Big stacks, better than normal-sized stacks?

Napisao Cpt.Magic, 25.03.2014 at 17:13

Napisao zombieyeti, 25.03.2014 at 17:10

While no 'roll' in this game is going to be 'random' (it is instead pseudorandom), I am curious:
- Who is 'we', and upon what are your beliefs based?
- How are battles resolved in the nonrandom universe?
- Do you have additional experimental data to share?

70 units with 1/1 vs 10 units with 1/1. no critical, no strategy, no city bonus.




Now 60 vs 20, same unit:



Edited: Same units, but now with 5 critical:

only 1 out of the 7 battles had a extra -1



Same units, but with 7 critical, now 3 out of 7 had extra damage





Here is the mind fuck:

Please note:
- A critical of '7' means that the unit has a 7% chance of doing an additional full amount of damage (whatever their attack or defense is) per round of combat.
- Combat is resolved on a unit-by-unit basis.

(Edited after you've explained your battle results) Awesome experiment.

- The 1/1 0 crit, 0 buff units are good to demonstrate that battle mechanics work as expected when 'dice' are not involved.

- If I am reading your results correctly, when crit. is involved, but normal damage is still restricted to '1' where the only additional damage comes from 'crit' there is still no demonstrable bias towards the player/large stack. Is that correct?
== in the first example (5 crit) there are at least 119 rounds of combat, and once, the player seems to score a critical.
== in the second example there are at least 117 rounds of combat, and three times, the player seems to score a critical.

- Finally, If I am reading the third set of data correctly, of the 6 battles, where one should expect the player to lose 60 units, on the average, the player loses only 35, interestingly, at least in this case, a 42% bias in favor of the player and/or stack of 5x larger than defenders.
My data, also a limited-but-larger set (124 data points) demonstrates a 38% bias towards the player and/or very large stack (15x) of attackers. I wouldn't expect such a strong possible correlation (about a 40% bias towards the player/large stack) in the small datasets.

I can't validate your results because I can't repeat your experiment.
I hope someone will!

Napisao zombieyeti, 25.03.2014 at 17:44

Napisao Cpt.Magic, 25.03.2014 at 17:13

Napisao zombieyeti, 25.03.2014 at 17:10

While no 'roll' in this game is going to be 'random' (it is instead pseudorandom), I am curious:
- Who is 'we', and upon what are your beliefs based?
- How are battles resolved in the nonrandom universe?
- Do you have additional experimental data to share?

70 units with 1/1 vs 10 units with 1/1. no critical, no strategy, no city bonus.




Now 60 vs 20, same unit:



Edited: Same units, but now with 5 critical:

only 1 out of the 7 battles had a extra -1



Same units, but with 7 critical, now 3 out of 7 had extra damage





Here is the mind fuck:

Please note:
- A critical of '7' means that the unit has a 7% chance of doing an additional full amount of damage (whatever their attack or defense is) per round of combat.
- Combat is resolved on a unit-by-unit basis.

I see you have what appear to be combat result summaries, but I don't understand what your findings are, what conclusions you're making based on your data, and what your data is.
Also, if I read your experiment correctly, both units do an attack/defense of 1/1?

the first battle are (70) units of 1 attack and 1 defense, attacking (10) units of 1 attack and 1 defense. [No Critical, Strategy, Upgrade or Defense Bonus involved]
the 3rd battle is the same units but with 5 critical, the 4th battle is the same units but with 7 critical.

the 5th battle are (70) units of 7 attack and 7 defense, attacking (10) units of 7 attack and 7 defense. [No Critical, Strategy, Upgrade or Defense Bonus involved]

This shows that attackers have a tendency to roll better rolls.

Napisao ifinishlast, 25.03.2014 at 18:13

This shows that attackers have a tendency to roll better rolls.

It's a small dataset, it may show absolutely nothing.
It may also demonstrate that players have an advantage over neutrals, that custom maps/units are different/the same as standard units etc. though the 1/1 unit with no critical was an interesting hack - but every unit that has an opportunity to attack or defend will always do at least 1 damage/round of battle (according to the FAQ).
Edit:
It may also demonstrate exactly the tendency described.

Napisao zombieyeti, 25.03.2014 at 15:37

I'll respond.
Please check the battle replays (settings > my settings > battle speed > Very Slow).

Battle replays are supposed to be a 'roll by roll' recounting of a battle. The most interesting (and we are not there yet) are when 2 players attack a third, or even more complex.
You get to see who's attacking, who's defending and how much damage each does.

