07-07-2005 05:00									#1
	Combat mechanics Support Apolyton, buy GURPS/ Alpha Centauri

I've performed an exhausting survey of the probabilities involved in combat in Alpha Centauri. Most important is the chance the attacker will win a single portion of combat, an exchange, where a point of damage is assigned to either the attacker or defender. Prior to these tests, I had believed that the odds the attacker would win an exchange were simply the ratio of the attacker's strength to the defender's strength (after all modifiers). From this, and the number of hit points each unit has, the odds of winning the combat can easily be calculated, and the mean damage to the expected victor is also at hand. My tests have shown that the odds of winning an exchange are definitely not a simple ratio of strengths.

Here is a plot of exchanges won / total exchanges against attacker's strength / (attacker's strength + defender's strength). The error bounds are each two sample standard deviations away from the sample mean.

Attachment: combat_graph.png
This has been downloaded 97 time(s).

	07-07-2005 05:00									#2
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The data used for that graph is as follows:

code:

Ratio         	Prob           	stddev
0.022624434	0.012040753	0.001916421
0.044247788	0.028831563	0.002915101
0.076923077	0.041916168	0.003467523
0.100000000	0.06020558	0.004076395
0.166666667	0.100346021	0.005589062
0.225739849	0.133473096	0.008990174
0.250000000	0.172699069	0.012155239
0.333333333	0.24906015	0.013258179
0.333333333	0.245994345	0.013221848
0.368421053	0.299239222	0.013313792
0.400000000	0.341176471	0.013743603
0.406779661	0.327620303	0.012050357
0.421686747	0.356759976	0.011690935
0.480000000	0.460965224	0.00663983
0.500000000	0.517477204	0.013774524
0.500000000	0.482522796	0.013774524
0.520000000	0.539034776	0.00663983
0.578313253	0.643240024	0.011690935
0.593220339	0.672379697	0.012050357
0.600000000	0.658823529	0.013743603
0.631578947	0.700760778	0.013313792
0.666666667	0.754005655	0.013221848
0.666666667	0.75093985	0.013258179
0.750000000	0.827300931	0.012155239
0.774260151	0.866526904	0.008990174
0.833333333	0.899653979	0.005589062
0.900000000	0.93979442	0.004076395
0.923076923	0.958083832	0.003467523
0.955752212	0.971168437	0.002915101
0.977375566	0.987959247	0.001916421

Last edited by Chaos Theory on 07-07-2005 at 05:17

	07-07-2005 05:09									#3
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The raw tests I used to generate this information are available in the attached zipped XLS file.

Attachment: combat_data.zip
This has been downloaded 5 time(s).

	07-07-2005 12:15									#4
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quote: raise both the attacker's and defender's strengths to the 4/3

Raise both?

x*a / y*a = x / y

	07-07-2005 12:26									#5
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quote: Originally posted by Chaos Theory I've performed an exhausting survey of the probabilities involved in combat in Alpha Centauri.

You know what they say about some analyses - it mightn't have exhausted the space to be analysed, but it sure did exhaust the analyser .

	07-07-2005 12:42									#6
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quote: Originally posted by binTravkin Raise both? x*a / y*a = x / y

(x*a) / (y*a) = x / y, yes.
x**a / y**a = x / y, no.
x*a / y*a = x / y, actually not either

	07-07-2005 13:57									#7
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Ok, I mistaked, it just seem obvious to me that one of the values need to be raised not both as it makes no diff to raise both:

(x*a) / (y*a) = x / y

	07-07-2005 15:36									#8
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(x^a)/(y^a) = (x/y)^a

	07-07-2005 16:11									#9
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Aha!

I have misread it probably.

So if odds are 2:1 then they actually are (2:1)^1.(3)



* binTravkin rushes to his Command Nexus..

	07-07-2005 17:52									#10
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Note that this is per exchange. If two healthy units with those strengths go at it, it's even more likely the stronger unit will win, and its expected damage would be (1/2)^(4/3) * 10 (assume fission reactor on defender) = 4.

	07-07-2005 23:12									#11
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somebody has way too much time on his hands this summer...

	07-07-2005 23:29									#12
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Actually, obtaining the raw data was pretty easy and fun. For some of the tests, I started an accelerated game and gave myself battle ogres and other units to destroy opposing factions. It was quite entertaining to hear Yang's and Miriam's screams of punishment sphere agony, repeatedly.

The exhausting part was making sense of the data, trying to fit it to a simple equation.

	07-07-2005 23:59									#13
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How did you keep track of the battle progress?

Draw lines for each fire-exchange ( on the side of the "winner" ) ? Then count them ?

Is the damage during each fire-exchange equal to the attack-strength (if the attack wins) or the defense-strength (if the defender wins) ?

