Author Topic: King's Guard Armor is mispriced  (Read 424 times)

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Fluffhead

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King's Guard Armor is mispriced
« on: July 19, 2017, 02:28:55 PM »
I promised Darcy on Twitch that I'd post some math backing this claim.  So here goes.




Royal Shield is arguably one of the best treasure cards in the game.  I argue that King's Guard Armor is nearly as good as one of the best treasure cards, but only costs 2 gold!

There are two differences between the Royal Shield and King's Guard Armor that would need to be looked at to evaluate how much better the Royal Shield is.  The first is the probability that the Royal Shield's ability would do more than what the King's Guard Armor already would.  The second is the value of that difference to the game outcome.  The first is easy to look at and the second could be analyzed in a hundred different ways.  We'll limit the discussion to the probability that the Royal Shield's additional benefit over the King's Guard Armor would kick in.

Basically the number of misses that convert to shields is a binomial distribution (n=number of starting dice, p=1/5).  This is because each die has any number of explosions and then ends on one of five options, one of which is a sun or moon miss.  Let's say you have n dice at the start of a battle and you roll, what's this probability? (the answer is 1-(p(0)+p(1)+p(2)) for the misses distribution)

n    |  Probability Royal Shield would be better than your King's Guard Armor
4    |  2.7%
5    |  5.8%
6    |  9.9%
7    |  14.8%
8    |  20.3%
9    |  26.2%
10 |  32.2%
11 |  38.3%

Notice "better" here just means more shields, but more shields might not actually imply a different outcome if you're already not taking damage.

So basically the chance of even seeing a difference with the treasure card is less than 1 in 6 battles when you have less than 8 dice to start with!

I strongly believe this card should cost 4 gold and be more rare... or really a better solution is making it only the first miss instead of first two.

What do you think devs?   
« Last Edit: July 19, 2017, 02:45:08 PM by Fluffhead »

Darcy Smith

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Re: King's Guard Armor is mispriced
« Reply #1 on: July 19, 2017, 03:35:46 PM »
Heyo Fluffhead! Thanks for writing this up, i've passed it through to the team for further discussion.


Cheers,

<3 Darcy

Kletian999

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Re: King's Guard Armor is mispriced
« Reply #2 on: July 19, 2017, 11:12:09 PM »
Comparing gear to treasure isn't a good metric for gold cost, because Treasure have 0 Gold cost. 

Instead we should ask, how does it compare to gear that costs money?  2 gold buys 1 static shield, 3 gold buys 2 shields-1 dice, or 2 shields on defense only, 4 gold buys a shield and a very likely reflection die.   Yes there's value "not missing" when attacking the king, but that's a particular corner case that can be negated with burns.  By the way UNLIKE Royal Shield, the armor DOESN'T work on burns- throwing your equation off.

What does King's Armor get?  6 rolled dice give a ~1/5 chance to roll offtime, so 1 shield is likely but not guaranteed.  2 shields is possible but nothing higher.

Fluffhead

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Re: King's Guard Armor is mispriced
« Reply #3 on: July 20, 2017, 05:22:18 AM »
Comparing gear to treasure isn't a good metric for gold cost, because Treasure have 0 Gold cost. 

Instead we should ask, how does it compare to gear that costs money?  2 gold buys 1 static shield, 3 gold buys 2 shields-1 dice, or 2 shields on defense only, 4 gold buys a shield and a very likely reflection die.   Yes there's value "not missing" when attacking the king, but that's a particular corner case that can be negated with burns.  By the way UNLIKE Royal Shield, the armor DOESN'T work on burns- throwing your equation off.

What does King's Armor get?  6 rolled dice give a ~1/5 chance to roll offtime, so 1 shield is likely but not guaranteed.  2 shields is possible but nothing higher.

The Royal Shield is particularly good and not easy to get so it seems unreasonable to have an equipment nearly as good that is only 2 gold, but I agree that it makes sense to compare it to other non-treasure equipment when considering its pricing.

