16-10-2003 14:19									#1
	The Probabilities of the War: The Evidence Against GCA Avatar Enlargement: We've got the solution

EARLIER COMBATS:

5/5 Horse killed 4/4 NM on flat land (35.5%)
5/5 Horse, fired on by cat, killed 4/4 NM on hills (15.1%)

VERSION 1:
5/5 Horse killed 3/3 Sword on mountain (37.1%), ends up 4/5 (2.9%)
Overall Chance = 2.9%

VERSION 2:
5/5 Horse dies attacking 3/3 sword on mountain, bringing it down to 2/3 (68.7%), no Apolytonian promotion (51.5%)
5/5 Horse killed 1/3 Sword on mountain, ending up 4/5 (30.8%)
Overall Chance = 15.9%

VERSION 3:
5/5 Horse dies attacking 3/3 sword on mountain, bringing it down to 1/3 (89%), no Apolytonian promotion (66.8%)
5/5 Horse dies attacking 2/3 sword on mountain, bringing it down to 2/3 (89%), no Apolytonian promotion (66.8%)
5/5 Horse killed 1/3 Sword on mountain, ending up 4/5 (30.8%)
Overall Chance = 13.7%

5/5 Horse killed 3/4 sword on grassland (73.3%), ends up 3/5 (27.8%)
Spawns a Great Leader (6.25% chance) once out of four elite victories during the war (25%)

---------------------

BATTLE OF ALGALORD:

VERSION A: The most likely in my belief
ARMY, fired on by cat, killed 4/4 NM (~90%)
5/5 Sword, fired on by cat, killed 4/4 NM (49.6%)
4/4 Sword, fired on by cat, killed 4/4 NM (35.3%)
5/5 Horse, fired on by cat, retreated 1/5 when 4/4 NM was down to 2/4 (51.4%, 66% chance retreat = 33.9%)
5/5 Horse, fired on by cat, retreated 1/5 when 3/3 NM was down to 2/3 (77.7%, 66% chance retreat = 51.3%)
5/5 Horse killed 4/4 MI (58.3%)
4/4 Horse killed 4/4 MI (44.8%)
4/4 Horse killed 4/4 MI (44.8%)
4/4 Horse killed 4/4 MI (44.8%)
Overall Chance = 0.144% (1 in 696)

VERSION B: 1 extra 5/5 Horseman
ARMY, fired on by cat, killed 4/4 NM (~90%)
5/5 Sword, fired on by cat, killed 4/4 NM (49.6%)
5/5 Horse, fired on by cat, retreated 1/5 when 4/4 NM was down to 3/4 (77.7%, 66% chance retreat = 51.3%)
5/5 Horse, fired on by cat, retreated 1/5 when 4/4 NM was down to 2/4 (51.4%, 66% chance retreat = 33.9%)
5/5 Horse, fired on by cat, retreated 1/5 when 3/3 NM was down to 2/3 (77.7%, 66% chance retreat = 51.3%)
4/4 Sword killed 3/4 NM (61.1%)
5/5 Horse killed 4/4 MI (58.3%)
4/4 Horse killed 4/4 MI (44.8%)
4/4 Horse killed 4/4 MI (44.8%)
4/4 Horse killed 4/4 MI (44.8%)
Overall Chance = 0.128% (1 in 784)

VERSION C: 2 extra 5/5 Horsemen
ARMY, fired on by cat, killed 4/4 NM (~90%)
5/5 Horse, fired on by cat, retreated 1/5 when 4/4 NM was down to 3/4 (77.7%, 66% chance retreat = 51.3%)
5/5 Horse, fired on by cat, retreated 1/5 when 4/4 NM was down to 3/4 (77.7%, 66% chance retreat = 51.3%)
5/5 Horse, fired on by cat, retreated 1/5 when 4/4 NM was down to 2/4 (51.4%, 66% chance retreat = 33.9%)
5/5 Horse, fired on by cat, retreated 1/5 when 3/3 NM was down to 2/3 (77.7%, 66% chance retreat = 51.3%)
5/5 Sword killed 3/4 NM (73.3%)
4/4 Sword killed 3/4 NM (61.1%)
5/5 Horse killed 4/4 MI (58.3%)
4/4 Horse killed 4/4 MI (44.8%)
4/4 Horse killed 4/4 MI (44.8%)
4/4 Horse killed 4/4 MI (44.8%)
Overall Chance = 0.097% (1 in 1034)

As you can see, adding even extra elite horsemen to "soak up catapult damage" actually makes the outcome of us not getting a Golden Age LESS likely, since it assumes that all of these extra horsemen ALL managed to retreat rather than even a single one dieing at the hands of an Numidian Mercenary.

