The Average Survival (AS) of Heroscape Figures

[Kroc]

Some characteristics of a Heroscape unit interact with each other in important ways. For example move plus range is what is called the “threat range” of a unit. The two characteristics where this interaction is most important are defense and life. Either alone has little value, it is the interaction that is important. For example DW9000 may have 9 defense dice, but he only has 1 life, so he is still pretty weak. The interaction tells you the probability of it surviving an attack, or in general the Average Survival (AS) of a unit.

We haven’t known this statistic because no one has figured out the exact formula before now. After discussions with mccombju (who helped a lot in this project, go +rep him) I have figured it out, and would like to show you the exact Average Survival of every Heroscape figure.

For these calculations I ignore any defense abilities like vanish (Isamu) or one shield defense (Warriors of Ashra). I also assume the attack is using 3 dice, since this is the sweet spot found by mccombju. For other levels of attack the table will be different, but this table is intended as a general strategy guide and so calculations based on the sweet spot are best. The interested party can easily generalize the results to take into consideration special abilities or different number of attack dice—see the end of this post.

In the table below the average survival for a figure with D defense dice and L life is calculated, for D or L up to 12. This is the average number of attacks you can expect that figure to survive—notice that each figure in a unit has the same average survival, thus the unit can be expected to survive #Figures*AS.

AS(D,L)                                                                                                   
D\L         1       2      3       4       5       6       7       8       9       10      11      12
0            .14     .63   1.3     2.0     2.7     3.3     4.0     4.7     5.3     6.0     6.7     7.3
1            .33     1.0   1.9     2.7     3.5     4.3     5.2     6.0     6.8     7.6     8.5     9.3
2            .6      1.5   2.6     3.6     4.6     5.7     6.7     7.8     8.8     9.9     10.9   11.9
3            .96     2.1   3.5     4.8     6.2     7.5     8.8     10.2   11.5   12.8   14.2   15.5
4            1.5     3.0   4.8     6.5     8.2     9.9     11.7   13.4   15.1   16.8   18.5   20.3
5            2.1     4.2   6.6     8.7     11.0   13.2   15.5   17.7   20.0   22.2   24.5   26.7
6            3.0     5.8   8.8     11.7   14.7   17.6   20.6   23.6   26.5   29.5   32.5   35.4
7            4.2     7.9   11.9   15.8   19.7   23.7   27.6   31.5   35.5   39.4   43.3   47.3
8            5.8     10.8 16.1   21.4   26.6   31.9   37.1   42.4   47.7   52.9   58.2   63.4
9            8.0     14.8 21.9   29.0   36.1   43.1   50.2   57.3   64.3   71.4   78.5   85.5
10          11.1   20.3 29.9   39.4   49.0   58.5   68.1   77.7   87.2   96.8   106.4 115.9
11          15.2   27.8 40.8   53.8   66.8   79.8   92.8   105.8 118.8 131.8  144.8 157.8
12          20.9  38.1 55.9   73.6   91.4   109.1 126.8  144.6 162.3 180.1  197.8 215.5 

If you round down each of these numbers that is the number of attacks you can expect a unit to survive. For example a full strength Major Q9 can be expected to survive 15 attacks on average. If you round up these numbers it is the average number of attacks that will take to kill it. For example on average it takes 2 attacks to kill a Deathreaver.
I would like to clarify a common error of probabilities here. Say that your Deathreaver survives an attack, then it still has an AS of 1.5, just like it did before the attack. It will still probably survive the next attack. All this statistic says is that on average a Deathreaver will die after 2 attacks.

Now, how can this be useful?
Order Marker Placement: Say that Ne-Gok-Sa is in the threat range of your opponent’s army. Should you put an order marker on him? If he has taken no wounds then on average he will survive 14 attacks so yes, with great confidence. If he has taken two wounds then there it may be risky to put your 3rd order marker on him—for example on average he will survive only 2 full rounds of attacks by the 4th Mass. using wait then fire.

Calculating a unit’s IGPV: UranusPChicago’s measure of how valuable a unit is doing at a given point during the game clearly depends on the units AS. Say, for example, someone has placed all 3 order markers on the same unit. If that unit has an AS of 6 then the IGPV of that unit is still fairly low. You need to attack it, but don’t spend too much of your energy on it.

