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Die modifier mechanics and "value"

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chowdah
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Joined: 12/31/1969

Question for those who are smrter than me - which is probably all of you =)

Working on a dice (6-sided) mechanic system for a wargame and need some help with the "value" dice modifiers.

The basics of the system: Both players roll X dice (X can be a different value for each player) and count any results of 4 or more as a success. You are attempting to roll more successes than your opponent. The more successes over your opponents the better.

Now I want to add modifiers to this system 4 different ways:

1.Bonus or Penalty to the Dice
This modifier ranges from -2 to +2, and modifies the number rolled on each die. For example a +1 modifier means you only need 3 or more for a success. The limit being that a 1 is never a success and a 6 always is a success.

2.Extra or Less Dice
This modifier changes the number of dice rolled.

3.Re-Roll Dice
This allows re-rolling of all or some dice. There is a lot of flexibilty with this modifier.

4.Automatic Success
Bonus successes added after dice are rolled. Like the re-roll modifier there are a lot ways this can be limited/used.

The question in what order are these modifiers "valued"?

I think the auto-success modifier is the most valuable and re-rolls the least, with the result modifier and extra dice falling somewhere inbetween. But I am no mathamatician so the answer escapes me =(

Anyone willing to give this a try?

zaiga
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Joined: 12/31/1969
Re: Die modifier mechanics and "value"

Hello,

I'm no mathematician either, but this is basic probability math:

As an example let's assume that both players roll 100 dice. On average both players will roll 50 "hits". Now, let's see what happens when we adjust the values.

chowdah wrote:
1.Bonus or Penalty to the Dice
This modifier ranges from -2 to +2, and modifies the number rolled on each die. For example a +1 modifier means you only need 3 or more for a success. The limit being that a 1 is never a success and a 6 always is a success.

A +1 means that you will score 4 out of 6 = 67 hits. A +2 means that you will score 5 out of 6 = 83 hits. A -1 will score 2 out of 6 = 33 hits. A -2 will score 1 out of 6 = 17 hits.

Quote:
2.Extra or Less Dice
This modifier changes the number of dice rolled.

A lot will depend on how many extra dice are rolled. To get the equivalent of a +1 modifier (67 hits) you will need 34% extra dice. So, rolling six dice with a +1 modifier is the same as rolling eight dice without modifier, both would score 4 hits on average.

Quote:
3.Re-Roll Dice
This allows re-rolling of all or some dice. There is a lot of flexibilty with this modifier.

Rerolling all dice, even the hits, would only make sense when you roll worse than you would statistically expect, or when you are desperate for some reason.

Being allowed to reroll all the failures would give you an average hit % of 75.

Quote:
4.Automatic Success
Bonus successes added after dice are rolled. Like the re-roll modifier there are a lot ways this can be limited/used.

Just add the bonus succeses to the average expected hits.

Quote:
The question in what order are these modifiers "valued"?

I think the auto-success modifier is the most valuable and re-rolls the least, with the result modifier and extra dice falling somewhere inbetween. But I am no mathamatician so the answer escapes me =(

I really think it depends on the numbers. The result modifier is pretty strong and the small difference between the numbers leave little "wiggle room", but I also find them the most elegant solution. Adding dice is also an elegant mechanic. The others don't seem so cool to me.

I hope this helps some. I'll be happy to help with any further questions.

Nando
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Joined: 07/22/2008
Die modifier mechanics and "value"

To roll some value (or greater) on a SINGLE ROLL of some quantity of dice:

<br />
1 Die<br />
==========<br />
6	16.67%<br />
5	33.33%<br />
4	50.00%<br />
3	66.68%<br />
2	83.35%</p>
<p>2 Dice<br />
===================<br />
6   30.56%	 2.78%<br />
5	55.55%	11.11%<br />
4	75.00%	25.00%<br />
3	88.90%	44.46%<br />
2	97.23%	69.47%</p>
<p>3 Dice<br />
============================<br />
6	42.14%	 7.41%	 0.46%<br />
5	70.37%	25.92%	 3.70%<br />
4	87.50%	50.00%	12.50%<br />
3	96.30%	74.09%	29.65%<br />
2	99.54%	92.61%	57.91%</p>
<p>4 Dice<br />
=====================================<br />
6	51.78%	13.20%	 1.62%	 0.08%<br />
5	80.24%	40.73%	11.11%	 1.23%<br />
4	93.75%	68.75%	31.25%	 6.25%<br />
3	98.77%	88.90%	59.28%	19.77%<br />
2	99.92%	98.38%	86.83%	48.26%</p>
<p>5 Dice<br />
==============================================<br />
6	59.82%	19.63%	 3.55%	 0.33%	 0.01%<br />
5	86.83%	53.90%	20.98%	 4.53%	 0.41%<br />
4	96.88%	81.25%	50.00%	18.75%	 3.13%<br />
3	99.59%	95.48%	79.03%	46.12%	13.18%<br />
2	99.99%	99.67%	96.46%	80.41%	40.23%<br />

Example:
To roll a 4+ on AT LEAST 2 of 3 dice (rolled on a single roll), you have a 50% chance of success.

This display seems to indicate that when considering the value of more dice or more modifiers, the odds favor modifiers except when you only have a single die. (But don't forget the fact that you can't get the hit if you don't throw a die for it.)

I have a spreadsheet that tells me this, so don't argue! ;)

markmist
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Joined: 12/31/1969
Die modifier mechanics and "value"

Here is my attempt to answer your question, with my limited probability skills:

For one die - the order is as follows from best to worst:

1) Automatic success gives you 1 auto success and 1 die roll. The chances of getting 2 successes is 50%, 1 success is of course guaranteed.

