2 ----- 27.78 -- 11.11 -- 0
3 ----- 34.72 -- 22.22 -- 12.50
4 ----- 38.58 -- 29.63 -- 25.00
5 ----- 40.19 -- 32.92 -- 31.25
6 ----- 40.19 -- 32.92 -- 31.25

Note, this is only for doing 6 points of damage. As you pointed out, a target score in multiples of 6 may add favor to the Archer of Knight. But as the target get higher, the Mage will generally be favored to win.

Interesting, now I have my answer. Thanks! It's been somewhat of a brain teaser.

BTW: It's interesting that turn 5 and 6 both have identical stats for all three players. It wasn't a typo.

rcjames14
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Practical Application

There are a number of game designs which have implemented a weighted die rolling mechanic. Axis & Allies, Memoir44 and Cookie Fu are three games that immediately come to mind for adjusting the probabilities of success on a six sided die.

In the Axis&Allies board game, they utilized six sided dice pips and assign a different number to each type of unit for attack and defense. Whereas Memoir44 created customized dice similar to your melee, range, magic mechanic. Infantry appears twice, cavalry and artillery once, a grenade once and a blank side once. Cookie Fu introduced the idea of collectible dice, so each die in the set has a slightly different weighting of basic punch, throw and kick sides.

Axis&Allies and Memoir44 resolve success slightly differently. A&A requires that you match the strength of the actor. Memoir44 requires that your die result matches the target. But, they both make a one to one equivalence between success and damage. If you hit, the opponent removes one unit. Cookie Fu allows you to take your die results and perform 'maneuvers' with them which interact roughly on a rock-paper-scissors level. But, some dice may be combined for more powerful maneuvers and each maneuver inflicts a different level of damage.

Both A&A and Memoir44 were remarkable hits. Cookie Fu is an obscure game that flopped completely based primarily upon its very limited strategic nature. But, for a while, I have been contemplating a way to reinvent a Fantasy RPG with weighted dice rolling as its principle mechanic. So, Cookie Fu remains an interesting failure to learn from and your topic reminds me of this long-standing backburner project.

As I see it, there could be some very interesting room to design in the area of customizable weighted dice rolling. By combining elements of Memoir44 and Cookie Fu with a more robust resource and maneuver mechanic, I think there's promise in building a weighted RPG.

So, here are my thoughts (for a fantasy war game):
For production purposes, let's keep the six sided dice conceit and run with all the ways you can modify a weighted system.
Let's assume there are five different energy resources. A-B-C-D-E
The most basic die could have one of each type and a wild. A-B-C-D-E-W
But, you could have a whole range of different possibilities. A-A-A-A-A-W, B-B-B-C-C-C, D-D-E-E-W-W, you can imagine a very large number of combinations.
Let's assume there are units which perform three different types of actions. But, in order to perform each action it must 'use' the appropriate resource.
The most basic unit would have a 1 resource cost action, a 2 resource cost action and a 3 resource cost action. example: A: Move 1 space in any direction. A-A: Move 3 spaces in any direction. C-C-?: Inflict 6 damage to a unit in the adjacent space.
Before the game begins, you would be able to select which dice and which units you play with. But, the idea is that all dice and units would be balanced with each other so you build your armies based off of total number of dice and total number of units. So, you might have a limit of 10 dice and 20 units.
The game could be played on board and the objective could be TA or destroy a certain number of units or capture the flag... whatever.
Each turn you would roll all your dice and then players would take turns activating the powers on their units (by discarding their dice) until both players pass.
Units dealt more damage than their health would be removed from the board (or captured) and there could be a variety of different bonuses to armor, damage, etc... based upon both the terrain as well as the activated powers.
But, when each turn is over, players would have the opportunity to throw their dice again and continue the battle.