I hadn't consider the additional benefit the Royal Shield has of being able to burn a sun/moon for a shield that would normally miss.  This could be a game changer when attacking the king if you intend to burn in battle and happen to draw those cards.  I wouldn't say it throws my math off, I just should have had a different label for those percents... maybe "chance of having more than two sun/moon misses" because that's what it really is.

I'm not sure I follow what you mean by " 6 rolled dice give a ~1/5 chance to roll offtime".  The King's Guard Armor is expected to give you an additional dice/5 shields each battle (this is the average; so with 6 dice that's an average 1.2 shields, better than 1).  Let's look at a table showing dice count vs. chance of getting at least one additional shield (equivalently having at least one sun/moon miss):

Dice  |  Chance of getting at least one sun/moon miss
4         |  59.04%
5         |  67.23%
6         |  73.79%
7         |  79.03%
8         |  83.22%
9         |  86.58%
10      |  89.26%
11      |  91.41%

This shows that the vast majority of the time, the King's Guard Armor is at least as good as the Trusty Shield.


How often is it strictly better than the Trusty Shield?
Dice  |  Chance of getting at least two sun/moon misses
4         |  18.08%
5         |  26.27%
6         |  34.46%
7         |  42.33%
8         |  49.67%
9         |  56.38%
10      |  62.42%
11      |  67.79%

There's also the potential massive advantage the King's Guard Armor has over the Trusty Shield during king battles of gaining defenses when you would otherwise give the king more dice.  And also consider that many of those times when you aren't getting a sun/moon miss you might not need that Trusty Shield defense (maybe you aren't taking damage anyway).  The King's Guard Armor specifically lifts you up during your unlucky battles packing more relevance when it is activated (hence maybe those scenarios are more impactful  to the game than the ones where you don't get an additional shield, but would with the Trusty Shield ... if that makes sense).

So thanks Kletian999 your post made me really think.  Maybe 3 gold is the right price for the King's Guard Armor?  What do you think?  Are you a firm believer that 2 gold is appropriate?

EDIT:  It occurs to me that since the distribution of attacks and defenses you roll is a function of the number of dice you have and the King's Guard Armor doesn't give you any extra dice, it has one glaring disadvantage when compared to the Trusty Shield.  The Trusty Shield leaves you with the same number of dice for producing attacks and shields while providing an additional free shield and the King's Guard Armor only allows current dice to become shields more easily.  I haven't really looked into quantifying this distinction.  At the very least it means that, all else equal, the cap on the number of shields you can have with the Trusty Shield is dice+1 and the cap with the King's Guard Armor is only your dice count.  This particular case would leave you with only explosions for attacks and probably isn't all that important or likely, but it illustrates an undeniable difference between the two in favor of the Trusty Shield.
« Last Edit: July 23, 2017, 09:24:43 AM by Fluffhead »

Kletian999

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Re: King's Guard Armor is mispriced
« Reply #4 on: July 20, 2017, 08:58:39 PM »
I may not have typed it elegantly, but I think we are on the same page.  I'm not 100% committed to 2 gold but definitely wouldn't go higher than 3, because there is a non trivial chance, especially for burners or low fight characters, that Kings armor gives you nothing and will only give you 2 shields some of the time (like other 3 gold armor).

Outside of Brun and Rotlords it's rare to have more than 6 dice actually rolled; while a good portion of the cast will never roll more than 4.

Fluffhead

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Re: King's Guard Armor is mispriced
« Reply #5 on: July 21, 2017, 07:44:05 PM »
because there is a non trivial chance, especially for burners or low fight characters, that Kings armor gives you nothing.