One unit attacks at a time.

There is always the possibility that the Battle Replays are nonsense, composed after a battle result, to 'justify' the results, but when you watch the Battle Replay, it all makes sense. If everyone were forced to watch a Battle Replay after losing a battle they think they shouldn't they'd realize that they had an epic run of bad luck, and the victor had good luck.

Thanks for responding. Now I understand! Referring me to the FAQ was useless as it does not mention that units fight one at a time...
So yeah, I've run a few battles on slow. It does look very much as if units fight one at a time. However, when attacking with large stacks, my experience suggests I tend to suffer much fewer losses. Either:
1) I am mistaken
2) The battle information on slow/very slow is false (as you suggested)
3) There is some sort of bonus conferred by large stacks. Perhaps the bonus takes the form of a penalty to the defender; the high number of "1" rolls by defenders suggests that this might be the case.

I have been playing for few months now and I noticed it is true when you have bigger stacks you get less casualties then the enemy.

Napisao Mercedes-Benz, 26.03.2014 at 03:14

I have been playing for few months now and I noticed it is true when you have bigger stacks you get less casualties then the enemy.

Contact me if you're interested in conducting or participating in a controlled experiment.

Seeking an Method of the Application of Bias in AW
Can anyone think of an algorithm that would apply the bias Capt.Magic and I have quantified?

Intro
Our limited, but independently-reached data indicates about a 40% bias in favor of either the player and/or a 6-15x numerical/cumulative attacker damage advantage (large stacks).

Discussion
I am interested in dice-rolls, because I was building a Battle Simulator, in semi-cooperation with AlexMeza. The Player/Attacker/Large Stack model worked well in forecasting results. Neutral/Defender/Small stack did not.

Cpt. Magic's (CM) experiments have detected a 42% bias in favor of either the player, and/or a 6x numerical/power advantage. In my experiments, a 38% bias in favor of the same properties of the attacking units was also detected. I will approximate this to be 40%.

Consider the dataset from before - the Defender's Damage rolls. Defend of 6, all critical results (2) removed.
Actual Results: 124 data points, 2.16 average damage, 268 cumulative damage.
Expected Results: (for 126 data points), 3.5 average damage, 441 cumulative damage. Any given number appears 21 times, with a 16.7% chance of any number appearing.

1: 56, 45% (translation "1 was rolled 56 times and appeared about 45% of the time)
2: 33, 27%
3: 12, 10%
4: 10, 8%
5: 8, 6%
6: 5, 4%

Can anyone come up with a model that might explain these results?
The model would have '1' result about 45% of the time, '2' about 27% of the time etc.

It isn't the easy way
-The 'easy' way to apply the 40% bias would be to permit the fair damage roll, apply a 40% penalty to that roll, round the result, if lower than 1, round to 1, and 'publish' the rounded, penalized result in the battle results.
- This method would result in no 6s or 5s. 4s would be rare, 3s and 2s would appear at the same rate.

It also isn't on 'the bottom end'
- Make 40% of the rolls 1s, then make the rest of the rolls 'fair'.
- 2s are still over-represented and 3-6s are still under-represented.

I've messed around with exponents/logarithms and square roots with no luck so far.

The model would have to permit all of the natural numbers from 1 to the maximum damage to be possible results, yet still yield about a 40% disadvantage to the cumulative damage, from expected.

Notes
- CM's data represents at least 60 rounds of combat, and a minimum of 120 dice rolls, likely more, so the 'dataset' is likely as large as the 124 datapoints I am recycling, but I can't drill down into the underlying data. It isn't less valid and isn't less usefull, just less useful for the purposes of this discussion.
- CM's data demonstrated bias only in the case where the normal damage roll for the defender/neutral/small stack occurs. In the cases where no bias could be applied (1/1 attack/defend, no critical) actual results and expected results meshed; and when critical was added, the results weren't skewed beyond expectation. Only when normal damage rolls (7/7) applied was bias detected.
- Completely unknown if the 40% bias remains at 40% if different power of units are used/magnitude of 'largeness' is changed.
- I stopped conducting experiments and research in this area about a month ago because observation of the AW universe isn't like the IRL universe. The AW universe can change at any time, and all the work we do may then be invalidated (for continued usefulness). Because of the renewed interest in the subject, I'll continue - I just want everyone involved to be warned. If you invest effort, don't expect that your findings will be universally applicable for all time. If you apply our findings in your battles, they may not always work.

