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Subject: Dice Probability, Expected Values, and a look at Conan and Valeria rss

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Zachary Ruiz
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Hey all,

I've seen some discussion on here about the relative strength of different characters, and while we all seem to have an intuitive opinion of the "true value" of each different die type, I decided to crunch the numbers and see what I could come up with.

I decided to look at the expected value (or weighted average) of each die, trivial stuff but vitally important in games where players can choose how many dice to roll. For dice with rerolls, I attempted to maximize the result - the player always rerolls if the initial result is below the expected value, 0 for Yellow and Orange, 0 and 1 for Red.

Here's what I came up with:

Yellow: .667
Yellow w/ Reroll: 1
Orange: 1
Orange w/ Reroll: 1.333
Red: 1.5
Red w/ Reroll: 2.25 (Pointed out below its actually 1.92, had my math wrong, changes some of the specific numbers but not the conclusions).

What does this tell us? A couple things immediately stand out. Yellow w/ Reroll and Orange are identical. Orange w/ Reroll and Red are fairly similar, and Red w/ Reroll is significantly more powerful than any other.

One of the most common complaints I've seen is about the balance between Conan and Valeria, both combat-centric characters. While Conan's limit of 5 is undeniable an advantage, with Ambidexterous Valeria can seriously narrow the gap.

Assuming two enemies in the same zone as the player, Valeria can make two separate attacks, one with 2 dice and one with 1. These attacks (assuming starting equipment) will have on average 4.667 and 3.333 hits. Enough to kill two Pict Hunters and more than enough to take out Pirates.

For the same 3 dice, Conan can deal 6.75 total damage, and with his ability this too can take out both hunters. Conan can tack on an extra die for insurance or in hopes of getting lucky, or tack on 2 more (up to his max) to have a good chance of taking out 3 Pict Hunters.

Thematically, I think this makes sense as well. As Valeria you have more control, with multiple smaller attacks, while Conan simply smashes his way through the entire group. And, in terms of fluctuations, the fact that all of Valeria's dice are rerollable adds something to the stability of the results.

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Max Laager
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seems red with reroll is 1.92 average...

see here the anyfice calculatore
http://anydice.com/program/8d74



what about n dice with m reroll ?
that would be nice
 
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Max Laager
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ok finaly more dice more rerolls

1 yellow 0 reroll (0.67 1 orange 0 reroll (1.00 1 red 0 reroll (1.50
1 yellow 1 reroll (1.00 1 orange 1 reroll (1.33 1 red 1 reroll (1.92
1 yellow 2 reroll (1.17 1 orange 2 reroll (1.44 1 red 2 reroll (2.13
1 yellow 3 reroll (1.25 1 orange 3 reroll (1.48 1 red 3 reroll (1.23

2 yellow 0 reroll (1.33 2 orange 0 reroll (2.00 2 red 0 reroll (3.00
2 yellow 1 reroll (1.83 2 orange 1 reroll (2.56 2 red 1 reroll (3.63
2 yellow 2 reroll (2.17 2 orange 2 reroll (2.81 2 red 2 reroll (4.04
2 yellow 3 reroll (2.37 2 orange 3 reroll (2.93 2 red 3 reroll (4.30

3 yellow 0 reroll (2.00 3 orange 0 reroll (3.00 3 red 0 reroll (4.50
3 yellow 1 reroll (2.58 3 orange 1 reroll (3.70 3 red 1 reroll (5.23
3 yellow 2 reroll (3.04 3 orange 2 reroll (4.11 3 red 2 reroll (5.80
3 yellow 3 reroll (3.29 3 orange 3 reroll (4.26 3 red 3 reroll (5.91

http://anydice.com/program/8d82


 
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Max Laager
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so what does it tell...

1 red dice with reroll is better then 2 yellow with 1 reroll...

3 yellow 2 reroll (3.04 is better then 3 orange 0 reroll (3.00 (about the same)

the more dice you have and the better the dice, the better the reroll

also the first reroll is the one that give the better improvement...
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Mad Halfling
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These figures may change now as you can make as many attacks as you can pay for, up to your exertion limit - Valeria's ability now allows her to add 2 1H weapons to her attack roll.