If so, when you are attacking with a 2-1-1 against an other 2-1-1 (of the same health etc), the attacker gets (2/3)^(4/3) chances to inflict 2 points of damage, and defender gets (1/3)^(4/3) chances to inflict 1 point of damage ?

BTW: how do you get to this (odds)^(4/3) ? If you do (2/3)^(4/3) you get a number LOWER than 2/3, this should be HIGHER than 2/3 right ?

	08-07-2005 00:03									#14
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quote: Originally posted by Chaos Theory Actually, obtaining the raw data was pretty easy and fun. For some of the tests, I started an accelerated game and gave myself battle ogres and other units to destroy opposing factions. It was quite entertaining to hear Yang's and Miriam's screams of punishment sphere agony, repeatedly.

Good work (i'm happy you found it to be funny)
Your data keeps me thinking it's a very good game - it doesn't happen here what happens in civ2 where simple warrior (1-1-1) can stand against much stronger units and beat them or survive their attacks (sometimes i called those warrior units Rambos )
of course we cannot be sure of the odds of the battle, but the probability is strongly supporting us and this is what i really like in this game

quote: The exhausting part was making sense of the data, trying to fit it to a simple equation.

	08-07-2005 00:16									#15
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quote: Originally posted by PJayTycy How did you keep track of the battle progress? Draw lines for each fire-exchange ( on the side of the "winner" ) ? Then count them ? Is the damage during each fire-exchange equal to the attack-strength (if the attack wins) or the defense-strength (if the defender wins) ?

I don't understand your suggestion, but what I did was pit two units against each other, and note how much damage each took. The losing unit received damage equal to its hit points prior to the battle, and the winning unit received damage equal to its pre-battle hit points minus its post-battle hit points. Each point received by one side is a point inflicted by the other.

quote: If so, when you are attacking with a 2-1-1 against an other 2-1-1 (of the same health etc), the attacker gets (2/3)^(4/3) chances to inflict 2 points of damage, and defender gets (1/3)^(4/3) chances to inflict 1 point of damage ?

No, the ratio of hits scored by the attacker to the hits scored by the defender would be about (2/1)^4/3 ~= 2.54:1. Then, the probability the attacker scores a hit each time a hit is inflicted equals 2.54 / (2.54+1) ~= .72. Tests indicate the probability is closer to .75, but it certainly isn't the 2/(2+1) ~= .667 I would have expected.

quote: BTW: how do you get to this (odds)^(4/3) ? If you do (2/3)^(4/3) you get a number LOWER than 2/3, this should be HIGHER than 2/3 right ?

Which is why you need to use odds (such as 10:1), which can be both greater than and less than 1. For a probability, use attacker^4/3 / (attacker^4/3 + defender^4/3) for the chance the attacker will score a hit.

	08-07-2005 00:29									#16
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quote: Originally posted by binTravkin Raise both? x*a / y*a = x / y

interesting, very interesting...
just wonder what doctors says about it

	08-07-2005 12:14									#17
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quote: interesting, very interesting... just wonder what doctors says about it

Yeah, go ask your doctors.
In their place I wouldn't let you to sit in front of computer.

	08-07-2005 13:44									#18
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this math looks too complicated to me in my old age.

What's the gist of it?

	08-07-2005 15:37									#19
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quote: Originally posted by Chaos Theory Thus, to calculate the true estimated probability of winning an exchange, raise both the attacker's and defender's strengths to the 4/3, and then calculate the ratio of your side's strength to the sum of the strengths.

In even simpler terms, the odds are somewhat further skewed towards the stronger unit than the ratio of their strengths would indicate.

	08-07-2005 16:58									#20
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Are you simply trying to calculate the odds of winning a battle? I have a spreadsheet that does this. You simply input the attacker value, the defense value, the attacker's pips and the defender's pips. Would this be of any help to you?

	08-07-2005 17:34									#21
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What formula does it use? It's probably not correct, but if it is I'm very interested to see it.

	08-07-2005 18:46									#22
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So harsh.

The odds that the game reports are not correct because the combat system is not a *** for tat turn. It is simply a chance to damage someone. Let me give you an example:

Attacker Value 10
Defender Value 5
Attacker Pips 1
Defender Pips 2

The game reports something like a 1:1 chance of victory for this attack. That is not the case. The following are the outcomes of the this battle, attacker takes no damage killing defender, attacker does 1 point to defender then dies, attacker does no damage to defender and dies.

The chances are calculated using the odds for the attacker or defender damaging the other, then repeating until one is dead. (There is no mutual destruction, otherwise the odds would be 50-50.)