2) Getting an extra die (now you have 2), gives you a 25% chance at 2 successes, and a 75% chance at 1 success.

3) +2 Modifier gives you a 83.3% of success - can only fail on a 1.

4) Re-rolling the die gives you a 75% chance at 1 success.

5) +1 Modifier gives you a 66.6% of success.

For 2 dice:

1) Auto success gives you 1 auto success and 2 die rolls. The chances of getting 3 successes is 25%, 2 successes - 75%.

2) Extra die (you have 3), 3 successes - 12.5%, 2 successes - 50%, 1 success - 87.5%

3) +2 Modifer to both dice: 2 successes - 69.4% , 1 success - 97.3%

4) +1 Modifier to both dice: 2 successes - 44%, 1 success - 88.9%

5) Re-rolling dice gives you a 37.5% of 2 successes, and an 87.5% of 1 success.

Without getting into complex mathematics and extrapolating these limited results outward, you can see that Auto success provides the biggest benefit because you always get an extra success.

The extra die always give you a chance to get 1 more success than your opponent - however, already at 3 dice you can see that the chances of getting 1 or 2 successes are better with the modifiers and re-rolling than just getting 1 extra die. Of course if you are to compare getting multiple extra dice - it will always be more powerful than the modifiers.

The +2 Modifier is always more powerful than re-rolling, and the +1 is always better than re-rolling except for 1 die in which re-rolling is the better choice.

Here is a good website that answers some basic questions on dice probabilities that might help you if you want to go more into detail.

http://mathforum.org/library/drmath/sets/select/dm_dice.html

Anonymous
Re: Die modifier mechanics and "value"

Quote:

Rerolling all dice, even the hits, would only make sense when you roll worse than you would statistically expect, or when you are desperate for some reason.

Unless of course an opponent can make you reroll your dice.

I personally always placed a lot of value on the ability to reroll (my own or an opponents) dice... its HUGE if someone has a great roll and you can force them to (hopefully) mess it up.

____

Nando:

1 Die
==========
6 16.67%
5 33.33%
4 50.00%
3 66.68%
2 83.35%

I may totally be misunderstanding what this chart reflects here, but you cant put percentages on a single die roll (how often one face will come up versus another)... it cant be quantified.

Each roll of a single die is independant and totally random... maybe I'm misunderstanding what you were showing though.

Anonymous
Re: Die modifier mechanics and "value"

Brahmulus wrote:
Each roll of a single die is independant and totally random... maybe I'm misunderstanding what you were showing though.

he was showing the chance of rolling 4 or higher

seo
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Joined: 07/21/2008
Die modifier mechanics and "value"

Brahmulus wrote:
I may totally be misunderstanding what this chart reflects here, but you cant put percentages on a single die roll (how often one face will come up versus another)... it cant be quantified.

What you can't tell is what will the outcome be, but you certainly can tell what's the likeliness of each result. If you flip a coin, it will fall either on one side or the other, and you can say each side has roghly 50% chance.

What is a bit confusing is that you can also predict that given a series of tosses, the coin will fall roghly 50% of the times on each face, yet no matter how many times it has fallen on each face until now, the next toss still presents 50% chance for each outcome.

But you definitelly can tell the % of chance of any given outcome before any toss of coin, dice, draw of a card, etc. That doesn't let you predict the actual result, just the chance for any given result. There's a big difference there.

Seo

Nando
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Joined: 07/22/2008
Re: Die modifier mechanics and "value"

Brahmulus wrote:
I may totally be misunderstanding...

Truth.

The chart shows the chance of rolling the value OR GREATER.

For the multiple dice percentages, it shows the same thing, but moving left to right, it shows the probability of rolling the result on AT LEAST the number of dice corresponding to the column number. So rolling a Yahtzee of 6s on 5 dice only has a .01% chance of succeeding, but rolling a 4 or greater on 3 of 5 dice has a 50% chance of succeeding.

chowdah
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Joined: 12/31/1969
Die modifier mechanics and "value"

Super thanks guys!

Completly answered my questions and actually cleared up some issues I was having with certain aspects of these modifier mechanics.

Anonymous
Die modifier mechanics and "value"

Quote:

yet no matter how many times it has fallen on each face until now, the next toss still presents 50% chance for each outcome.

Bingo, this is what I was trying to explain... each roll resets so to speak... independent and random everytime you pick it up and throw it, you can not in anyway determine the percentage chance of a single outcome of any single face.

I understand now due to another post that he was showing a percentage chance of it landing on a 4 or higher... i gets it.

seo
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Joined: 07/21/2008
Die modifier mechanics and "value"

Brahmulus wrote:
Bingo, this is what I was trying to explain... each roll resets so to speak... independent and random everytime you pick it up and throw it, you can not in anyway determine the percentage chance of a single outcome of any single face.

But you can determine the percentage chance, it's just that the percentage will never be 100% or 0% for any single face, thus you can't predict the outcome, just the percentage chance for each outcome.

And while flipping a coin once gives you 50% chance for each face, doubling the number of flips doesn't double the percentage chance for any of the faces. Chances will get closer to 100% for each face to win at least once as you repeatedly flip the coin, but will never reach 100%.

Seo

Anonymous
Die modifier mechanics and "value"

Quote:

But you can determine the percentage chance,

Hmmmm...

I'll have some comments on this later.

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