Choosing a Glyph Holder: You might want to choose a unit with high move to get to the glyph quickly, but other than that you will want a figure with a high AS to hold a glyph. This explains the attraction of Deathreavers, with an AS of 1.5 and scatter it’s obvious they are great glyph holders.

How to use Defense bonuses: In the augmenting strengths or bolstering weaknesses thread I said that it is always better to bolster weaknesses. This table shows that for defense this is absolutely not true. The benefit of adding one defense dice to a figure is always increasing in the number of defense dice a unit has, and also in the life (the AS in short). So you should always augment the defense of already strong units. For example if you add Thorgrimm to Major Q9 you get an AS of 21.4 versus 15.8—he can now survive 5 more attacks. If you add it to a Venoc Viper squad they can not be expected to live longer at all—on average each figure will die in the first attack. (Yes, I know you can’t bolster the VV with Thorgrimm, it’s an example.)

Sweet Spots: These are critical values for D and L that could make a big difference to the figure’s survival.
For squads this is just D, and one critical value is 4D—that’s the number the Deathreavers have—with that value on average the unit will survive one attack. Every additional defense dice after that essentially means the figure can survive one more attack on average.
Another sweet spot is when it is better to augment Defense than Life. I know that (currently) one can’t augment life, but the sweet spot is so clear it is interesting. If the unit has 2L or less augment life, if the unit has 4L or more augment defense, if the unit has 3L it is better to increase defense unless it is 2 or lower. This may be due to the fact that the attacker has 3 attack dice—in the future I will check this with other level of attack dice.

Comparing and Evaluating Units: Consider the classic glass-jaw hero, Deathwalker 9000. If you look at the table he can be expected to survive 8 attacks, not too bad you might say. But that is two attacks by a full squad of 4th Mass using wait then fire. No wonder people are under-impressed. While that's an above average AS DW9K costs 140 points. You can get Ne-gok-sa for a mere 90 points and get an AS of 14.7. (I have heard people comment on this virtue of NGS.) This might also explain why we don't really like Mimring all that much; he has an AS of only 6.2 for a cost of 150? If he didn't have decent offense and flying he would be worthless.
Another example was a recent thread comparing Su-Bak-Na vs. Tor-Kul-Na. One comment was that TKN seems to stay around for longer, TKN has an AS of 13.2, versus SBN's AS of 6.2. So on average TKN will survive twice as many attacks. This person's insight was right on the money.
Which common squad will survive the longest? The Minions of Utgar with 6D, but how much is that 6D worth? It gives a per-figure AS of 3, so to wipe out a squad will take 9 attacks on average. Compare this with the Deathreavers, with an AS of 1.5. This means that wiping out a squad will take 8 attacks on average—and they cost almost a third as much. No wonder we love those little ratties.

Statistics:
These statistics were calculated using Bixby’s latest sheet—which only goes up to wave 8. When he updates the sheet I’ll update these calculations. The average is 4.5, or your average figure can withstand one full round of 4th Mass wait then fire attacks--but just barely. The average for heroes is 6.7, and the average for non-heroes (1 life) is 1.1. But these averages in general are deceiving because there is a heck of a lot of variance.
In general what is the top 10% in each category? Here I rank them by the average number of attacks necessary to kill off the entire unit.

Best Common Squads— Deathstalkers, Marrden Hounds, and Minions of Utgar (9); Deathreavers and Knights of Weston (8 )
Best Unique Squads—Zettian Guards (10); Izumi Samurai and Tagawa Samurai (9); Tarn Viking Warriors (8 ) (Notice the Samurai have Counter-Strike, so the probability of them being attacked is lower.)
Best Unique Heroes—Charos (20); Major Q9 (16); Ne-Gok-Sa (15); Tor-Kul-Na (14); Brunak, Jotun and Valguard(12); Braxas and Krug (11)

Notice that there are some favorites on that list but there are some very bad units and some that are just average. Average Survival is not the only thing that determines a unit's value.