Just for the sake of supporting your argument:
Dice  |  Chance of zero sun/moon misses
1         |  80.00%
2         |  64.00%
3         |  51.20%
4         |  40.96%
5         |  32.77%
6         |  26.21%
7         |  20.97%

(dice count being rolled dice as opposed to burned, hence I have reported as low as 1 here; really observant to notice that Kletian999)
« Last Edit: July 21, 2017, 07:49:34 PM by Fluffhead »

Kletian999

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Re: King's Guard Armor is mispriced
« Reply #6 on: July 22, 2017, 10:28:48 AM »
I don't get to playing (and posting) this long without being observant :)

Fluffhead

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Re: King's Guard Armor is mispriced
« Reply #7 on: July 22, 2017, 12:55:39 PM »
I had an idea.  Why not calculate the average shields you'll get with n dice and the Trusty Shield vs. with n dice and the King's Guard Armor?  I was able to do this analytically and here is how they compare (assuming no burning):

(1) With n dice and Trusty Shield equipped your average total # of defenses are:


(2) With n dice and the King's Guard Armor equipped your average total # of defenses are:


Let's look at (2)-(1) which tells us how many more shields you expect from the King's Guard Armor than the Trusty Shield [rounded to nearest hundredth]:
n    |  (2)-(1)
1    |  -0.80
2    |  -0.60
3    |  -0.41
4    |  -0.23
5    |  -0.06
6    |  +0.08
7    |  +0.21
8    |  +0.33
9    |  +0.43
10 |  +0.52

At 5 and 6 dice, they have nearly the same average and the King's Guard Armor is a bit better with more dice.  However, the difference is underwhelming.  I always play characters that are 4 fight and up so I think this makes the King's Guard Armor better for me... but not by a huge margin when examined this way.

Interesting huh?  Is this how the developers chose to price at 2 gold? 

Actually after this analysis I question if calling the Royal Shield one of the best treasure cards is accurate.  The Royal Shield would have a larger average total defenses than the King's Guard Armor [namely 2n/5].  Subtracting from the Trusty Shield would give n/5-1.  Hence the difference is 0 for 5 dice and the averages are symmetric around 5 dice (4 dice is -0.2, 6 dice is +0.2 etc.).  When you take (Royal Shield - King's Guard Armor) the difference is quite unimpressive (0 @ 1 die, 0, 0.01, 0.03, 0.06, 0.12, 0.19, 0.27, 0.37, 0.48 @ 10 dice).

@Kletian999 I think you were accurate in saying that comparing to the Royal Shield doesn't indicate what the cost of the card should be very well.

I think this is good evidence to make the cost 3 or keep at 2 (assuming the price of the Trusty Shield is appropriate haha)

Any devs reading this?  Thoughts?
« Last Edit: July 22, 2017, 12:59:18 PM by Fluffhead »

Fluffhead

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Re: King's Guard Armor is mispriced
« Reply #8 on: July 23, 2017, 09:20:49 AM »
Given an opponent's # of attacks and your health, the question of the likelihood of survival is answered by looking at the chance of having at least a certain number of defenses.

I was thinking it might be interesting to dig into the cases of having 4, 6 or 9 dice and look at these probabilities for a character equipped with the King's Guard Armor only vs. one equipped with the Trusty Shield only.

The following looks at "(chance of having at least n shields with King's Guard Armor) - (chance of having at least n shields with Trusty Shield)"



So with 4 dice it is not only the average number of defenses that is lower with the King's Guard Armor, but the actual survival probabilities are all less as well.

However with 6+ dice, your survival probabilities are better UNLESS you only need 1 shield.  This is good evidence that the King's Guard Armor is superior for 6+ dice.  Thoughts?

Shrapnel

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Re: King's Guard Armor is mispriced
« Reply #9 on: July 23, 2017, 09:52:31 PM »
The current price for this card is fine,  imo, as it only gives shield when there is at least one moon/sun roll. Which could be disastrous if most of our rolls miss as rots or go to attack.

This equipment works best when we have high fight, another shield as contingency, and wyld cards to burn.