CARNAGE!!!!
 
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Stephan Beal
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swammeyjoe wrote:

Here's what I came up with:

Yellow: .667
Yellow w/ Reroll: 1
Orange: 1
Orange w/ Reroll: 1.333
Red: 1.5
Red w/ Reroll: 2.25


Since nobody's mentioned it so far: this makes a great starting point for calculating relative point values for the units.

e.g. if a beastie attacks with 2 orange, each with a reroll, thats 2 x 1.33 points. Add some points for its skills, movement range, and armor, and it "should" be good enough for basic scenario balance calculations. (Keep in mind that a movement or armor of 2 is notably better than 1, and 3 is notably better than 2, so those point scales should not be linear.) Essentially all that's really missing is relative values for the skills, and only experience with the game will reveal what those should look like.

Calculating points for Heroes is far more difficult because they have many more factors involved, some of them complex (e.g. spellcasters aren't spellcasters any more if they're carrying too much weight, and that limit can vary by character).
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zerg wrote:
[..]

also the first reroll is the one that give the better improvement...


That's good, because you can only reroll once
 
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Simon Croquet
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Barlimann wrote:
zerg wrote:
[..]

also the first reroll is the one that give the better improvement...


That's good, because you can only reroll once


No, you can reroll several times.
source: Hero's rule book french version p.10 top of the page.
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Stephan Beal
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Shoum wrote:
Barlimann wrote:
zerg wrote:
[..]

also the first reroll is the one that give the better improvement...


That's good, because you can only reroll once


No, you can reroll several times.
source: Hero's rule book french version p.10 top of the page.


Kind of: free rerolls (those listed directly on item cards) can only be applied once. You can pay Stamina to reroll any number of times, limited only by your Stamina.
 
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Paul F
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Here are the full probabilities for the dice. In addition to 0, 1, 2 and 3 hits, I've put down the probability of any hit (H) and the average (A) which is already posted above.
RR1 means red dice re-rolling anything less than equal to 1.

Y: 0.50(0), 0.33(1), 0.17(2), 0.50(H), 0.67(A)
YR0: 0.25(0), 0.50(1), 0.25(2), 0.75(H), 1.00(A)
O: 0.33(0), 0.33(1), 0.33(2), 0.67(H), 1.00(A)
OR0: 0.11(0), 0.44(1), 0.44(2), 0.89(H), 1.33(A)

R: 0.17(0), 0.33(1), 0.33(2), 0.17(3), 0.83(H), 1.50(A)
RR0: 0.03(0), 0.39(1), 0.39(2), 0.19(3), 0.97(H), 1.75(A)
RR1: 0.08(0), 0.17(1), 0.50(2), 0.25(3), 0.92(H), 1.92(A)
RR2: 0.14(0), 0.28(1), 0.28(2), 0.31(3), 0.86(H), 1.75(A)

I thought it was useful to see this as some people draw incorrect conclusions from just looking at the averages. I saw someone say a yellow with re-roll was the same as an orange. You can see above that although they have the same expected value of 1, the distribution is very different. A yellow with re-roll is less likely to miss and will mostly end up with a 1, whereas orange has even chance of 0, 1 or 2 hits.
I also included the different types of re-rolling for red. As you can see re-rolling less than the normal red average of 1.5 gives the highest average (RR1 - 1.92), but only re-rolling misses is better if you want the highest chance of any type of hit (RR0) and re-rolling 0,1 and 2 gives the highest chance of rolling a 3.
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Ray Gans
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And (for cheaters) the layout of the hits and blanks on the dice allow one to roll a yellow or orange die out of his or her hand along one axis such that the hit probability is maximized. whistle

I would never do this of course, but it may be worth requiring that people shake the dice up first to ensure that once thrown the axis it rolls along is relatively random.
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C B
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goblin wrote:
And (for cheaters) the layout of the hits and blanks on the dice allow one to roll a yellow or orange die out of his or her hand along one axis such that the hit probability is maximized. whistle

I would never do this of course, but it may be worth requiring that people shake the dice up first to ensure that once thrown the axis it rolls along is relatively random.

