Big stacks, better than normal-sized stacks?

OK, sifting through the last few posts, I think I have a better understanding of battle mechanics.

Simple question: how do you get the rolls for each unit?

@Alex: good idea using mili. This makes it much less complicated. I'm curious to see the results.

Note: Using Militia vs. Infantry doesn't simplify the process, it complicates it.
Buffs and strategy aside, Infantry has a defense of 6. Bombers have an attack of 6 etc. So, for those not tracking at a per-roll basis, it makes it difficult to get anything but an impression, which is uncontroversial: In combat, defending militia are not as powerful as attacking bombers.

If you're tracking each roll, it is easy to eliminate the effect of critical (on the rolls, anyway). Any critical hit will exceed the maximum damage value. If there is indeed some kind of nerf on defending neutrals (etc.) the fact that the militia have less distribution (1-4 vs. 1-6) means that outliers will be more difficult to detect, because the militia normal damage done data subset has only 4 values available (1,2,3,4) vs (1,2,3,4,5,6) of infantry.

To see rolls:
Settings | My Settings | Battle Speed = Very Slow. Save Settings.
Don't try this except in casual games, however!

Kirjutas Grimm, 11.02.2014 at 18:14

OK, sifting through the last few posts, I think I have a better understanding of battle mechanics.

Simple question: how do you get the rolls for each unit?

@Alex: good idea using mili. This makes it much less complicated. I'm curious to see the results.

Larger stack seem to have more critical because i believe each unit give diffrent amount of roll. You might have more roll(because of more unit) meaning critical are higher when combine. Critical improve or add both your defence and attack, I think. when your attack destroy an enemy its transfer to the next unit and if there are still spare it transfer to the next until it runs out. It like a united we stand(more troup) vs alone and the only(less troup). From what we know the number mostly win. Like a big bully buff strong troup vs weak by numerous troup.

Kirjutas Htin, 11.02.2014 at 18:58

Larger stack seem to have more critical because i believe each unit give diffrent amount of roll. You might have more roll(because of more unit) meaning critical are higher when combine. Critical improve or add both your defence and attack, I think. when your attack destroy an enemy its transfer to the next unit and if there are still spare it transfer to the next until it runs out. It like a united we stand(more troup) vs alone and the only(less troup). From what we know the number mostly win. Like a big bully buff strong troup vs weak by numerous troup.

No mate..Critical = roll + maxDmg
Critical has nothing to do with these rolls. And what you said has nothing to do with this? xD

Bump.
Doing tests with no upgrades, militias v militias, I get -3 -4, -6 -7, etc most of the time. Sometimes, it's -3 -3 or so. Though, Militias have a low damage, and maybe that's why the results aren't that different.

Only assholes will tell you size does not matter but trust me if you have a big baby arm holding those apples it sure beats one that don't hold nothin

Kirjutas 4Chan, 17.02.2014 at 09:32

Only assholes will tell you size does not matter but trust me if you have a big baby arm holding those apples it sure beats one that don't hold nothin

LOL'D
You're a pervey. Stahp doing this to important threads plez.

Yes. Also noticed this issue several games when pd attackers lose fewer troops in comparison to pd defenders (both inf) when attacking stack is bigger (e.g. 150 attacking side lose 86, defending side 102). Also happened today.

If you make a list of 100 rolls, with 100 units attacking 100 units, and record the outcome. I will calculate for you, the distribution they fall under and give you their statistical probability. Otherwise, I suggest you guys do not worry much about it, because their are chances that you will get high casualties and low casualties, you just don't know the probability.

[The more data, the better]

I have a feeling they will fit nicely under a normal distribution:
X ~ N(mu, sigma^2).
Mu is the mean of the data.
Sigma^2 is the variance.
To find this, we need data.

Once someone reports the result, you can do this:
Z score = (x - mean) / sigma, where z score probability can be found on any tables online. x represents # of casualty you choose.

For example:
(50 - 50) / 1 = 0; the probability of a zero z-score is 0.5000
Which would mean, there is 50% probability that only 50 units will survive the battle.