Adding some additional 4/4 horsemen to die against our MIs is impossible, since if an MI were damaged, we would still have it and the next horseman would have attacked a healthy MI.

So, actually, every additional unit that was NOT killed by a Numidian Mercenary makes the liklihood that GCA victories were caused by random chance alone LESS likely rather than more! (since the chance that each additional unit successfully retreated reduces overall probabilty more than additional units causing damage to our NMs improves it)

Therefore, I'll use the conservative estimate that GCA only used the 9 offensive units they already had and no more. Two of those units successfully retreated and the rest successfully killed our units.

GCA's luck during their turn was so good that there's only a 1 in 696 chance that it could have been that good or better, meaning the other 695 out of 696 possibilies were ALL better for Apolyton.

----------------------

RESULTS FOR ENTIRE WAR SO FAR (APOLYTON-GCA WAR ONLY)

VERSION 1A:
3.88 x 10^-8
(1 in 25,772,309)
1.55 x 10^-7
(1 in 6,443,077)

VERSION 2A:
2.13 x 10^-7
(1 in 4,700,610)
8.51 x 10^10-7
(1 in 1,175,153)

VERSION 3A:
1.83 x 10^-7
(1 in 5,455,452)
7.33 x 10^-7
(1 in 1,363,863)

Personally, I believe the 1 in 6.4 million scenario because it fits the pattern - GCA would then have never lost a unit on offense, having only lost units when defending and thus unable to control the outcome. If we believe that even they would have lost a unit to our fortified swordsman on the iron mountain, the odds improve to about 1 in 1.2 million, but I think that if they cheated on the rest, there's no reason they didn't cheat to storm the iron mountain with a single horseman rather than multiple units.

I believe we have sufficient evidence to approach Mongoose and request that he use his admin password to access GCA's save and test with their turn whether (without RNG-changing gimicks) he can obtain the victories of this turn. I would be additionally interested in him testing the 110 B.C. turn in which we lost 2 swordsman in the south and NM(s) in the north to see if he can replicate GCA's results without RNG gimmicks.

If Mongoose can replicate their results without playing with the RNG, then we might accept the 1 in 1.2 million or 1 in 6.4 million chance that GCA's luck is this good. I'm betting, however, that those odds are so extreme that anyone will have a hard time reproducing GCA's results... afterall... given random probability, 695 times out of 696, Mongoose should be unable to do as well or better than GCA did just this turn. Those are pretty good odds. If GCA hit the jackpot (literally!), I'm willing to let them win the "you killed 10 Apolyton units" lottery.

That said, as much as I'm willing to be convinced by evidence that it was, I don't currently believe this is a case of GCA hitting the jackpot on the lottery...

Last edited by Arnelos on 16-10-2003 at 14:47

	16-10-2003 14:21									#2
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I just realized that I should have averaged the chance of spawning the great leader out across all elite victories in the war...

Calculating that now... it will only depress the numbers a little

	16-10-2003 14:28									#3
	Increase the size of your Attachments

Edit complete. Probabilities are now 1 in ~1.2 million and 1 in ~6.4 million

	16-10-2003 14:32									#4
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(nevermind.... I thought I made an error with that GL recalculation, but it's right when I double-checked it manually)

Last edited by Arnelos on 16-10-2003 at 14:37

	16-10-2003 14:44									#5
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EDIT: spelling, grammar.

	16-10-2003 14:57									#6
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The least reliable number above is the army. The chance that it beat the 4/4 NM by random chance if it was at FULL HEALTH is 97.7%. That said, I don't believe the army was at full health when it attacked.

I assumed 9/13 or 10/13 hp for its attack to give ~90%, which is probably too low. Does anyone have a better idea of what it is so I can provide a better figure for the army victory?

Minor issue, reallly. If we assume the army was at full health, overall probilities change only to 1 in ~1.1 million and 1 in ~5.9 million. It's somewhere in between those if the army was less damaged when it attacked.