I hear you! I've played with people who just pick the dice up and drop them, which often leads to the same results. It was a really casual group, so I didn't want to say anything, but man did it ever bother me.
 
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Brad Ocharuk
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OK, this is driving me nuts...

I can't figure out how it is that the chance of rolling a '2' on a yellow die given that a single reroll is allowed, ends up having a probability of 0.25.

How is that calculation made? I have very limited probability knowledge (I'm trying to get more, hence the question) and from what I understand the probability should be closer to 0.31.

Thanks,

bocharuk

EDIT: I think I've got it now. The 0.25 probability of getting a 2 is based on re-rolling only if the first die is a complete miss. My calculation was based on a re-roll after a miss or a 1 being rolled. Please correct me if that's not right.

Also, how does one calculate the average result when a re-roll is allowed? I thought I had it, but the numbers don't quite match...

Thanks again

bocharuk
 
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Zachary Ruiz
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bocharuk wrote:
OK, this is driving me nuts...

I can't figure out how it is that the chance of rolling a '2' on a yellow die given that a single reroll is allowed, ends up having a probability of 0.25.

How is that calculation made? I have very limited probability knowledge (I'm trying to get more, hence the question) and from what I understand the probability should be closer to 0.31.

Thanks,

bocharuk

EDIT: I think I've got it now. The 0.25 probability of getting a 2 is based on re-rolling only if the first die is a complete miss. My calculation was based on a re-roll after a miss or a 1 being rolled. Please correct me if that's not right.

Also, how does one calculate the average result when a re-roll is allowed? I thought I had it, but the numbers don't quite match...

Thanks again

bocharuk


This is several months late, but this is my thread anyways so...

You should only "use" a Reroll if your result is less than the expected value of the Die (with 1 Reroll) (0 for Yellow, 0/1 for Orange and Red). Otherwise, across many rolls you will likely end up with a lower total than if you didn't Reroll. However, in the game you're likely to have rolls where you know you need a 3 on a Red, for instance, and Reroll the 2.
 
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Niki Gyoshev
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swammeyjoe wrote:


Thematically, I think this makes sense as well. As Valeria you have more control, with multiple smaller attacks, while Conan simply smashes his way through the entire group. And, in terms of fluctuations, the fact that all of Valeria's dice are rerollable adds something to the stability of the results.



Vanyr Valkyrie is much stronger then Valeria. I think much stronger then Conan himself. Just see:
- 4 attacks, orange die with reroll.
- evasive, Ambidexterous, precious strike, bodyguard.
- manipulation - red die.
- speed - 3.
- penalty for encumbrence: -1 after for 7 and more, -2 for 9 and more.

This is much much better then Valeria. Real imba this Vanyr!!!
 
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Max Maloney
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Red Sonya doesn't mess around.
 
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John Mauney
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In another thread I was concerned about the usefulness of the Circular Strike skill if the Overlord is allowed to guard against the excess damage after the first attack. This comes from me not interpreting the rule like most people here do.

Anyway, I didn't see this thread so I performed some math specifically around Conan using Circular Strike, to find out if it should influence the player's decision: Should I make 1 strong attack or 4 weak attacks? It turns out that it should, but not in a way that is obvious.

If you're interesting, take a look!
https://docs.google.com/document/d/1hHmB51Dd2NKcRJ57sPFE_aKb...

Cheers

Edit: You can also take a look if you're not interesting...but interested
 
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John Mauney
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Hm, I'm not the one who owns the game in our group. Can someone let me know: does the basic Conan hero get a 4 or 5 gem attack? If it's 5 I need to re-crunch the numbers a bit. My memory said 4.
 
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Doug Nordwall
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Big J Money wrote:
Hm, I'm not the one who owns the game in our group. Can someone let me know: does the basic Conan hero get a 4 or 5 gem attack? If it's 5 I need to re-crunch the numbers a bit. My memory said 4.


pretty sure he can dump 5 gems into an attack. one of my players was pulling 6 with a battle axe friday - at least I think. I don't have the game handy
 
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Tom Zsolt
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Conan's limit for melee attacks is 5 gems.
 
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