Kirjutas Cthulhu, 20.03.2014 at 23:07

If you make a list of 100 rolls, with 100 units attacking 100 units, and record the outcome. I will calculate for you, the distribution they fall under and give you their statistical probability. Otherwise, I suggest you guys do not worry much about it, because their are chances that you will get high casualties and low casualties, you just don't know the probability.

[The more data, the better]

I have a feeling they will fit nicely under a normal distribution:
X ~ N(mu, sigma^2).
Mu is the mean of the data.
Sigma^2 is the variance.
To find this, we need data.

Once someone reports the result, you can do this:
Z score = (x - mean) / sigma, where z score probability can be found on any tables online. x represents # of casualty you choose.

For example:
(50 - 50) / 1 = 0; the probability of a zero z-score is 0.5000
Which would mean, there is 50% probability that only 50 units will survive the battle.

Before we waste anyone's time here, let me clarify a few points.
1. Thank you!

2. The reason there is a question: I, and Alex, both have conceptions of how the AW universe is supposed to work.
Based on those conceptions, we have both solicited further input from the community to refine our understanding. This was an iterative process; many improvements were made as our understanding was refined.
We then applied these rules to develop expectations (a unit with an attack of 6 and a crit of 5 will attack for 1 - 12 damage, with an average damage of 3.8).
We then independently developed Battle Simulators with RNGs. In my observations, it generally took 100,000 simulated battles, and millions of 'rolls' for simulated results to converge on the expected results.

3. Using different approaches, we both pit our simulations and expectations against the AW Universe, matching observations against our models. Alex looked at rolls. I looked at large player bomber stacks against small neutral infantry stacks, and was only interested in the number of player survivors. He observed micro, I, macro.
Fail.

It is possible (highly unlikely) that we both had a run of 'bad' luck - that our expectations were valid, but our observations were anomalous. Sample sizes were not huge.
Since we had already exhausted the community with appeals for Battle Mechanics knowledge the next step would be to perform tests and aggregate observations to further refine our understanding of battle mechanics.

I stopped, because understanding the AW universe isn't like understanding the IRL universe.
=The underlying laws governing physics of the IRL universe don't change, our understanding and interpretation of them does.
=The underlying laws governing battle mechanics of the AW universe change on the developer's whim - and may even include goal-seeking behavior, additional chaos, or some other capriciousness, rendering our observations meaningless for future predictive value.
God could tell us how the AW universe works now, but chooses not to. The AW Gods, usually quite just, are in this matter excessively cruel.

You don't need to understand how it works, just understand the probability of the events happening :) To do that, one must first observe, and then can draw upon the observation.

• Attack: 100 Inf // PD
  
• Defend: 100 Inf // PD

Best result: Def remaining units 54
Worst result: Def remaining units 39
Average: Def remaining units 46,76

Now try with twice the attacking force. 10 +- 10 attackers should be left alive.

Attackers Attackers' (bomb) casualties Defenders' (inf) casualties

75 7 9
75 8 9
75 6 9
75 5 8
75 8 8
75 8 9
75 8 11
75 3 9
75 4 9
75 5 6
75 9 12
75 4 6
75 4 5
75 6 10
75 7 10
75 6 11
75 5 10
75 3 7
75 4 10
75 5 8
75 7 10
75 4 7
75 5 8
75 4 10
75 6 10
75 10 13
75 8 13
75 9 13

30 6 8
30 4 7
30 4 6
30 5 8
30 5 8
30 7 10
30 3 5
30 5 6
30 2 5
30 1 3
30 8 11
30 3 4
30 3 3
30 6 8
30 7 9

Kirjutas Columna Durruti, 22.03.2014 at 10:27

• Attack: 100 Inf // PD
  
• Defend: 100 Inf // PD

Best result: Def remaining units 54
Worst result: Def remaining units 39
Average: Def remaining units 46,76

I also need the standard deviation or variance xD

Which is :
variance = summation of (xi - mean)^2 / n - 1

Kirjutas ifinishlast, 23.03.2014 at 05:52

Attackers Attackers' (bomb) casualties Defenders' (inf) casualties

75 7 9
75 8 9
75 6 9
75 5 8
75 8 8
75 8 9
75 8 11
75 3 9
75 4 9
75 5 6
75 9 12
75 4 6
75 4 5
75 6 10
75 7 10
75 6 11
75 5 10
75 3 7
75 4 10
75 5 8
75 7 10
75 4 7
75 5 8
75 4 10
75 6 10
75 10 13
75 8 13
75 9 13

30 6 8
30 4 7
30 4 6
30 5 8
30 5 8
30 7 10
30 3 5
30 5 6
30 2 5
30 1 3
30 8 11
30 3 4
30 3 3
30 6 8
30 7 9

Can you help explain this data? like how you get it, and what is 75, 30? also is it units left? or units that died?

