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  #1  
Old August 22nd, 2021, 01:38 AM
Tauros105 Tauros105 is offline
 
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Hero Defensive Values

I have made a google sheets spreadsheet to determine how many attacks of 3,4 or 5 it will take to kill a hero of a given defensive value and life. I have used sissyphus's probability tables for this. My methodology is quite flawed, but is a loose indicator of how heroes stack up against each other in the realm of pure attacks. I am open to criticism and questions. Special thanks to dysole for getting my account able to post!
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Last edited by Tauros105; August 22nd, 2021 at 11:21 AM.
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Old August 22nd, 2021, 09:18 AM
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Re: Hero Defensive Values

Hi Tauros. It looks like you posted images hosted directly on your computer. You'll need to put the images online on an image hosting website like imgur for other people to see them. Excited to see your analysis though.
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Old August 22nd, 2021, 12:17 PM
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Re: Hero Defensive Values

Looks like you got the images working. Great first post!

Can you talk a little bit about the math you're doing behind the scenes? The numbers intuitively seem okay to me, but would just like to hear a little more about the process.

What this is great for is analyzing the concept of effective survivability of figures. I've talked before on the forums about the concept of using Life times Defense as a proxy for this sort of detailed analytical work, to get the general effective survivability for figures. So a figure with 3 defense 3 life like MBS would have the same life times defense as Deathwalker 9000, with 1 life 9 defense, since 3x3 = 9 and 1x9 = 9. So we can use this table of effective survivability to see how accurate this heuristic really is.

Major Q9 and Tandros Kreel (4 life 7 defense vs. 7 life 4 defense) (28 life defense)

Q9 will survive for:
5.39 attacks of 5
8.58 attacks of 4
15.74 attacks of 3

Tandros will survive for:
5.24 attacks of 5
7.51 attacks of 4
12.06 attacks of 3

Nilfheim and Braxas (6 life 4 defense vs. 8 life 3 defense) (24 life defense)

Nilfheim will survive for:
4.39 attacks of 5
6.74 attacks of 4
11.86 attacks of 3

Braxas will survive for:
5.03 attacks of 5
6.95 attacks of 4
10.66 attacks of 3

There aren't any real figures in this range, but let's check out the perfect square number 16 life defense to see how the various extremes handle:

2 life 8 defense will survive for:
3.34 attacks of 5
5.50 attacks of 4
10.52 attacks of 3

4 life 4 defense will survive for:
2.99 attacks of 5
4.29 attacks of 4
6.89 attacks of 3

8 life 2 defense will survive for:
4.27 attacks of 5
5.70 attacks of 4
8.35 attacks of 3

The obvious conclusion that you probably knew before even reading this thread is that heroes with higher life do better against higher attacks, and heroes with higher defense do better against lower attacks, proportionally. The math plays that out pretty clearly.

Before reading this table, I would have guessed that among figures where their life and defense was equal, it was better to have similar life and defense values ie: I would have thought it was better to be 4 life 4 defense than 8 life 2 defense or 2 life 8 defense. But the math doesn't play that out. The math clearly favors either extreme life or extreme defense value for heroes. Maybe that's intuitive; we don't have the math here, but I think it makes sense that Deathwalker would survive better than MBS. Would be fun to do a Deathwalker 9K vs. MBS vs. Feral Troll survivability comparison, if we can get 1 life added to this table .

(And obviously, this is all theory. In practice, Braxas has worse survivability than Nilfheim even though this math says she has slightly better, because her lower ranges forces her to worse spots. But it's fun to know the theory, and makes you think about the gameplay in practice in better ways, imo.)
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Old August 22nd, 2021, 12:34 PM
Tauros105 Tauros105 is offline
 
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Re: Hero Defensive Values

Thanks and great Reply! My method of getting the numbers was dividing life by average damage to a figure with a given defense. I used 3,4 and 5 attack dice because my method techically creates a 3 dimensional space of values, so I just wanted to use common attack values. I have excluded one life because wound chance is a better approximation than average damage, and I also just think the variance is so high that I don't even want to touch it. I have thought about using defense times life as well as an approximation of how well a character can take damage, and it seems like defense plus life isn't a terrible way to approximate it either. For example:

Brunak (3 life + 7 defense = 10)
11.8 attacks of 3
6.4 attacks of 4
4 attacks of 5

Seige (5 defense + 5 Defense = 10)
11.7 attacks of 3
6.7 attacks of 4
4.5 attacks of 5

Seige loses in attacks of 3, and only leads attacks of 4 and 5 by 5% and 12% respectively. If you use life times health you get 21 for Brunak and 25 for Seige, which would imply a greater advantage for Siege. I was defenitely a little surprised by this. The main application of this in my mind is that if you want to kill Raelin, You need to commit at least 7 attacks of 3 to get the job done.
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Old August 22nd, 2021, 02:47 PM
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Re: Hero Defensive Values

Very creative method. It's a nice way to get an approximation of the answer, but the end answer is somewhat skewed. Essentially, it conflates discrete and continuous probability. A figure receiving enough wounds to die or not is a discrete binary yes or no event, while a figure receiving an "average" amount of wounds is a continuous event. Your method would work well for figures with massive amounts of life (like 30+ life) but at low numbers of life like most Heroscape figures have, it will give skewed answers.

The result of the way you're doing the math is you give credit for "overkill", while real Heroscape gives no credit for overkill. If a hero is one wound from death and receives many additional wounds all in one blow, in average damage terms, you're giving credit for killing him "faster", but in reality, in the discrete world, that huge overkill was the same result as if he had received one wound to die.

The real way to do this math is through probability state diagram. Let's look at the question of how many 3v3 attacks it takes to kill Me-Burq-Sa. MBS can either be at 0 wounds, 1 wound, 2 wounds, or 3 wounds (in which case he dies and the sequence ends).

We need to trace through all the states of this probability tree, slowly unfolding as each attack hits MBS. Let's say your first attack against MBS fails; you're basically back at the start of the tree, but you took a turn to do it. This is a pretty meaningful portion of the tree. After 4 attacks on MBS there's a 5% chance you still haven't done a wound to him. On the flip side, you could drop MBS at any time with a single attack of 3 skulls met with a whiff. You have a 3.7% of ending the tree after your very first turn.

Below is the screenshot of my state diagram math. I think I did it right, but it's possible I made a mistake. There may be some slight rounding error, since I only started with 3 digits and I really needed more than that. I clicked into one of the cells so you can see what I'm doing at each state: I'm taking the probability of each state (the number of wounds MBS currently has) from the previous row, and then applying the same repeated 3v3 probabilities to this current incoming attack, which then gives the probability of each state. When we enter the orange cell, the tree ends, as MBS is at 3 wounds, so you can see the total cumulative probability of each row fall off, since it's less likely we're even still rolling dice. The orange cell is the most complicated to calculate. If MBS is at 1 wound, receiving 2 wounds OR 3 wounds will kill him. If MBS is at 2 wounds, receiving 1 OR 2 OR 3 wounds will kill him. So many different states can all lead to the 3 wound state. Only one path can can lead to the 0 wound state: if MBS is already at 0 wounds and successfully blocks again.



My end result is that it should take 4.54 turns to kill MBS with attacks of 3v3, while your table says a flat 4 turns (3 life /.750 average damage per attack with 3v3 = 4 attacks). So, not an end of the world error, but significant enough to note.

Last edited by vegietarian18; August 22nd, 2021 at 03:22 PM.
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Old August 22nd, 2021, 03:08 PM
Tauros105 Tauros105 is offline
 
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Re: Hero Defensive Values

This is exactly why I said my methodology was flawed. I knew that what I was doing was not the theoretically correct answer. I did not know that it would be that far off the actual result. Thank you for the input, and I'm honestly impressed with the stats know how you have applied here. I think your approximation is much closer to the true value. A theoretical answer can certainly be achieved using probability analysis, but would include infinite sums and some stats work that I am not exactly begging to do. Thanks again!
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