Credits and Acknowledgements:
This project was started by Killercactus’s idea of evaluating units based on their lifense which is D*L. Aldin quickly pointed out that such a simple formula couldn’t capture the interaction properly, and then mccombju came up with a lot of better formulas that captured the interaction better. In a discussion with mccombju in the lifense thread I realized that the precise formula wasn’t that hard to calculate, and then ran off to figure out what the exact AS of a figure was. While constructing the AS I had mccombju check my math, and I thank him a lot for his help, so aim your lil’ ol rep cannon in his direction. Of course he didn’t check my basic formula (below) so there is a small chance that all of these calculations are garbage—in which case you should put the blame squarely on me.

The Formula:
This is only for the math geeks out there. I want to tell you my formula so you can check my work but for most of you this probably requires math that you are not familiar with. So, for the few weirdoes out there who really know more math than is good for them.

What we want to figure out is the AS of a figure with D defense dice and L life, denoted AS(D,L). This obviously depends on the number of attack dice used against them, in this calculation I assume that there are 3 attack dice—the formula can easily be generalized but the resulting AS probably will follow similar trends to the calculations above.
To understand the general formula it is best to first work on figures that have 1 life, AS(D,1). Let P(w|D) be the probability of w wounds given the figure has D defense. The notation follows Sisyphus’s notation for calculating the probability of a wound. Then with P(0,D) the figure survives one attack, with P(0,D)* P(0,D) (=P(0,D)^2) it survives two, and so on. So the formula is:

AS(D,1)= P(0,D)+ P(0,D)^2+ P(0,D)^3+ P(0,D)^4+…

And using a well known rule for summing infinite sequences this is equal to:

AS(D,1)= P(0,D)/ [1-P(0,D)]

You could also calculate this using a value function formula:

AS(D,1)= P(0,D)(AS(D,1)+1)

The logic for the +1 in this formula is that if the figure survives the first attack its survival has increased by 1 because of this. Now to present the general formula in the simplest manner let AS(D,L)=-1 if L is 0 or lower. Then the formula is:

AS(D,L)= P(0,D)(AS(D,L)+1) + P(1,D)(AS(D,L-1)+1) + P(2,D)(AS(D,L-2)+1)
+ P(3,D)(AS(D,L-3)+1)

And if you solve for AS(D,L) you get:

AS(D,L)= [ P(0,D) + P(1,D)(AS(D,L-1)+1) + P(2,D)(AS(D,L-2)+1) + P(3,D)(AS(D,L-3)+1) ]/
[1- P(0,D)]

This formula can then be calculated starting with L=1 and carried on as high as you want for given D. Calculating P(w,D) for w in {0,1,2,3} can be done with Sisyphus’s formula.
If the figure has special defense abilities this will alter P(w,D), and with these modifications you can figure out the survival given that ability. If you want to consider what happens if the attacker does not have 3A you can easily add or subtract terms on the right hand side of this equation and calculate the correct values of P(w,D) to generalize this formula.

[mccombju]

Interesting work, Kroc.

The following graph summarizes the data presented by Kroc in the post above. The values shown represent the Average Survival (AS) or 'Survival Factor (SF)' of the defending figure. That is, the graphs show how many attacks a defending figure will survive based upon the combination and interaction of Defense and Life when confronted with an attack of 3A.





The following graph highlights a number of the survival powerhouses that Kroc outlined in his discussion. Green dots represent the Average Survival of heroes with powerful combinations of Life and Defense (vs. 3A).



Heroscape Survival Powerhouses:

D	L	SF	
5	9	19.955	Charos
7	4	15.819	Major Q9
6	5	14.630	Ne-Gok-Sa
6	8	23.452	Abomination
7	6	23.693	Silver Surfer
6	8	23.452	Hulk
4	7	11.691	Valgaurd
7	6	23.693	Thanos
7	3	11.878	Brunak
4	6	9.958	Sir Gilbert
3	8	10.180	Krug
5	6	13.207	Tor-Kul-Na

[killercactus]

Guys,

Thanks for this great work. It's awesome to see my lifense concept taken up to the next level. Though, I still like it some because it's easy and doesn't focus on one attack level , but this is far better.

The only things left to do, I think, is 1) do this table for each level of attack dice, and then do some kind of average and 2) factor in Special Powers somehow.

Still though, excellent work! I'll have to RunaRep mccombju again later since I just did a short while ago.