	16-10-2003 14:58									#7
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Good job Arnelos.

Shall we take this to Mongoose, or do we want to wait for a second CFC investigation before we present our case?

	16-10-2003 15:50									#8
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BigFree brought up a good point. It's possible (though it runs in the face of everything else that's happened), that GCA attacked our MIs and lost so badly that we did not suffer a single HP hit while their unit died or a horseman retreated in the process.

I'm assuming that a team which tries to cheat in order to maximize the damage inflicted by their units and seems to never lose a unit on offense would never do this, but I'm willing to humor the possibility for the sake of argument (because it's a good point, just in case).

The chance of a veteran swordsman dying against a veteran MI without causing any damage to the MI is 3.2%

The chance of a veteran horseman either dying OR retreating at 1 HP against a veteran MI without causing any damage to the MI is 12.06%

Now, if you assume, for a moment, that it's possible that GCA could have scored ALL victories against our NMs without losing a SINGLE unit and then scored an amazing series of losses against our MI's to make up for it without our MIs losing a SINGLE hitpoint (other than the 4 killed), you come to something as equally unlikely as GCA winning their battles and losing none. Even if you average their losses and victories together to receive a washed result there, the bifurcation of results between NM battles and MI battles (not to mention killing MIs or MIs not losing a single hitpoint) would be so extreme that it's probably also in the ballpark of 1 in 1,000 for the turn. Either way, the result of the turn is absolute absurd.

To track down the other potential complaint (our victory in OUR attack later the same turn), I'm going to calculate that as well for a comparison.

	16-10-2003 16:28									#9
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Quick question, and it only matters a little for my opinion: How long has GCA been taking with the save? Particularly last turn, but also on average during the war.

	16-10-2003 17:14									#10
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I don't think it matters. They only had to cheat 7 battles; with the above-mentioned techniques to alter the RNG, it shouldn't take suspiciously long to cheat these battles if the player is experienced, takes notes while he works and does it fast enough.

	16-10-2003 17:21									#11
	Support Apolyton, buy Galactic Civilizations: Deluxe Edition

IIRC, if a horse retreats from a NM, it counts as a victory for the NM and we would be in a GA. Since we are NOT in a GA, then they did not retreat from the NM. They had to damage them another way...

	16-10-2003 17:31									#12
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quote: Originally posted by donegeal IIRC, if a horse retreats from a NM, it counts as a victory for the NM and we would be in a GA. Since we are NOT in a GA, then they did not retreat from the NM. They had to damage them another way...

I don't believe that's true, I think a retreating horse will not cause a GA. but then again, I rarely use defensive UU's.

	16-10-2003 17:33									#13
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Do you think Mongoose will actually believe this information? I don't. He checked out the immortal info, but thought GCA was just very lucky. He's already rigged this game. Why would he become impartial now? I don't trust him as a neutral administrator.

	16-10-2003 17:41									#14
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I'll go check. Hold on....

Test showed that a horse retreating from a NM did not trigger a GA, however, I am by no means a modder expert and may have screwed something up.

Multiplayer hotseat. Carthage and India. Elite horse vs Vet NM. NM won with horse retreating. Hit enter to end India's turn and went to Carthage. No GA. However, it might have been skewed because it was Carthages first turn.

Last edited by Donegeal on 16-10-2003 at 17:59

	16-10-2003 18:38									#15
	Avatar Enlargement: We've got the solution

From my experience, retreats don't count as victories in all aspects (e.g. you don't get promotions for holding off but not killing fast units, and you can't get GLs).

	16-10-2003 19:34									#16
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Retreats are not a "victory" for the defender, and thus do not trigger a GA.

Neither, incidently, does killing a scout or capturing workers count (I was playing nasty with the Aztecs the other day, and it occurred to me that Jags are great for whacking scouts and stealing workers. No GA).

-Arrian

	16-10-2003 19:35									#17
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Oh, and if my opinion isn't clear: I think they are cheating their asses off.

-Arrian

	16-10-2003 20:05									#18
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Arnelos, there are some problems with your analysis:

quote: 5/5 Horse, fired on by cat, retreated 1/5 when 4/4 NM was down to 2/4 (51.4%, 66% chance retreat = 33.9%)

1. You're assuming that all our Catapults scored hits.

2. Where does the '2/4' figure come from? It's a pure guess. You cannot use just this guess to calculation a joint probability, especially when calcuations down the line depend upon your assumption; you must exhaustively consider each possibility (1/4, 2/4, 3/4 in this case) to get an accurate result.

