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.