There is no Bias favoring Player in Player vs. Neutral
I just wanted to eliminate or accept this proposition, to focus on the 'big stack bias' proposition.

Summary:
- Player selected 'no strategy'. Player attacked neutral cities with bombers (6/6) defended by infantry (4/6).
- Player attacked neutral cities with the equivalent number of bombers, as defending infantry.
- When infantry survived, Player would attack the survivors in matching numbers (e.g. 2 surviving infantry would be attacked by 2 bombers).
- 36 battles were fought. To eliminate 105 defending neutral infantry in 11 cities required 123 player bombers.
- Each round of battle was observed. Neither attacker or defender 'rolled' below a 1, or above a 6 during the normal damage phase.
- More player bombers were required to defeat a smaller quantity of infantry. While there might be a claim of bias towards neutrals (defending infantry in cities), a '7' was never observed. Attacker required more units than defender, because when the attacker won a battle, there were no more defenders in that city. When the defender won a battle, the attackers would need to attack again.

Napisao zombieyeti, 26.03.2014 at 05:48

- Completely unknown if the 40% bias remains at 40% if different power of units are used/magnitude of 'largeness' is changed.

This.
I suspect the defender's roll is impacted by the ratio of attackers : defenders. If 10:1 gives a 40% bias, you would expect a lower bias for 5:1 and a greater one for 20:1 (unless the bias maxes out).

CM's dataset, while small, should be sufficient to validate the bias. It is way outside the normal distribution you'd expect and I suspect a statistics test would give us a confidence above 99% if we compared to a predicted (no bias) simulation.

UREKA

what if the roll dice is like risk? and the person with more units has extra dices? that would explain why the person with more units always gets better rolls and criticals.


You see, because when the units had 1 attack and 1 defense, the is no dice as the attack/defense will always be 1

Napisao Grimm, 26.03.2014 at 11:51

Napisao zombieyeti, 26.03.2014 at 05:48

- Completely unknown if the 40% bias remains at 40% if different power of units are used/magnitude of 'largeness' is changed.

This.
I suspect the defender's roll is impacted by the ratio of attackers : defenders. If 10:1 gives a 40% bias, you would expect a lower bias for 5:1 and a greater one for 20:1 (unless the bias maxes out).

CM's dataset, while small, should be sufficient to validate the bias. It is way outside the normal distribution you'd expect and I suspect a statistics test would give us a confidence above 99% if we compared to a predicted (no bias) simulation.

My original data (looking at rolls, not kills) was a (minimum) 15 to 1 advantage, and a 38% bias seemed to be experienced with rolls.
CM's data (looking at kills) was a 7 to 1 advantage, and a 42% advantage seems to have been conferred.

I don't know when the advantage begins, maxes out, if it increases or decreases, if it is numerically based, or a function of (qty x damage). Current data suggests no difference in 7-15 to 1.

Napisao Cpt.Magic, 26.03.2014 at 17:00

UREKA

what if the roll dice is like risk? and the person with more units has extra dices? that would explain why the person with more units always gets better rolls and criticals.


You see, because when the units had 1 attack and 1 defense, the is no dice as the attack/defense will always be 1

Well, the advantage conferred doesn't seem to give the attacker any additional damage when you see the replay, right?
- My initial experiment also seems to indicate that the big stack attacker doesn't do more damage than we'd expect - the 'bias' is instead the lower damage rolls by the small stack defender.
- In your experiment (7 to 1) we don't know if it was attacker bonus damage, or defender weaker damage, we only know that 42% more defenders lost, than attackers, but there's no evidence to challenge the tendency for 1s and 2s to appear well out of expected proportion when the small stack defends.

Napisao zombieyeti, 26.03.2014 at 17:14

Napisao Cpt.Magic, 26.03.2014 at 17:00

UREKA

what if the roll dice is like risk? and the person with more units has extra dices? that would explain why the person with more units always gets better rolls and criticals.


You see, because when the units had 1 attack and 1 defense, the is no dice as the attack/defense will always be 1

Well, the advantage conferred doesn't seem to give the attacker any additional damage when you see the replay, right?
- My initial experiment also seems to indicate that the big stack attacker doesn't do more damage than we'd expect - the 'bias' is instead the lower damage rolls by the small stack defender.
- In your experiment (7 to 1) we don't know if it was attacker bonus damage, or defender weaker damage, we only know that 42% more defenders lost, than attackers, but there's no evidence to challenge the tendency for 1s and 2s to appear well out of expected proportion when the small stack defends.