3. You're assuming it's more probable for the Elite Horsemen to attack then retreat, and therefore you're not considering the possibility that they actually won their respective combats. Elite Horsemen still have a 35% chance of winning a combat against Vet. Mercs. Again, this is a case that does not appear in your analysis and therefore skews your joint.

4. Although it may seem that this is just a straightforward "multiply them all together" problem because the trials are independent, it's actually better to model it as a conditional probability problem. Either this, or put in a factor to your joint (which you've forgotten).

All these factors (the last one is a more general comment, mind you) argue against your result that GCA pulled off the extremely improbable. I'm not saying I'm convinced GCA is playing legit, I'm just trying to get our calcuations down straight.


Dominae

	16-10-2003 20:18									#19
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quote: Originally posted by Dominae Arnelos, there are some problems with your analysis: 1. You're assuming that all our Catapults scored hits.

No, I'm not. Chance of the catapult hitting a horse (defense 1) is 78.4%. Chance of the cat hitting a sword is 64.5%. That is factored into the percentage value posted.

	16-10-2003 20:23									#20
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quote: Originally posted by Dominae 2. Where does the '2/4' figure come from? It's a pure guess. You cannot use just this guess to calculation a joint probability, especially when calcuations down the line depend upon your assumption; you must exhaustively consider each possibility (1/4, 2/4, 3/4 in this case) to get an accurate result.

I honestly don't know just what you're getting at with this comment. Do you mean the suggested other scenarios that may be possible? Yes, I chose to add units that I do not believe to actually exist in order to illustrate what might happen if they do exist. I fully recognize (and believe) that such additional units actually do not exist and were not involved. Others, however, have suggested their possible existence.

As for knocking units down to 1/4, 2/4, 3/4, this was also chosen to illustrated possible examples of other things, not what I believe actually took place. Given the trend in the only 2 cases I looked at, I didn't feel it was necessary to look at every single possibility, which would undoubtedly take me hours figure every single probabiliy of possible GCA forces and possible hits by possible GCA forces... taking a few examples that illustrate the point should be sufficient.

	16-10-2003 20:28									#21
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quote: Originally posted by Dominae 3. You're assuming it's more probable for the Elite Horsemen to attack then retreat, and therefore you're not considering the possibility that they actually won their respective combats. Elite Horsemen still have a 35% chance of winning a combat against Vet. Mercs. Again, this is a case that does not appear in your analysis and therefore skews your joint.

I'm not assuming that it's more probable, Dominae, I know that that is the only possibility other than what I believe took place. Only so many units were killed and we did NOT get a GA. There are a very limited number of possibilities that could have lead to that. If more GCA horsemen had had VICTORIES rather than retreats, we'd have less units. Since we know how many units we lost, we already know how many victories GCA won. The only question is how many and which GCA units were involved in those GCA victories and what additional units may have been sacrificed or retreated to contribute to them.

I fail to see your point, at all, about me not considering ADDITONAL GCA victories... GCA had 7 victories this past turn... we know this because we have 7 dead units. Is there something I'm missing?

As for my supposed assumption that Elite horses would not attack NMs, the given possible scenario where they DO is the one I believe is most likely, as is stated in my post. Again, I'm failing to see what you're getting at here.

	16-10-2003 20:37									#22
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quote: Originally posted by Dominae 4. Although it may seem that this is just a straightforward "multiply them all together" problem because the trials are independent, it's actually better to model it as a conditional probability problem. Either this, or put in a factor to your joint (which you've forgotten).

I know how to do this, but it's really not worth the trouble. Under the case I believe is correct (leading to a chance of 1 in 1.5 million), GCA did not lose a single unit. As such, it is easy to calculate the chances of them winning every single combat (with some adjustment for the cases of retreat, which I did) and just multiply them together. Where it would certainly be possible that you could churn this through a larger probability system (which I've done just now with the results for our attack on Algalord on our own turn because it's NECESSARY with those results), it is entirely unecessary. The reason is that if you make some adjustments for retreat cases, you can calculate all of the factors together as chances of GCA success and find the total chance that GCA would meet with such overwhelming success. Had I considered strict win/loss of combats, this would not be possible, but instead I am considering the % chance that GCA would "succeed" in each case to the extent they did.