This same mechanic could be adapted to a fantasy RPG, by giving each player a set of dice. At the beginning of each round of a combat/conflict, each character would have a number of dice available based upon his class and level and then use them to perform the actions.
Ostensibly, each of the five resources types would correspond to a different type of attribute. So, things that take Strength to complete, would require more strength type dice to be rolled. But, each time a player 'levels up', he would be given the opportunity to gain one more die of his choosing based upon the 'class' he wants to take. So, each character would be identified by the type and number of classes they have acquired throughout the game.
Instead of rolling dice all at the beginning of each round... each player has a pool of dice each round and when he wishes to perform an action, he chooses to roll any number or type of dice from his pool. That way, if a player really wants to get one action performed, he can roll all his dice at once, but he would be drained for the rest of the turn (and defenseless). Defensive players may choose to roll a minimum number of dice at the beginning to wait until others are drained to counter-attack, etc...
But, essentially this, first choose your action then roll customized weight dice to see if you succeed mechanic, combined with a much larger pool of dice would give players a lot of discretion over how they hedge their risk. Specialists (players with a lot of dice of one or two energy types) could count on doing certain things more consistently and more powerful things, but they could also likely be unable to do things outside their field. So... as players level up, they need to choose their dice wisely.
In fact, in this mechanic, all actions could be known to all players. So, really what makes your character yours is the dice you own. You may tell the GM what you want to do and he may say that requires 1 strength and 1 dexterity die and you would then choose which of your dice you want to roll.
In combat situations, the dice available to you would be time sensitive so you do not get a chance to refresh your pool until everyone passes. But, the requirements for performing any action may be low (to allow for defensive responses). Whereas, in non-combat situations, you may be able to use all your dice at once and the die results needed may be much higher.

As a third possibility, this mechanic could be adapted into a Euro game. Each turn, players would roll all their dice and they would then have to dice how they want to allocate them amongst a number of different tasks, with different tasks requiring a different number/type of dice. However, the number and type of dice that players start with may be remarkably smaller than what they end up with, so a lot of the mechanic might revolve around collecting dice.
As a typical Euro, economic, engine, the players may use dice to acquire access to more dice... in which case the choice of what dice they acquire each turn will act as a form of strategic commitment over the game.
You could even layer another currency mechanism onto the game and use cards to roll different dice. So... there may be six to ten different dice and cards that allow you to roll D number of X,Y,Z dice. You would have to select at the beginning of each round which cards you want to play. Once everyone has selected their cards, they roll those dice and use them to perform other actions. Since cards can be discarded, they might be the fundamental currency and the limiting resource... So players have to decide when best to use what cards to do what they want to do.

But, essentially, each of these mechanics pushes the design possibilities of using a weighted die system with the possibility to customizing the results and controlling the number of dice you have access to.

releppes
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The magic number

I thought the weighted d6 {2,2,2,3,3,6} was interesting because 3 unique probabilities could be achieved with a single die, and all three weights had an equal average:

(2+2+2)/6 = 1
(3+3)/6 = 1
6/6 = 1

So if you had (n) dice, the average of getting a 2, 3, or 6 was (n).

I incorrectly thought this also meant that all three probabilities (1:2, 1:3, 1:6) with weighted payouts would also be equal, however I was wrong. Still, I figure there's got to be some merit to such a weighted mechanic.

A similar scheme can be achieved with a d10 die {3,3,3,3,4,4,4,6,6,12}. Here I have 4 unique probabilities being represented by 4 unique weights. Like the d6 example, the average weight is the same:

(3+3+3+3)/10 = 6/5
(4+4+4)/10 = 6/5
(6+6)/10 = 6/5
12/10 = 6/5

It's not as interesting as the d6 example, but unique all the same. As far as I know, only a d6 and d10 can be divided this way.

releppes
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releppes wrote:...As far as I

releppes wrote:
...As far as I know, only a d6 and d10 can be divided this way.