It could be that "extra dice" are rolled for the defender and that the lowest roll is used.

Napisao Grimm, 26.03.2014 at 17:16

Napisao zombieyeti, 26.03.2014 at 17:14

Napisao Cpt.Magic, 26.03.2014 at 17:00

UREKA

what if the roll dice is like risk? and the person with more units has extra dices? that would explain why the person with more units always gets better rolls and criticals.


You see, because when the units had 1 attack and 1 defense, the is no dice as the attack/defense will always be 1

Well, the advantage conferred doesn't seem to give the attacker any additional damage when you see the replay, right?
- My initial experiment also seems to indicate that the big stack attacker doesn't do more damage than we'd expect - the 'bias' is instead the lower damage rolls by the small stack defender.
- In your experiment (7 to 1) we don't know if it was attacker bonus damage, or defender weaker damage, we only know that 42% more defenders lost, than attackers, but there's no evidence to challenge the tendency for 1s and 2s to appear well out of expected proportion when the small stack defends.

It could be that "extra dice" are rolled for the defender and that the lowest roll is used.

It could be a lot of things.
When casting 2 fair dice, the chances at least one will be a '1' is about 30.6% (11 in 36).
When casting 2 fair dice, the chances that the lowest number will be 2 is about 25% (9 in 36)
When casting 2 fair dice, the chances that the lowest number will be 3 is about 19% (7 in 36)
When casting 2 fair dice, the chances that the lowest number will be 4 is about 14% (5 in 36)
When casting 2 fair dice, the chances that the lowest number will be 5 is about 8% (3 in 36)
When casting 2 fair dice, the chances both will be a '6' is about 2.7% (1 in 36)

The results for 1-6:
1: 56, 45% (translation "1 was rolled 56 times and appeared about 45% of the time)
2: 33, 27%
3: 12, 10%
4: 10, 8%
5: 8, 6%
6: 5, 4%

The hypothesis you and Capt. Magic provide has zero grounding in Battle Mechanics-as-understood but strong explanatory power (evaluating the expected rolls of two fair dice, and picking the lowest of the two) - results are not far off.
It needs to be tested.

Hypothesis:
When faced with an attacker with at least a 7x numerical/attack damage advantage, the 'big stack' advantage is conferred on the attacker as the defender, when determining normal damage, rolls two dice, with the lower value being the damage inflicted.

Background:
The 'bias' towards large-stack attacking players seems to convey about a 40% advantage to the attacker in the form of less damage inflicted by the defender. Capt. Magic (CM) used defenders with a defense of 7, and ZombieYeti (ZY) used defenders with a defense of 6.

The advantage of about 40% may not be 40% at all, but instead whatever advantage would be conveyed if the defender, facing a superior force, accepted the lower of two 'rolls' for the normal defense damage.

A damage of 1 conveys no advantage to the attacker, as experimentally observed by CM
If the damage were 2, the advantage would be only about 17%
- The attacker's average normal damage would be: 1.5
- The defender's average normal damage would be: 1.25

If the damage were 4, the advantage would be about 25%
- The attacker's average normal damage would be 2.5
- The defender's average normal damage would be 1.875
4's would occur 1/16 times, 6.25%
3's, 3/16 times, 18.75%
2's, 5/16 times, 31.25%
1's, 7/16 times, 43.75%

Defending Militia should do an average of 1.875 damage, and attacking GW Militia should do an average of 2.58 including critical, conferring an attacker advantage of about 27%

Proposed Test:
- Set up a casual game, strategy GW, 50k budget. Ensure no buffs are applied to Militia which would change attack. GW is chosen as defending neutral militia have a defense of 4 and a crit of 0. Attacking Infantry have an attack of 4 and a crit of 5. Attacking GW militia have an attack of 4 and a crit of 2 - so GW militia are the best, not the perfect, choice.
- Choose a nation near concentrations of militia. You will attack only militia, and with only militia.
- Build a stack of at least 50 militia, and when attacking, attack with no fewer than 7x and no more than 15x the enemy stack.

Option 1: More attacks, less replay. This is easier to do.
- Record the number of militia units you attack with, the number of defending neutral militias, and how many militia you lose. Perform at least 20 attacks.

Option 2: Fewer attacks, more replay-rewinding, This will give us more accurate data.
- Record the rolls made by the defender. Perform exactly 126 measurements.