Chance that the attacker does 1 point of damage is 10/(10+5) = 0.6666
Chance that the defender does 1 point of damage is 5/(10+5) = 0.3333

So, the attacker wins the above scenario w/o taking damage 0.6666^2 = 4/9 = 0.44444
The defender wins taking one point of damage = 0.6666*0.3333 = 2/9 = .22222
The defender wins w/o taking damage 0.3333 = 1/3 = 0.3333

The attacker wins this battle 44.444% of the time. The math gets more involved as you have more pips on each side, but luckily Pascal developed his famous triangle to solve this.

So for the following
Attacker Value 4.5
Defender Value 3
Attacker Pips 2
Defender Pips 3

Attacker wins w/o damage = [4.5/(4.5+3)]^3 = 0.216
Attacker wins taking 1 point of damage [4.5/(4.5+3)]^3*[3/(4.5+3)]*3 = 0.2592
This final factor of three is from Pascal’s triangle, because this outcome can be achieved three ways, the attacker can:
do 1 point, do 1 point, take 1 point, do 1 point;
do 1 point, take 1 point, do 1 point, do 1 point;
take 1 point, do 1 point, do 1 point, do 1 point
(Note the sequence do 1 point, do 1 point, do 1 point, take 1 point does not exist since the defender is dead after taking 3 points)
Defender wins taking 2 points of damage [4.5/(4.5+3]^2*[3/(4.5+3)]^2*3 = 0.1728
Again this can be achieved three ways, the attacker can:
Do 1 point, do 1 point, take 1 point, take 1 point
Do 1 point, take 1 point, do 1 point, take 1 point
Take 1 point, do 1 point, do 1 point, take 1 point
(Note the sequences take 1 point, do 1 point, take 1 point, do 1 point & take 1 point, take 1 point, do 1 point, do 1 point & do 1 point, take 1 point, take 1 point, do 1 point do NOT exist because the attacker is dead after taking 2 point)
Defender wins taking 1 point of damage [4.5/(4.5+3]*[3/(4.5+3)]^2*2 = 0.192
This can be achieved two ways:
Attacker does 1 point, defender does 1 point, defender does 1 point
Defender does 1 point, attacker does 1 point, defender does 1 point
(Note the sequence defender does 1 point, defender does 1 point, attacker does 1 point does not exist because the attacker is dead after taking 2 points).
Defender wins taking no damage [3/(4.5+3)]^3 = 0.064

0.216 + 0.2592 +0.1728 + 0.192 + 0.064 = 1

Attacker wins 0.216 + 0.2592 = 0.4752
Defender wins 0.1728 + 0.192 +0.064 = 0.5248

Any questions?

	08-07-2005 18:55									#23
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I'm not trying to be harsh, merely blunt. I'm aware of the effects of multiple exchanges of damage on the odds of winning a battle.

quote: Originally posted by Deathsinger Chance that the attacker does 1 point of damage is 10/(10+5) = 0.6666 Chance that the defender does 1 point of damage is 5/(10+5) = 0.3333

This is exactly what I've shown is not correct. In this scenario, the chance the attacker does 1 point of damage is about 3/4, not 2/3, and the chance the defender does 1 point of damage is about 1/4, not 1/3. The rest of your math is sound, and until recently, I believed the probabilities were as you said, but the numbers tell a different story.

	08-07-2005 19:14									#24
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confused here what is ment by pips? Thanks

	08-07-2005 19:23									#25
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Pips is the number of health points of each unit, a fission reactor provides 10 pips, a fusion 20 pips, a quantum 30 pips, a singularity reactor 40 pips.

	08-07-2005 19:25									#26
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Ah ok thanks, how exactly did we get to call that pips as opposed to health...?

	08-07-2005 19:37									#27
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After re-reading and downloading your spreadsheet I now understand the point. I originally misunderstood your line "calculate the ratio of your side's strength to the sum of the strengths" interpreting that to mean ratioing the health (pips). What is somewhat troublesome with your results is the dips around 0.4 and 0.6 strength ratios.

	08-07-2005 19:42									#28
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I have two points near .4 (reflected to .6) with slightly different x values and different, though not statistically different, y values that cause a blip on the chart. Consider the plot to be a smeared line about as wide as the upper and lower error bounds, rather than the somewhat jagged central line.

	08-07-2005 19:49									#29
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How did you create this test in the game? Did you create two units with 1 health point each and have them fight over and over again. Then vary their attack/defense values and repeat?

	08-07-2005 19:52									#30
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I had bunches of units, from accelerated-start games and from old saved games, combined with the scenario editor. I pitted them against each other after noting each unit's strength and initial health, then looked at which unit won and with how much health, and recorded its damage as exchanges won by the defender, and the losing unit's initial health as exchanges won by the attacker. I varied both attack and defense across a wide spectrum, in the range .4375 to 45.