[mccombju]

Originally Posted by killercactus

...1) do this table for each level of attack dice...

The calculations are complete. You can access them in the following MS Excel File, including the tables and graphs for the Survival Factor; Probabilistic Damage Inflicted; and Probabilities of Wounding.

Originally Posted by killercactus

and then do some kind of average

The so-called "metagame attack power" was my attempt at simulating the average attack power (above and beyond simply selecting 3A as the 'magic attack number').

The metagame attack power worked adequately well for Survival Point Economy (SPE); Tactical Battlefield Positioning (TPB) (aka Survival %); and Overall Tactical Survival and Point Economy (OTSPE).

However, I have found the metagame attack power to be much less useful when concerning the Survival Factor (aka Average Survival aka Probabilistic Number of Attacks Survived).

[ollie]

As with all the lovely stats stuff that gets generated on the site, the first question to ask is: Is this useful in-game or is it just theoretical fun? (Believe me, as a pure mathematician I have nothing whatsoever against theoretical fun.)

I don't usually playtest much (or at all) for tournaments, but I do sometimes run batches of numbers that I think are potentially good to know.

When defending, you don't need to remember all these numbers (as well as killercactus's hypothetical ones), you only need the relevant ones. So, for example, before I took DW9K to an event I worked out these numbers (or very similar ones) for nine defence against various attack levels and then summarised it in a handy package that I could remember: DW9K has about a one in three chance of succumbing to three attacks of 3, two attacks of 4 or one attack of 5. This gives an anchor from which to make in-game intuitive calls.

Similarly, when attacking, you only care about what you can achieve and what is likely to happen with the attack values in your army.

In short, tabulations and calculations such as this are interesting in an abstract way and useful once some more situation-specific work is put in. I guess my main interest is in that second stage. Does anyone (else) do mathematical prep work for games/tournaments? What form does it take? How can these numbers be best used?

Good stuff Kroc and all. Thanks.

[Kroc]

Originally Posted by ollie

As with all the lovely stats stuff that gets generated on the site, the first question to ask is: Is this useful in-game or is it just theoretical fun? (Believe me, as a pure mathematician I have nothing whatsoever against theoretical fun.)

Well, ollie, my man, thou are so dissmissive. Like I just said above in the Charos example and as I said in the original post, placement of order markers should be affected by this statistic. AS changes during the game, like last weekend I managed to whipe out 3 of Q9's life in my first attack. Knocking him down from an AS of 15.8 to an AS of 4.2. Quite a blow, ehh?
My opponent lost an order marker because I then whiped out Q9 pretty quickly and he didn't check the table.
Of all the posts I've written on strategy I think this is potentially the one that is most useful. It's only a statistic, but it can be an important one.

Originally Posted by ollie

...In short, tabulations and calculations such as this are interesting in an abstract way and useful once some more situation-specific work is put in. ... How can these numbers be best used?

I think looking over the AS of your army could be very important. I haven't figured it out but you could try to maximize the AS of your 500 point army, then try to win every game on points. I wouldn't want to do that, and wouldn't want to play against someone who has, but that would be interesting.

But outside of tournament preparation, I think these tables do make one important point. They show that we should always augment the defense of already strong units. I didn't realize that before, and I'll definitely be using that in the future.

[ollie]

Originally Posted by Kroc

Well, ollie, my man, thou are so dissmissive.

Sorry for seeming dismissive---that certainly wasn't my intent. My reaction was more "wow, this is really cool, what can we do next".

Like I just said above in the Charos example and as I said in the original post, placement of order markers should be affected by this statistic. AS changes during the game, like last weekend I managed to whipe out 3 of Q9's life in my first attack. Knocking him down from an AS of 15.8 to an AS of 4.2. Quite a blow, ehh?
My opponent lost an order marker because I then whiped out Q9 pretty quickly and he didn't check the table.
Of all the posts I've written on strategy I think this is potentially the one that is most useful. It's only a statistic, but it can be an important one.

I agree with all this---rephrasing my "is this useful in-game" question: Assuming that I'm not going to commit all of this (and the additional Excel file that I haven't looked at yet) to memory, can something useful in-game be extracted?