Grimm and Capt.Magic, go fnck yourselves lol. Wild-ass guess *appears* to fit experimental results.
I'll write it up as soon as I can, but the defending militia (I chose Option 2) did 232 damage in 124 rounds of combat, an average of ... 1.8709677419. The 'lowest of two dice' forecast an average of 1.875. If the dice were 'fair' the average should be 2.5.

Napisao zombieyeti, 26.03.2014 at 19:18

Grimm and Capt.Magic, go fnck yourselves lol. Wild-ass guess *appears* to fit experimental results.
I'll write it up as soon as I can, but the defending militia (I chose Option 2) did 232 damage in 124 rounds of combat, an average of ... 1.8709677419. The 'lowest of two dice' forecast an average of 1.875. If the dice were 'fair' the average should be 2.5.

I did seem to recall some mention of multiple dice. So maybe this is what it was. It's interesting that this seems to fit the observations. If this is true, I wonder what prompted the designers to set it up this way... It's the opposite of Risk Maybe it's just easier to code this way for some reason?

I'd also be curious to see whether the # of dice rolled is a function of the attackers : defenders ratio. Or if, as zombi suggests, it's simply a bonus dice added once you reach something around 7:1.

Method for Approximating Big Stack Advantage

Abstract:
For large-stack vs. small stack battles, when faced with an attacker with at least a 7x numerical/attack damage advantage, the 'big stack' advantage is conferred on the attacker, as the defender, when determining normal damage, rolls two dice, with the lower value being the damage inflicted.

Intro:
Large Stack Bias (apparent advantage when attacker has a superiority of 7-15x the quantity of defender units) has been experimentally observed. Previous experiments, where 'small stack' defenders had a defense of either 6 or 7 resulted in either defender dice rolls being 38% less than expected (6) or defender suffered 42% more casualties than expected (7).

In related experiments, Cpt. Magic (CM) and ZombieYeti (ZY) demonstrated that the large stack advantage springs from a defender disadvantage in the normal damage roll phase of a round of battle, and that the advantage is tied to large stacks. The observed advantage of ~40% could not be modeled in a manner aligned with experimental results; specifically, defending damage wasn't 'taxed' 40% across the board, at times, the maximum defending damage would be incurred, but at a rate much less frequent than 'classical' battle mechanics would forecast. ZY observed that the bias of ~40% may only apply to units defending at 6-7, and CM and Grimm (G) conjectured that the method used to calculate defending 'small stack' damage was that instead of one die of 1 to defense, two dice of the same range were rolled, the lower of the two being the Normal Damage Roll.

Methods:
1) Previous results were evaluated against this new model
Previously recorded data:
124 rolls, average 2.16.
1: 56, 45% (translation "1 was rolled 56 times and appeared about 45% of the time)
2: 33, 27%
3: 12, 10%
4: 10, 8%
5: 8, 6%
6: 5, 4%

New Model
Average 2.53
1: 31%
2: 25%
3: 19%
4: 14%
5: 8%
6: 3%

Units with a defense of 4, according to classic battle mechanics, would have an average damage of 2.5. Under the New Model would have an average normal damage of 1.875.
1: 43.75%
2: 31.25%
3: 18.75%
4: 6.25%
Militia were chosen as the new Defense Target, GW Militia as the attacking unit. Neutral Militia have a Critical of 0, GW Militia a Critical of 2. The player will attack each city with between 7 and 15 times the number of defending militia.

Results:
In 124 rounds of combat, 'New Model' defending militia are expected to do 232.5 cumulative damage, and an average of 1.875 damage per round. The actual results were 232 cumulative damage, and 1.871 damage per round.

Attackers lost 28 Militia for 39 Defender Militia killed, attackers killed approximately 28% more defending militia than defending militia killed attackers.

[Edit - number of times rolled, % table added]
1: 61 49.2%
2: 32 25.8%
3: 17 13.8%
4: 14 11.2%

Discussion:
The results for the New Model were suspiciously, eerily, close to the forecast, in terms of cumulative damage and damage per round. In terms of overall enemy killed vs. friendlies lost, the New Model forecast varied from the actual result by about 10%. When comparing individual dice rolls, New Model vs. Results, there is some deviation, but well within the expected margin for error.

Conclusions:
0. They Hypothesis is validated.
In cases where the attacker has a numerical superiority of 7-15x the defender, the defender's normal damage is the lesser of two rolls, each virtual dice in a range of 1 to defense.

1. Besides the forecasting value the New Model has over Classic Battle Mechanics for the small stack damage component, the implementation of 'two dice' is easy to accomplish programmatically.