Low fight heroes don't benefit much from this card, so 3 golds would be too expensive for them.
« Last Edit: July 24, 2017, 06:59:49 PM by Shrapnel »

Fluffhead

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Re: King's Guard Armor is mispriced
« Reply #10 on: July 24, 2017, 06:24:19 AM »
I ran 10,000,000 simulations for 4 dice, 6 dice and 9 dice and sent the probabilities of each (block,attack) combination for both a player with the King's Guard Armor equipped and one with the Trusty Shield equipped.

I then spit out the differences.  See below (click to zoom):


It certainly seems like, for the most part, the King's Guard Armor takes probabilities away from less favorable outcomes and moves them to more favorable outcomes (when compared to the Trusty Shield).  This is not the whole story because of course there is always a positive probability of no shields with the King's Guard Armor and this is not possible with the Trusty Shield.

Also realize that with no equipment the distribution looks exactly the same as the Trusty Shield except translate the table 1 column to the left (since it's just adding 1 to blocks).

I gotta say after all this analysis I'm thinking 3 gold is a better price.


If we were to compare the average # of defenses by dice count between the King's Guard Armor and Battle Armor, we would see that Battle Armor is strictly better until about 15 dice!  After this they remain extremely close out to infinity.  In fact, even Royal Shield produces a lower average until 9 dice (but then shoots way ahead after this).

It would seem that the King's Guard Armor is somewhere between the Trusty Shield and Battle Armor.  It also seems that Battle Armor is much better than I previously gave it credit for.  Of course here Battle Armor was only judged based on average defenses and it certainly has a negative impact to # of attacks as well. 

I've said this in another thread: 1 die has an impact to average # attacks of 0.6 and average # of blocks of 0.2.  So the Battle Armor, on average, nets out -0.6 attack and 2-0.2=1.8 block.  This is probably the best way to value items that add or subtract dice.


Bastard Sword then gives 1.6 attack and 0.2 block.


The Reaper's Trident gives 3-0.6*2=1.8 attack and -0.2*2=-0.4 block. etc.  Interestingly, the impact to your averages of this card is nearly identical to a card that says "Gain +3 dice and lose 1 shield if you have any" because 3 dice is 0.6*3=1.8 attack and 0.2*3=0.6 block... so to get 0.6 block to -0.4 block you subtract 1 shield.

All things considered, I think the King's Guard Armor is closer to a 3 gold item than a 2.
« Last Edit: July 24, 2017, 10:51:34 PM by Fluffhead »

Kletian999

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Re: King's Guard Armor is mispriced
« Reply #11 on: July 25, 2017, 12:10:40 AM »
It's been fun watching all these graphs.   The question of burns does make your numerical assumptions a bit off (The Royal shield helps burns a lot;  Burns tend to make 9 pool rolls not happen, etc ).  A lower total dice pool also loses the chance to burn.  And the loss of attack dice can be either inconsequential or disastrous depending on the state of the opponent.  Depending on the rarity of the armor, even if 3 is a "fairer" price than 2, it could be "allowed" to be a better for cheap item because of its rarity.

Fluffhead

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Re: King's Guard Armor is mispriced
« Reply #12 on: July 25, 2017, 12:28:58 AM »
It's been fun watching all these graphs.   The question of burns does make your numerical assumptions a bit off (The Royal shield helps burns a lot;  Burns tend to make 9 pool rolls not happen, etc ).  A lower total dice pool also loses the chance to burn.  And the loss of attack dice can be either inconsequential or disastrous depending on the state of the opponent.  Depending on the rarity of the armor, even if 3 is a "fairer" price than 2, it could be "allowed" to be a better for cheap item because of its rarity.

All great points.  Rarity could also be thought of as part of the cost for sure.  Yes of course I agree on the Royal Shield making burns more useful.  However as for how burns affect other things, I think my "dice" in all my tables and graphs above should be considered "rolled dice" so you just lower your dice count when you burn.  In this way the marginal advantage seen still changes in line with my numbers (ignoring that the battle outcome may not).  You can't really look at things in terms of battle outcomes unless you take into account HP, rot, other equipments, amulets and rings right?  I think it'd be difficult to really quantify the advantage of burning.