Kirjutas Cthulhu, 23.03.2014 at 06:17

Kirjutas Columna Durruti, 22.03.2014 at 10:27

• Attack: 100 Inf // PD
  
• Defend: 100 Inf // PD

Best result: Def remaining units 54
Worst result: Def remaining units 39
Average: Def remaining units 46,76

I also need the standard deviation or variance xD

Which is :
variance = summation of (xi - mean)^2 / n - 1

This?

75 7 9

75 - number of attackers
7 - how many attackers have perished
9 - how many defenders were killed by those 75 attackers

Kirjutas Columna Durruti, 23.03.2014 at 13:51

Kirjutas Cthulhu, 23.03.2014 at 06:17

Kirjutas Columna Durruti, 22.03.2014 at 10:27

• Attack: 100 Inf // PD
  
• Defend: 100 Inf // PD

Best result: Def remaining units 54
Worst result: Def remaining units 39
Average: Def remaining units 46,76

I also need the standard deviation or variance xD

Which is :
variance = summation of (xi - mean)^2 / n - 1

This?

graph

You also need sample size. I'm assuming this is 1000 simulations you ran in Excel (not actually in-game)?

Now, what we have is what looks like a nice normal distribution. This gives us an idea of the variation for 100 INF attacking 100 INF battles. We could now run a few test battles 100 vs. 100 and should not observe a significant difference between expected vs. observed (a simple statistics test would do it).

The initial question was big stacks vs. normal sized stacks. To answer this, we would now need to run a simulation with a smaller number of INF attacking 100 INF (maybe 10 vs. 100?). Then, again, compare test battle results vs. simulation.

Would be fairly simple to do.

Kirjutas ifinishlast, 23.03.2014 at 14:22

75 7 9

75 - number of attackers
7 - how many attackers have perished
9 - how many defenders were killed by those 75 attackers

What's the number of defenders? Also, this is round-by-round data right?

Kirjutas Cthulhu, 21.03.2014 at 18:43

Tsiteeri:

You don't need to understand how it works, just understand the probability of the events happening :) To do that, one must first observe, and then can draw upon the observation.

Well, strictly speaking, that's bad science. You ask the question, formulate falsifiable hypotheses, predict, design and execute tests, and perform analysis.
Our primary flaw is that we *assumed* elementary battle mechanics followed specific rules, namely, an 'Attack' or 'Defense' Rating (e.g. 6) produces base damage of 1 to the rating (6) with an equal chance of (1,2,3,4,5,6), and any qualities that would modify this base damage would only increase, not decrease the damage.
Our assumptions were not capriciously established; we researched, spoke to peers etc.
In the tested case, large stacks of attackers vs. small stacks of defenders, the *attackers* damage rolls were as assumed.
Defenders were not.

Kirjutas Cthulhu, 20.03.2014 at 23:07

If you make a list of 100 rolls, with 100 units attacking 100 units, and record the outcome. I will calculate for you, the distribution they fall under and give you their statistical probability. Otherwise, I suggest you guys do not worry much about it, because their are chances that you will get high casualties and low casualties, you just don't know the probability.

[The more data, the better]

I have a feeling they will fit nicely under a normal distribution:
X ~ N(mu, sigma^2).
Mu is the mean of the data.
Sigma^2 is the variance.
To find this, we need data.

Once someone reports the result, you can do this:
Z score = (x - mean) / sigma, where z score probability can be found on any tables online. x represents # of casualty you choose.

For example:
(50 - 50) / 1 = 0; the probability of a zero z-score is 0.5000
Which would mean, there is 50% probability that only 50 units will survive the battle.

There are liars, damn liars and statistics. Rather than look at bombers in India, or beggars in Spain, I looked at the data we already have. I don't have data (anymore) for attacks, but we do have data for 124 die rolls, the 'defender' rolls from earlier. (126 rolls, 2 critical results dropped).