So I understand your concern and normally I'd agree, but this is a case where it's really not needed as far as I can see... it's like flipping a quarter 20 times and scoring heads every single time... it's a LOT easier to calculate this way if you're blessed with an opporunity for doing so.

	16-10-2003 20:40									#23
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quote: Originally posted by Dominae All these factors (the last one is a more general comment, mind you) argue against your result that GCA pulled off the extremely improbable. I'm not saying I'm convinced GCA is playing legit, I'm just trying to get our calcuations down straight.

I'm not clear on what you're objecting to in a number of these, which is probably just that I'm not reading what you wrote the way you intended it. As for the last one, I think I understand what you're getting at and I just would rather not go to the trouble unless it's actually necessary.

	16-10-2003 20:49									#24
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Our attack on GCA...

quote: Originally posted by notyoueither 5 cats bombard. The 4th takes down the walls, all others miss. mi steelglove attacks and kills army *mi Scourge attacks and kills spear *mi Slash attacks spear and dies. 2 pips on spear *mi Theseus attacks and kills spear *mi E_t attacks and kills spear *mi Arnelos attacks and kills spear mi wervdon attacks and kills spear mi g attacks and kills spear mi e attacks spear and dies. 2 pips on spear ho Mingapulco attacks and kills spear ho VonHorseman attacks 3/5 spear and dies. 2 more pips on spear.

1 catapult hit the walls while 4 catapults missed everything (18.25% success each)
MI kills the 7/13 army (33.4% success)
MIs with 4 victories with only 1 defeat against fortified 4-hp Spears (70.4% success each)
MIs with 2 victories with only 1 defeats against fortified 3-hp spears (81.6% success each)
Horsemen with 1 victory, 1 defeat against fortified 3-hp spears (50.9% success each)

Overall Chance of Apolyton being this lucky or better:
44.02%
(1 in ~2.27)

In other words, Apolyton was about as lucky as we should have been.

PLEASE NOTE: I know I don't have in the catapult equation the chances of hitting an improvement versus hitting civilians or a military unit. For the last catapult shot (made after the walls came down), I believethis introduces some very minimal error I'd really rather not iron out If someone wants to recalculate based upon minisculely different numbers for that last catapult shot (assuming the patch of PTW we're working under still behaves in this fashion), be my guest, but I really don't think it'd be worth it at this point The probability that we'd do as well as we would be as lucky as were is ~44%.

	16-10-2003 21:06									#25
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quote: Originally posted by Arnelos So I understand your concern and normally I'd agree, but this is a case where it's really not needed as far as I can see... it's like flipping a quarter 20 times and scoring heads every single time... it's a LOT easier to calculate this way if you're blessed with an opporunity for doing so.

How do you know it's not necessary until you calculate it out? The coin-flipping analogy is a good one: what are the chances of getting exactly a HHTTHTT result out of 7 coin flips? 0.5^7, which is pretty low. But then is there anything particular about this sequence that should get us worked up? Not really.

Here's an alternative way of approaching the analysis you just did: calculate the odds of GCA's combat results assuming all results were in their favor (for example, our Catapults never hit, their Horsemen took our NMs down to 1HP before retreating, etc.); in other words, determine how probable the worst-case (for us...best-case for them) scenario is. If this scenario is still highly improbable, we have good grounds to believe they're fiddling with their combat orders. Without knowing the specifics, we should not draw any conclusions (I'm not convinced as you are that you can fudge the analysis and still come up with something reasonable).

I would feel far more comfortable accepting such an analysis that the one you just did (which is only a calcuation of one non-special path through the combat). If we can "prove" that their feat was highly improbable even under the best circumstances, then we'll have them with their pants down (so to speak).


Dominae

	16-10-2003 21:50									#26
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This just makes me depressed.

There comes a point where probability approaches truth. I can no longer believe that they are just lucky. The complaint and the case against GCA needs to be made. What worries me is, "what next?"

Either we take horrible and improbable losses and just have faith that our industry will overtake them (if we ever get a GA...)