2. At least one more set of experiments needs to be run, hopefully with units that defend at 8+.
Ideally, attacking units will have the same attack as the defense of the defending units.

3. A new round of experiments should be arranged to establish at what point 'normal combat' becomes 'big stack vs small stack'

4. Independent reproduction of the results described here should be performed, especially because the results are so close to the Model forecast.

5. Besides independent reproduction of the results, player v. player, and pvpvp mechanics should be studied.

Napisao Grimm, 26.03.2014 at 20:19

Napisao zombieyeti, 26.03.2014 at 19:18

Grimm and Capt.Magic, go fnck yourselves lol. Wild-ass guess *appears* to fit experimental results.
I'll write it up as soon as I can, but the defending militia (I chose Option 2) did 232 damage in 124 rounds of combat, an average of ... 1.8709677419. The 'lowest of two dice' forecast an average of 1.875. If the dice were 'fair' the average should be 2.5.

I did seem to recall some mention of multiple dice. So maybe this is what it was. It's interesting that this seems to fit the observations. If this is true, I wonder what prompted the designers to set it up this way... It's the opposite of Risk Maybe it's just easier to code this way for some reason?

I'd also be curious to see whether the # of dice rolled is a function of the attackers : defenders ratio. Or if, as zombi suggests, it's simply a bonus dice added once you reach something around 7:1.

sauce please on multiple dice, and I too am curious where a battle becomes a fat stax vs slim change proposition.

It seems like this proves the attacker attacks first

Napisao Cthulhu, 27.03.2014 at 01:08

It seems like this proves the attacker attacks first

I am hopeful that someone will want to participate in pvp and pvpvp experiments so we can determine unit battle order.
Under the most recent understanding (thank you, Old One) if 2+ players attack each other, both will take turns attacking and defending.
- Battles are still resolved one combat at a time, between two units.
- In the + case (or where two players attack a neutral city) it *appears* that the 'sides' fighting are selected randomly.
- As to who is attacker and defender, in any particular round of battle, it *appears* that neutrals will always be defending, player units that didn't attack the other/didn't move will always be defending. Player units in cities *might* always be defending (but TB priority may apply here).

Call for Verification
I can't test this, I don't have premium. I would if I could, so we can get this settled and move foward.

If the 'small stack defends with the worst of two dice' damage rule is correct, then in 7-15x combat, attackers and defenders with 10 damage, no buffs and no crit ....

Average Normal Attacker Damage: 5.5
Average Normal Defender Damage: 3.85
Attacker Advantage: 30%
Attack defender with between 7-15x the number of units. Record defender dice rolls.

If someone runs 100 battles, the distribution of defender rolls should be something like the below. At 100 battles, the occurrence is the same as the % forecast.

1 19
2 17
3 15
4 13
5 11
6 9
7 7
8 5
9 3
10 1

EDIT: For a number of reasons, the dice roll method is the only rigorous method for measuring.
Next EDIT: Will someone with premium please perform this test?

Thanks in Advance!

Hi it's me.
First of all, I wouldn't want to read 1-4 pages ;_; (considering most of them are huge text walls) so, what is the dices theory thing? Also, correct me if I am wrong below:
- The initial idea is correct. Stacks get better results if they are big against significatively smaller stacks. Sometimes the differences are huge (-2 -8, for example) and sometimes they are not, Id say, it is still about 30-50% difference in average. Didn't do statistics, just an aproximation. The main reason to this, may be the high amount of 1 damage rolls by the smaller stack.
- It does no matter who defends nor attacks.
- Neutrals seem not to have anything to do with this; player vs player gives same results.
- The issue has something to do with the rolls "luck" (specially with the 1 rolls), not an actual mistake when taking down HP/units amount. (A test of 1 att/def was made by CM, every result was the same, with or without big stacks.

Also, we should consider that ARB was replaced by CRIT before, so maybe some ARB code is still there. ARB were important when deciding rolls. At the moment, we can't take any "theory" like this as a fact, but we could discuss if the "theory" can be correct or not. (Note the "can be" part, we can not discuss wether if it is true or not, but we can discuss if it CAN BE true or not)

I will next make a test; 10 tanks (with 6 att) plus 80 infantries (4 att) against 10 infantries (6 def), to see if the issue happens when a stack is big or when one unit type is big.