If I come up against Charos, but haven't memorised the numbers, do I play less than optimally? I'm not sure. A wider context is needed---what alternatives do I have? So, suppose I come up against Stingers and Charos. I can imagine sitting down with these numbers, considering both attack and defence including whatever units I have, to fine-tune decisions and make marginally better decisions. I can't imagine doing this in-game, for two reasons: I won't have memorised the necessary starting data and I won't have the time or mental arithmetic ability to make the necessary calculations even if I knew the data.

Before this is useful in-game I think it needs condensing to rules of thumb (such as my Deathwalker example above or your claim that it's always better to augment the defence of already strong units---see below).

Originally Posted by Kroc

I think looking over the AS of your army could be very important. I haven't figured it out but you could try to maximize the AS of your 500 point army, then try to win every game on points. I wouldn't want to do that, and wouldn't want to play against someone who has, but that would be interesting.

But outside of tournament preparation, I think these tables do make one important point. They show that we should always augment the defense of already strong units. I didn't realize that before, and I'll definitely be using that in the future.

I think you're going beyond what can be deduced from the stats here. There is more to the game than AS. Extreme case: high AS but attack values of 1 across the board. Good army? It's an important part of a bigger picture, but that bigger picture needs consideration.

Again, with the "always augment strong units" rule. I'll go with a "usually augment strong units" rule of thumb, but exceptions based on in-game considerations, or special abilities, or glyphs, or... are not hard to imagine.

Heh, I went all negative again, sorry. As I said at the start, I think this is cool, just not the ending point of the questions it raises.

Originally Posted by Kroc

But, eh hem, I do have to correct a possible error in your post, Sarpedon, it seems like your saying:
a) Deathreaver dies
b) Other deathreaver covers glyph on the scatter.
I thought this at one time, but was corrected. You scatter BEFORE you take any dead deathreavers off of the board. So if one dies you can't just move right onto his space.

I thought that conversation ended the other way round and that Sarpendon's version was the correct one.

Edit. Link.

Edit 2. The important post:

Originally Posted by Grungebob

1) The rat is first killed and removed from the board
2) Then you scatter with 2 other rats

[Aldin]

Impressive work. I do some math prep, but it's pretty minimal. Mostly it'll be something like "how rapidly can my army go downhill if I lose a key figure or two"? Charts and tables like this are useful for questions like "if I want a damage sump Warlord do I use Valguard or NeGokSa"? It clarifies impressions based on play that both do a nice job of clogging up the middle.

~Aldin, who notes yet again that the Tarn are the best squad that hardly anyone uses

[killercactus]

Originally Posted by Aldin

~Aldin, who notes yet again that the Tarn are the best squad that hardly anyone uses

A true statement. It's just that the Marro Warriors are so good for 50 points. I wonder if an army using them both is feasible.....

[NecroBlade]

Though I didn't have an exact formula (not much of a mathematician), I've been calculating "survivability" for a long time based on Life/Defense. I'll have to give this a more thorough read sometime, but it looks like what I've always wanted/tried to do with those numbers! It also appears my early deduction that Defense > Life, at least when they're relatively equal, was correct (see TKN/NGS). Very cool stuff, everyone involved. (We'll see how many of you I can get before my rep runs out.)

[Xn F M]

Originally Posted by killercactus

A true statement. It's just that the Marro Warriors are so good for 50 points. I wonder if an army using them both is feasible.....

Sounds like a pretty good 100 point army to me . Seriously though, they do play pretty well together at low point values.

Back on topic, thanks for putting the work into this guys. While I do see this kind of stuff as sort of a fun abstract kind of thing, it is something that seems like it would be handy to have in the back of my mind when putting together an army. It's also nice to see Valgaurd finally get some respect.

[GreenLanturn]

Does this value change during the course of the game? Meaning, if Charos has 9 life you wouldn't waste your time, but if he has 1 life he is a great 210 point target.

I would think that these numbers would begin to scale down as the game progresses. Or would we all still be looking at Charos as a 20AS hero?

That said I would also think that the original AS of any given hero (with more than one life) has a significantly greater AS than you are mathematically reasoning. Sure if I was forced to attack him these numbers would be true, but I'm not. Seeing that 9 life 5 defense is like a mental defense and adds greatly to his AS (in this case it could be read Assumed Strength)