Range: (1,2,3,4,5,6)
Results:
1: 56 (translation "1 was rolled 56 times)
2: 33
3: 12
4: 10
5: 8
6: 5 (translation "6 was rolled 5 times)
Total Rolls in Sample: 124
Expected mean: 3.5 (1+2+3+4+5+6)/6
Actual mean: 2.16
Variance: 8.08
Std. Dev: 2.84

Here are hypothetical results for a statistically 'perfect' expected result
Range: (1,2,3,4,5,6)
Results:
1: 21
2: 21
3: 21
4: 21
5: 21
6: 21
Total Rolls in Sample: 126
Expected mean: 3.5 (1+2+3+4+5+6)/6
Actual mean: 3.5
Variance: 10.21
Std. Dev: 3.20

A scientist with only the Total Rolls, Actual Mean, Variance, and Standard Deviation available would conclude that Set B, the perfect set, had more volatile data than Set A, the observed data.

As our current problem is large stacks losing less troops when attacking smaller stacks, all defenders in test cases are killed.

Excel case (if it is simulated) is useless. We need std. deviation from real data, not simulated one.

As you can see from my data, defenders always lose more.

Based on Columna's Data, you would have this would be the probability of the remaining INF:

68% would fall under 30-44
95% would fall under 26-48
99.7% would fall under 23-51

so while it is possible to get units different then those ranges, it is extremely rare.

As you can see from this limited set of data, attackers have an advantage and our hypothesis that bigger numbers helps might be true.

Perhaps this is where we should have started.

Anomalies in Expected AW Neutral Defender Battle Results

Abstract:
Observational data indicates that Battle Mechanics do not operate as expected in the case where large quantities of attacking player units battle one-or-more-orders of magnitude fewer neutral defending units. Defending neutral units defend at about 40% of the effectiveness the current models would indicate.

Intro:
A.Meza conjectured that large stacks of attackers perform better than expected against small stacks of defenders, based on anecdotal observations. Z.Yeti disputed this conjecture, attributing AM's impressions as being based on small sample size, 'gambler's myopia' etc. Both AM and ZY were looking to build stochastic-based battle simulators, and agreed that if such a bias towards effectiveness of player attackers did exist, the current understanding of Battle Mechanics did not incorporate such bias.

Methods:
AM and ZY performed exhaustive searches for source material and received peer and 'mod' contributions to explanation of Battle Mechanics. Early on, the community agreed that a unit v. unit paradigm existed; large vs. small had no additional effect on the conduct of the battle. Obviously the outcome of the battle, attrition-wise, favored the combatant with the most-numerous and powerful units.

Deterministic analysis of the elementary aspects of Battle Mechanics allowed the researchers to determine the average expected damage of attackers and defenders. Respectively, a set of 1 to n, composed of natural numbers (with n being the Attack or Defend value) was assembled.
- The average base damage was determined to be the sum of the members of the set, divided by the number of members in the set. A unit with a damage of 6 would have a set of (1,2,3,4,5,6), and an average base damage of (1+2+3+4+5+6) / 6, or 3.5.
- Critical damage was ascertained by using the Critical attribute as a percentage, and applying this to the maximum base damage the unit could achieve. Maximum base damage was multiplied by the Critical percentage. A unit with a damage of 6 and Critical of 5 would have an average critical damage of .3
- Total Average Damage was the sum of average base damage and average critical damage. A unit with a damage of 6 and a critical of 5 would possess a Total Average Damage of 3.8

Stochastic Modeling of the battle mechanics concluded that, over a series of 100,000 simulated battles, the Total Average Damage forecast by both models were in agreement. Both predictive and stochastic models forecast an average damage of 3.5 for a unit with a damage of 6.

The models were compared to AW Battles for validation. 126 rounds of combat would be recorded (21 x 6), with attacker and defender rolls tracked, and the Battle result compared to the Modeled Result ('actual' vs. 'expected').
In the battles recorded, the player would ensure no strategies were chosen, bombers sans generals would be the attacking units, and never would the attacker/defender ratio fall below 12 to 1.