Napisao Guest, 31.03.2014 at 18:40

Hi it's me.
First of all, I wouldn't want to read 1-4 pages ;_; (considering most of them are huge text walls) so, what is the dices theory thing? Also, correct me if I am wrong below:
- The initial idea is correct. Stacks get better results if they are big against significatively smaller stacks. Sometimes the differences are huge (-2 -8, for example) and sometimes they are not, Id say, it is still about 30-50% difference in average. Didn't do statistics, just an aproximation. The main reason to this, may be the high amount of 1 damage rolls by the smaller stack.
- It does no matter who defends nor attacks.
- Neutrals seem not to have anything to do with this; player vs player gives same results.
- The issue has something to do with the rolls "luck" (specially with the 1 rolls), not an actual mistake when taking down HP/units amount. (A test of 1 att/def was made by CM, every result was the same, with or without big stacks.

Also, we should consider that ARB was replaced by CRIT before, so maybe some ARB code is still there. ARB were important when deciding rolls. At the moment, we can't take any "theory" like this as a fact, but we could discuss if the "theory" can be correct or not. (Note the "can be" part, we can not discuss wether if it is true or not, but we can discuss if it CAN BE true or not)

I will next make a test; 10 tanks (with 6 att) plus 80 infantries (4 att) against 10 infantries (6 def), to see if the issue happens when a stack is big or when one unit type is big.

Dude, seriously?
If you can't be bothered to read, I certainly can't be bothered to write.

Napisao zombieyeti, 01.04.2014 at 16:21

Napisao Guest, 31.03.2014 at 18:40

text

Dude, seriously?
If you can't be bothered to read, I certainly can't be bothered to write.

I actually read pages 2-3 but didn't find it ;_;

Napisao Guest, 31.03.2014 at 18:40

Hi it's me.
First of all, I wouldn't want to read 1-4 pages ;_; (considering most of them are huge text walls) so, what is the dices theory thing? Also, correct me if I am wrong below:
- The initial idea is correct. Stacks get better results if they are big against significatively smaller stacks. Sometimes the differences are huge (-2 -8, for example) and sometimes they are not, Id say, it is still about 30-50% difference in average. Didn't do statistics, just an aproximation. The main reason to this, may be the high amount of 1 damage rolls by the smaller stack.
- It does no matter who defends nor attacks.
- Neutrals seem not to have anything to do with this; player vs player gives same results.
- The issue has something to do with the rolls "luck" (specially with the 1 rolls), not an actual mistake when taking down HP/units amount. (A test of 1 att/def was made by CM, every result was the same, with or without big stacks.

Also, we should consider that ARB was replaced by CRIT before, so maybe some ARB code is still there. ARB were important when deciding rolls. At the moment, we can't take any "theory" like this as a fact, but we could discuss if the "theory" can be correct or not. (Note the "can be" part, we can not discuss wether if it is true or not, but we can discuss if it CAN BE true or not)

I will next make a test; 10 tanks (with 6 att) plus 80 infantries (4 att) against 10 infantries (6 def), to see if the issue happens when a stack is big or when one unit type is big.

Ok, old friend. This is a summary:
- *Large stacks do have a bias in their favor*. At 7-15 to one numerical attacker advantage over defender, the defender's rolls are about 40% less than one would expect. If their defense is 6, 6s will still be rolled, but the sum of all dice rolls seems to be about 40% less.
- *The bias is for large stacks, not player v. neutral*. 124 dice rolls recorded seems to indicate that in an even to even battle, defending neutrals do as the 'classic model' would forecast.
- *Recording dice rolls is the best way to measure these biases*. Counting enemy losses alone incurs a greater margin for error. Record all critical hits, just remove the maximum damage from the damage, so the underlying 'roll' is still accounted for (e.g. if defense is 6 and damage is 10, roll was 4).
- *The large stack advantage seems to modify defender normal damage roll*. Cpt. Magic (CM) demonstrated that units with 1 damage do not suffer a bias on damage rolls.
- (unpublished) *Small stacks seem to start being affected at 2-1* Unpublished preliminary data indicates that the defender's rolls get affected at 2-1, worsen at 3-1, improve slightly at 4-1.

The mechanism for defender small stack damage being nerfed is not well understood.
A theory put forth by CM and Grimm that defenders rolled 2 dice for damage, instead of one, with the lower of the two being counted happened to well explain the damage rolls for units defending at both 6 and 4, which is a significant finding.
However, the results for 2-1, 3-1, 4-1 are not consistent with the 'worse of two dice' theory.