Or we formally bow out after the powers that be agree with our conclusions. Our results and analysis would be posted, and backed up by CFC & CDG. A new game might be started with C3C. Maybe.

I dunno. There really cannot be any happy ending to this. We ultimately just have to do what we feel is right.

--Togas

	16-10-2003 21:59									#27
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quote: Originally posted by Dominae How do you know it's not necessary until you calculate it out? The coin-flipping analogy is a good one: what are the chances of getting exactly a HHTTHTT result out of 7 coin flips? 0.5^7, which is pretty low. But then is there anything particular about this sequence that should get us worked up? Not really. Here's an alternative way of approaching the analysis you just did: calculate the odds of GCA's combat results assuming all results were in their favor (for example, our Catapults never hit, their Horsemen took our NMs down to 1HP before retreating, etc.); in other words, determine how probable the worst-case (for us...best-case for them) scenario is. If this scenario is still highly improbable, we have good grounds to believe they're fiddling with their combat orders. Without knowing the specifics, we should not draw any conclusions (I'm not convinced as you are that you can fudge the analysis and still come up with something reasonable). I would feel far more comfortable accepting such an analysis that the one you just did (which is only a calcuation of one non-special path through the combat). If we can "prove" that their feat was highly improbable even under the best circumstances, then we'll have them with their pants down (so to speak). Dominae

The point here is that I'm not actually out to definitiavely proove that they cheated. I just need enough for the preliminary hearing, as it were. The goal here is to present at least enough of credible evidence that Mongoose will check their ACTUAL save to see what they ACTUALLY did. Me spending the additional time it would take to nail down all of the alternative possibilities seems wholly unecessary if Mongoose can just check what really happened himself and definitiavely answer, one way or the other, whether they cheated.

If we really need the extra work because this isn't good enough for Mongoose, then we can go to the trouble, but I just don't want to spend the TIME if this is enough for preliminary hearing

	16-10-2003 22:02									#28
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Speaking of presenting what we have to Mongoose, I would very much like to get approval to approach Mongoose with this.

While we're at it, anyone yet have information from CFC or CDG?

Also, If anyone else (Dominae?) would like to compute these numbers with a different method, you're more than welcome

I just want to get this to Mongoose and start the ball rolling on that end of things already.

	16-10-2003 22:20									#29
	Inflate your Upload Space

The alternative I proposed is just as time-consuming as what you did, and would IMO be more informative.

I'll try to this analysis myself when I return tonight.

(By the way, Arnelos, I'm not downplaying your contribuation here. All I'm saying is that some people are getting all worked up because of your "one in a million" result, which is not entirely accurate...it could be off by at least a few factors. Like I said, calculating the odds of a particular sequence of combats is not the same as calculating the joint probability of the evidence we've seen. My opinion is that your "credible" evidence is misleading.)


Dominae

	16-10-2003 22:32									#30
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Not having computed, or checked anything, I do have one question to ask: How do you exactly alter the RNG? I thought this game had preserve seed on, meaning that they can try different combinations of fights, to get the best sequence, but they can only improve this much.

Let's assume the RNG sequence is something like .85 .54 .74 .32 .01 .(1 meaning success, 0 failure). a certain battle can have a 40% chance of winning, meaning that if the number generated for that given battle is larger then .60, it will be win the battle. If it's lower, it wil lose. That battle can be shifted by reloading, until it reaches positions 1 or 3, as there that given battle will succeed. In the other positions, no matter how many times you try, it will lose.
And, in the fifth position, every battle will lose, except those that have higher then 99% chance of succeeding.

I know this is a simplification, in reality each pip is such a calculation, and so the numbers can be tweaked and shifted using different promotion levels of attackers, but the thruth is that there aren't that many combinations to try. Once in a while, even with reloading, they have to give up a unit which will be lost. Now, they can reload so that such a unit is lost against a MI instead of against an NM, but even with reloading, I don't see a possibility for them to get 7 kills and 2 damages done without losing a battle.

Except... when they just are plain lucky.

I don't want to suggest your numbers are off, but I guess that if you account for some random losses in between (losses, or retreats), you will get more possibilities, and the huge chance will lower. Not that it means they aren't cheating, but for me, without knowing exactly what units they lost, this probability does not border certainty. Yet.

If you do the same calculation for our last battle, not taking into account the two units we lost, don't you get similar results?

DeepO