I stand corrected. There's a few more interesting weight mechanics. Here's some for d12 and d20.

d12: {12,12,12,12,12,15,15,15,15,20,20,20}
The probabilities are: 5:12, 4:12, 3:12
The weighted average is: 5

d20: {10,10,10,10,10,10,12,12,12,12,12,15,15,15,15,20,20,20,30,30}
The probabilities are: 6:20, 5:20, 4:20, 3:20, 2:20
The weighted average is: 3

Not that it's important, but I find these weight schemes interesting because each weight is unique as well as it's respective probability. The d6, d12, and d20 weight schemes produce a weighted average that's an even integer. In all weight schemes, the probabilities are sequential.

treyalsup
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releppes wrote: It's not as

releppes wrote:

It's not as interesting as the d6 example, but unique all the same. As far as I know, only a d6 and d10 can be divided this way.

Interesting thread. But I think you can probably do this with just about any sized dice, its just that the results become less clean. I started playing around with a d8 and came up with this example.

{3,3,3,3,4,4,4,12}

(3+3+3+3)/8=12/8
(4+4+4)/8=12/8
(12)/8=12/8

treyalsup
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etc

d12 3/3/3/3/3/3/6/6/6/9/9/18 (though 4 types of damage)
or
d12 2/2/2/2/2/2/3/3/3/3/6/6 (3 types- just doubling the 6 really)

d3 1/1/2 (though 2 types)
d4 1/1/1/3 (2 types)
d5 2/2/2/3/3 (2 types)

etc. unless I am not understanding what makes the d6 and d10 "unique."

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releppes wrote:releppes

releppes wrote:
releppes wrote:
...As far as I know, only a d6 and d10 can be divided this way.

I stand corrected....


You edited! :)

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treyalsup wrote:d12

treyalsup wrote:
d12 3/3/3/3/3/3/6/6/6/9/9/18 (though 4 types of damage)
or
d12 2/2/2/2/2/2/3/3/3/3/6/6 (3 types- just doubling the 6 really)

d3 1/1/2 (though 2 types)
d4 1/1/1/3 (2 types)
d5 2/2/2/3/3 (2 types)

etc. unless I am not understanding what makes the d6 and d10 "unique."

Yes, I tried to catch myself after making such a blanket statement.

When I first started toying with carving up a die into multiple probabilities, I was fixated on sequential distribution (ie: 1 of one weight, 2 of a second weight, 3 of a third weight, ...) Hence I only had examples for d6 and d10.

I considered your second example for the d12, but tossed it out for the same reason you pointed out. The probabilities of each weight was the same as the d6 example. However, I like that first example for the d12. Very nice! I gave another example for d12 as:

{12,12,12,12,12,15,15,15,15,20,20,20}

I don't know what a d3 or d5 is. I figured a d4 is the smallest practical die.

Of the d4 example, I considered it, but didn't give it merit. It's not really two unique probabilities. It's more like one probability and it's inverse.

I considered some d8 and d10 examples, but the one difference between d6/d12/d20 and the d4/d8/d10 is that the prior give averages that turn out to be whole numbers (not fractions). It's not a big deal, but I gravitated towards the d6 example {2,2,2,3,3,6} because the average of getting a 2, 3, or 6 is always 1. It works out nicely, but in the end doesn't mean much ;)

As for the d12 and d20 examples I gave, the d12 produces an average of 5 when rolling for a 12, 15, or 20, and the d20 produces an average of 3 when rolling for a 10, 12, 15, 20, or 30. Again, it's nothing special other than being a whole number instead of a fraction like you get with d8 and d10.

My investigation into weighted mechanics may seem pointless. I was considering cheap ways to get varying probabilities using less dice. In particular, what the options were when restricted to one die.

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I don't think its a pointless

I don't think its a pointless discussion at all. I think it has a pretty broad application too.

For example, the early part of the discussion I thought could be highly useful in rethinking MMO design. Character Min/Maxing often focused on DPS but this example points out how an average of damage can be misleading. Throw in healing into a combat situation and the larger "chunk" of damage becomes even more valuable.

