Author Topic: Guide to Burning (Gain/Loss)  (Read 358 times)

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Fluffhead

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Guide to Burning (Gain/Loss)
« on: July 28, 2017, 06:49:10 AM »
Kletian999 got me thinking about burning and the marginal benefit/cost from each particular type of burn.

One way (arguably the best way) to quantify the marginal benefit and cost of a burn is to look at the impact to your total average # of attacks and total average # of blocks for the battle.  There is a quantifiable impact to losing one die (namely your average # of attacks is reduced by 3/5 and your average # of blocks is reduced by 1/5 [this takes explosions into account]).  Then you can add the risk-free gain of the burn.

The result is the below marginal benefit/cost for each type of burn:


What are some takeaways from this?

(1) Burning two attacks has less of a benefit to your average # of attacks than burning one explosion (only 4/5 instead of 1), and an additional cost of 2/5 shields!
(2) Burning for the same number of blocks as attacks is a cost to your average attacks (+3/5 block at cost of 1/5 attack per pair)
(3) Burning for 2 blocks and 3 attacks is equivalent, from the perspective of these averages, to equipping a Trusty Shield and rolling

Keep in mind that although these are the exact impacts to your average # of attacks and average # of blocks, burning clearly reduces the variance of your battle outcomes.

I hope you found this interesting and useful!
 


Darcy Smith

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Re: Guide to Burning (Gain/Loss)
« Reply #1 on: July 28, 2017, 04:02:13 PM »
Can't express how much I dig this stuff Fluffhead! Really rad, I'm sharing it with the team.


Cheers,

<3 Darcy

Kletian999

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Re: Guide to Burning (Gain/Loss)
« Reply #2 on: July 29, 2017, 10:15:19 AM »
Glad I can be your muse.

Fluffhead

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Re: Guide to Burning (Gain/Loss)
« Reply #3 on: July 29, 2017, 10:43:04 AM »
Just to elaborate on that last point about reducing the variability of your outcomes when you burn.  All burns except explosion burns reduce your variance by the same amount (fixing your dice count) because you're simply losing one die.  Explosion burns do not change the variability of outcomes.

Below is a graph of the reduction to your standard deviation in attacks/blocks (the unit for standard deviation is attacks for the red and blocks for the blue) when you burn while having a certain number of dice (x-axis).

(any type of burn except explosion burns)

Notice each successive burn incrementally reduces greater amounts of variability from your outcome making it more and more likely that the outcome you achieve is close to the average outcome computed with the methods above (3/5*dice+benefit/cost of burns for attacks, dice/5 + benefit/cost of burns for blocks).  I chose to show standard deviation specifically because the units are your actual # of attacks (for red, and # of blocks for blue).  You can think of this standard deviation as the approximate number of attacks (blocks) away from the average that the majority of outcomes will be within.  This is the value that is reduced by the amounts shown in the graph.

It is a bit abstract but this is the other main cost/benefit to consider when burning.  Usually when you burn you want to lock in your possible outcomes in a specific way and reduce this variability in your favor so this is generally an advantage.  However, it can be a disadvantage if you need to get lucky anyway.   

The other huge advantage to burning that is not discussed here is the advantage of getting rid of cards you don't want to make room for draws.  This and cards that let you burn rots for poison etc. is why I put the miss burn up.  You can actually calculate the cost of getting rid of cards or having the poison effect added by looking at the loss to your averages and reduction to variability in outcomes.

________________________________________  EDIT ________________________________________

Let's look at a quick example so all this math talk makes sense.

Suppose you have 2 dice.  Without any burning you have the following:
Average attacks: (3/5)*2=1.2 attacks
Standard Deviation of attacks: ≈ 0.98 attacks
Average blocks: (1/5)*2=0.4 blocks
Standard Deviation of blocks: ≈ 0.57 blocks


Now say you burn a sword.  We know how that affects both values.  In my original post I say it adds 2/5 attack and subtracts 1/5 block from your averages and the graph above (at 2 dice) shows that it subtracts 0.29 attack standard deviation and 0.17 block standard deviation (0.29 red, 0.17 blue).  After one attack burn we now have:
Average attacks: 1.2+2/5=1.6 attacks
Standard Deviation of attacks: ≈ 0.98-0.29=0.69 attacks
Average blocks: 0.4-1/5=0.2 blocks
Standard Deviation of blocks: ≈ 0.57-0.17=0.4 blocks


Now we've reduced our variability by locking one sword in and our average is now 1.6 attack and 0.2 block.  Why?  Because our outcome of the last rolled die will either be a block (or any number of explosions and then a block) or that roll will bring our attacks to 2 attack (or maybe 3 with an explosion first, or 4 with 2 explosions first, etc., or maybe an explosion and then a miss etc.).  All of these outcomes and probabilities land us at an average 0.2 blocks and 1.6 attacks.  Naturally we have more variability in the attacks (0.69 standard deviation) than blocks because of those possible explosions (and the fact that the non-explosion attacks represent 2 and not just 1 of the possible non-explosion rolls).

Okay so what if we burn our last die instead of rolling?  Let's burn for 1 shield.  The values in my original post for the averages are that this subtracts 3/5 from our average attacks and adds 4/5 to our average blocks.  We only had one die to roll and we're burning it so we look at the values in the graph for 1 die now (0.69 red, 0.4 blue).  So our averages and standard deviations become:
Average attacks: 1.6-3/5=1 attacks
Standard Deviation of attacks: ≈ 0.69-0.69=0 attacks
Average blocks: 0.2+4/5=1 blocks
Standard Deviation of blocks: ≈ 0.4-0.4=0 blocks
 

So our average is 1 attack and 1 block, and we have 0 variability because we've burned for everything!

So the point is that each successive burn reduces more variability from your outcomes than the last and makes your outcomes more and more likely to be close to what the averages predict.

I hope that makes sense.
« Last Edit: July 29, 2017, 05:54:50 PM by Fluffhead »