Results:
Attackers were expected to do an average of 3.5 normal damage, and total average damage of 3.8 per attack. Attackers inflicted an average of 4.2 damage, about 9.5% more than expected, but well within the expected range considering the size of the sample.

Defenders were expected to do an average of 3.5 normal damage, and a total average damage of 3.8 per defend. Defenders inflicted an average of 2.16 normal damage, and a total average damage of 2.22 per defend.
Focus was shifted to the 124 'normal' damage rolls.

As the cumulative damage of 268/average normal damage of 2.16 over 124 rolls was thought to be exceptionally rare where the expected average was 3.5, 124 rolls were simulated 100,000 times. Not once was an average of 2.16 or lower obtained.

Discussion:
The current understanding of battle mechanics and the deterministic and probablistic models forecast attacker damage adequately well.

The defender actual average damage was 37.14% less than expected, over a series of 124 runs. The odds of 124 'fair dice' rolls (1-6) with a total of 268 or less (an average of 2.16 or less) is of a rarity approaching zero.

Conclusions:
0. As the likelihood of the 2.16 average damage of being a Black Swan event under our current understanding of Battle Mechanics is essentially zero, I can speak confidently to the conclusions. In 100,000 simulated sets of 124 rolls, the lowest average roll over a discrete set of 124 was 2.87.

1. There is an apparent in-game bias towards either Player efficacy overall, and/or towards Large Player Attacking stacks vs. Small Neutral Defending stacks.

2. Formally, Elementary Battle Mechanics are not well understood, or the understanding is flawed or at the very least, incomplete. Frustrating, as the current understanding of Battle Mechanics forecast Player Attacker damage adequately.

3. The experiment was not designed to test whether the bias is towards large stacks of units or towards player units (or both).

4. Further experiments should be conducted to isolate if the 'exception' to the current understanding of Battle Mechanics is limited to the special case of Large Player stacks vs. Neutral stacks, or is a general case where large stacks perform well against small stacks, and/or Player effectiveness is biased when attacking neutral units.

5. Obviously, independent verification of these results is necessary. There was no intent to deceive, but it is possible that results were incorrectly recorded.

6. The possibility exists that Battle Replays are not intended to replicate Battle Dice throws, but are approximations of the results achieved, simulated for entertainment value; if this is the case, only a small shift in approach would be necessary to re-validate the findings of this study; forecast battle losses for attacking units (macro level) based on the current understanding of Battle Mechanics and reconcile observed losses against expected losses.

EDIT:
Oh yeah, and while we can't say Alex was right (yet) we can say I was wrong. Something Strange is going on.

Kirjutas ifinishlast, 24.03.2014 at 05:43

As our current problem is large stacks losing less troops when attacking smaller stacks, all defenders in test cases are killed.

Excel case (if it is simulated) is useless. We need std. deviation from real data, not simulated one.

As you can see from my data, defenders always lose more.

Experimental results from you, me and Alex indicate a definite bias.
Is the bias towards large stacks, or player vs. neutral though?

Kirjutas zombieyeti, 24.03.2014 at 08:08

As the cumulative damage of 268/average normal damage of 2.16 over 124 rolls was thought to be exceptionally rare where the expected average was 3.5, 124 rolls were simulated 100,000 times. Not once was an average of 2.16 or lower obtained.

Heh. Combat system is bugged. This explains my losses, where pure maths would indicate a win (bigger combined defense should usually win against smaller combined offense both sides having same hp units, e.g. 100 attacking infs have 400 offense, 50 defending infs in a city - 450).

Kirjutas zombieyeti, 24.03.2014 at 08:10

Kirjutas ifinishlast, 24.03.2014 at 05:43

As our current problem is large stacks losing less troops when attacking smaller stacks, all defenders in test cases are killed.

Excel case (if it is simulated) is useless. We need std. deviation from real data, not simulated one.

As you can see from my data, defenders always lose more.

Experimental results from you, me and Alex indicate a definite bias.

Is the bias towards large stacks, or player vs. neutral though?

I need a fellow player to test against non-neutrals then. From a limited set of data, defenders have a disadvantage which increases with numbers.