I could see you take this idea and do an expansion to James Ernest's Button Men for example.
http://www.boardgamegeek.com/boardgame/17/button-men
its also an iphone app.

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More d12 variations

4/4/4/4/4/5/5/5/5/10/10/20
1/1/1/1/1/1/2/2/2/3/3/6

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Is this really any different

Is this really any different from saying something like:

Magic hits on a 6
Pierce hits on 5+
Melee hits on a 4+

? It seems like it's just plain simpler to state it that way. I am working on a game with that very same combat mechanic (though using 2 dice for a bell curve). So I have, for instance, Weapon A hits on a 7+ (~58%) and does 1 Damage, while Weapon B hits on 8+ (~42%) and deals 2 damage. There are other contributing factors in deciding a weapon, but still.

Is there a point to saying "1-3 = X, 4-5 = Y, 6 = Z"?

releppes
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stormywaters wrote:Is this

stormywaters wrote:
Is this really any different from saying something like:

Magic hits on a 6
Pierce hits on 5+
Melee hits on a 4+

? It seems like it's just plain simpler to state it that way. I am working on a game with that very same combat mechanic (though using 2 dice for a bell curve). So I have, for instance, Weapon A hits on a 7+ (~58%) and does 1 Damage, while Weapon B hits on 8+ (~42%) and deals 2 damage. There are other contributing factors in deciding a weapon, but still.

Is there a point to saying "1-3 = X, 4-5 = Y, 6 = Z"?

Funny, I just stumbled across this "simpler" mechanism yesterday. I created a blog entry for myself and named it "Simplified weighted dice mechanic". :)

Yes, there's little difference in this mechanic over other schemes.

As for the point of {X,X,X,Y,Y,Z}; it was just a way to get three different probabilities on one die. It could have been stated X=4+, Y=5+, and Z=6. But I was thinking along the lines of customized dice. Have a d6 with three sides depicting X, two sides depicting Y and one side depicting Z. From a visual perspective, I thought it'd be easier to read the results. Like Heroscape dice where they used 3 sides to show attack and 2 sides to show defense.

The rest of my long winded discussion was about balancing probability with weights such that on average all events were equal. That's where choosing X=2, Y=3, and Z=6 came from.

I then branched out and looked at other dice (ie: d8, d10, d12 and d20) to see what sort of "evenly" weighted schemes could be developed. That investigation eventually led to how d6, d8, d12, and d20 dice could be combined to form a unified balanced weight scheme. I found it fascinating how it all worked out.

Then yesterday I came across the same conclusion you did. For 5 event types (A,B,C,D,E), why not just use a d6 and define A=2+, B=3+, C=4+, D=5+ E=6. I already had the C,D and E part from where I started. So what's the difference?

The multi-dice mechanic is a NOVEL approach to balancing 5 events. It allows for custom dice (ala Heroscape) with easy to read output. The simpler d6 example provides a similar scheme with using a standard die with slight interpretation of the results (ie: counting pips). The other difference between the multi-dice example and the standard d6 example is the assigned weights:

custom 3d6 + 2d8 + 2d8 + 3d12 + 5d20 uses the weights A(10) B(12) C(15) D(20) E(30)

standard d6 uses the weights A(12) B(15) C(20) D(30) E(60)

The multi-dice example has a smoother distribution.

If interested, check out my Blog entry for the standard d6. You may like it for your combat. It's simple and it works great with any number of dice. For example:

3d6 will provide damage in the range (0-3) for all five events (A,B,C,D,E). The A event has a high chance of success while E has a low chance of success. The balance is:

A/E = 60/12 (ie: A is weighted as 12, and E is weighted at 60).

Meaning event A will score on average 60 hits for every 12 hits scored by event E (ie: 5:1 ratio). It's a nice system because if you want damage in the range (0-5), just use 5 dice. Use the weights to balance character stats (ie: armor + health).

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