Preliminary Results 2v1, 3v1, 4v1
124 dice rolls were recorded for units with a damage of 6 where the attacker numerical advantage was 2v1, 3v1, 4v1.
The results were inconsistent with each other, and possibly inconsistent with the 'lower of two dice' (>2d) damage theory.
*This is preliminary, raw data* and it is possible that all findings are consistent with the >2d damage theory.

Conditions: All attacking units were unbuffed bombers with an attack of 6, all defending units were neutral infantry.
- If defending units defended at 'classical' rates, one would expect an average total normal damage of 434, average normal damage roll of 3.5

- If defending units defended at >2d rates, one would expect an average total normal damage of about 265, average normal damage roll of about 2.53

2v1
350 total damage, average damage roll of 2.822.

1 rolled 35 times, 28.22%
2 26, 20.96%
3 23, 18.54%
4 18, 14.51%
5 11, 8.87%
6 10, 8.06%

3v1
286 total damage, average damage roll of 2.306.

1 48, 38.71%
2 24, 24.42%
3 18, 14.52%
4 10, 8.06%
5 9, 7.26%
6 4, 3.23%

4v1
309 total damage, average damage roll of 2.491.

1 49, 39.52%
2 27, 21.77%
3 13, 10.48%
4 16, 12.9%
5 11, 8.87%
6 8m 6.45%

*I have not yet run the stochastic analysis (100k simulated battles) on these results to determine how frequently each result might come up in the normal course of battle, whether under 'classic' or >2d damage models.*

It is entirely possible that at 2v1, 'classic' damage rules apply, and the observations I have were merely on the low end of expectations.
At 3v1 and 4v1 it is entirely possible that >2d damage models apply. 3v1 results seem to be near expectations, and 4v1 results may merely be on the high end of expectations.

Also possible:
- A different mechanism than >2d applies to the special case of 2v1.
- The >2d model is the wrong approach. Its consistency with defenders with damage of 4 and 6 is a fluke.

I am more likely to start the research on the verification if someone runs the '10 damage' experiment above.
*I cannot*.

Assessment of 2v1-4v1 results
All results were using units defending at 6, and are the results of 124 rolls.
Under 'classic' rules the expected average cumulative damage is 434, and the average roll is 3.5
Under 'new model' rules (>2d) the expected average cumulative damage is about 314, and the average roll is 2.53

2v1
350 total damage, average damage roll of 2.822.
In 10,000 'classic' simulations of 124 rolls, never was 350 or less reached. If 2v1 battles use classic rules, the results obtained are <1 in 10,000. In 100,000 simulations the result was achieved just once, .001%
In 10,000 'new model' simulations of 124 rolls, 350 or more was reached 95 times, so about 1% of the time.
(please note: these results 'seem' crazy, because 350 is only 10% more than 314, and only 19% less than 434, but 434 and 314 themselves represent 124 'average' results)


3v1
286 total damage, average damage roll of 2.306.
In 10,000 'new model' simulations of 124 rolls, 286 or less was reached 491 times, so about 49% of the time.

4v1
309 total damage, average damage roll of 2.491.
Not run (see results for 3v1) as the results were in line with expectations.

So,
With a few experiments compared to simulations, even at 3v1 levels, we see that defenders suffer a definite loss, and the >2d 'new model' isn't contradicted.

At 2v1, we can say the results are very unlikely to be evidence for the 'classic' model. There is a possibility that the 'new model' applies to 2v1 battles - about 1% of simulated battles have the same results as my observation.

Conclusions
- The 'new model' has explanatory value for 3v1 and larger results.
- The 'new model' may or may not apply to 2v1 results.
- The 'classic' rules almost certainly do not apply to 2v2 results.

Either:
- The 'new model' is a false model that matches our small set of results, and a different set of rules applies -or-
- There is a different set of rules governing the battles when the ratios of attackers to defenders exceed 1v1 and are less than 3v1 -or-
- The 'new model' applies even in 2v1 battles, but the results observed were anomalous.

Recommendations:
- *someone else* run the 10/10 attack/defense experiment above, at a ratio of 10v1 so we can see if the new model forecast applies.
- *someone else* re-run the 2v1 battle results to better understand if the results I observed are anomalous.

Until I get more external assistance, I'm not likely to publish any more findings, except in IB cln forums.
Cpt. Magic, Cthulu, Alex, due to your contributions and leadership, I'll keep you posted on major findings.

Anyone wanting updates will need to join IB, or start contributing. PM me if interested.