Kirjutas ifinishlast, 24.03.2014 at 08:29

Kirjutas zombieyeti, 24.03.2014 at 08:10

Kirjutas ifinishlast, 24.03.2014 at 05:43

As our current problem is large stacks losing less troops when attacking smaller stacks, all defenders in test cases are killed.

Excel case (if it is simulated) is useless. We need std. deviation from real data, not simulated one.

As you can see from my data, defenders always lose more.

Experimental results from you, me and Alex indicate a definite bias.

Is the bias towards large stacks, or player vs. neutral though?

I need a fellow player to test against non-neutrals then. From a limited set of data, defenders have a disadvantage which increases with numbers.

I'll describe the ideal scenario for the test, implement as best you can.
1. Set up a cas. game, 48h, 50k world, PW protected, no upgrades permitted, allow joining until 99, make it at least 500 turns. Choose whatever strat gets your bombers to Attack 7. Choose a China, and build a big stack of bombers and an economic base. THEN message me with the PW, and head to India.
2. I will log in, choose India, and select strategy NONE. My infantry should defend at 7.
3. First test: Equals against equals.
I will End Turn as often as possible.
Attack each Indian city with the equivalent number of bombers as they have infantry.
Record results. In the (frequent) cases where you do not win (say, 3 infantry remain), attack the remainder with the equivalent number of bombers.
Make sure that your bombers do indeed attack at 7.
I will make sure my infantry do indeed defend at 7 (Settings > My Settings > Battle Speed > Very Slow)
Never capture a city.
Make sure you are attacking Infantry (not Militia).

4. Repeat until you have 100 battles recorded, and tell me to ally end.
---
Next experiment will be similar, except it will be exactly 15x the bombers as infantry attacked.

Kirjutas zombieyeti, 24.03.2014 at 09:38

Kirjutas ifinishlast, 24.03.2014 at 08:29

Kirjutas zombieyeti, 24.03.2014 at 08:10

Kirjutas ifinishlast, 24.03.2014 at 05:43

As our current problem is large stacks losing less troops when attacking smaller stacks, all defenders in test cases are killed.

Excel case (if it is simulated) is useless. We need std. deviation from real data, not simulated one.

As you can see from my data, defenders always lose more.

Experimental results from you, me and Alex indicate a definite bias.

Is the bias towards large stacks, or player vs. neutral though?

I need a fellow player to test against non-neutrals then. From a limited set of data, defenders have a disadvantage which increases with numbers.

I'll describe the ideal scenario for the test, implement as best you can
1. Set up a cas. game, 48h, 50k world, PW protected, no upgrades permitted, allow joining until at least 99, make it at least 500 turns. Choose whatever strat gets your bombers to Attack 7. Choose a China, and build a big stack of bombers and an economic base. THEN message me with the PW, and head to India.
2. I will log in, choose India, and select strategy NONE. My infantry should defend at 7.
3. First test: Equals against equals.
I will End Turn as often as possible.
Attack each Indian city with the equivalent number of bombers as they have infantry.
Record results. In the (frequent) cases where you do not win (say, 3 infantry remain), attack the remainder with the equivalent number of bombers.
Make sure that your bombers do indeed attack at 7.
I will make sure my infantry do indeed defend at 7 (Settings > My Settings > Battle Speed > Very Slow). That means I will observe at least 1 7 in defending replays, and nothing higher.
Never capture a city.
Make sure you are attacking Infantry (not any other unit).

4. Repeat until you have 100 battles recorded, and tell me to ally end.
---
Next experiment will be similar, except it will be exactly 10x the bombers as infantry attacked. 100 battles.
Third experiment will be the same as 2nd, except 20x bombers.

Kirjutas zombieyeti, 24.03.2014 at 09:39

^^^^^^^

It seems to me that you guys are trying to address two questions in one experiment:
1) how rolls are calculated
2) whether large numbers of troops influence rolls

Perhaps it would be better (and easier) to first address question #1. I seem to remember reading somewhere that rolls were NOT equal probability of obtaining results within the range of the attack/defense rating (i.e. equal chance of 1,2,3,4,5,6 for attack 6), but rather were somehow the compounded result of two rolls (higher chance of 3 and 4 - lower chance of 1 and 6). Removing criticals from experiment #1 could be useful to facilitate interpretation of results (by using militia for example).