Combat results

[Verrucosus]

I'm afraid I don't even know what a "state search" is. Could you explain, please?

Since I cannot do any testing at the moment (no access to SMAC), my idea was to look for a theoretical explanation for the oddity in MariOne's test results that seem to indicate that a victorious attacker is more likely to leave the battlefield with an even number than an uneven number of remaining hitpoints. I cannot imagine how a variable modifier could explain this, but don't know what the calculation would look like when thousands of distinct event sequences have to be added up.

I agree that the calculation is simpler when the units have less hitpoints to begin with, particularly when the weaker unit has just 1 hitpoint left. In that case we know that to win it must win all rounds and that to lose it must lose the last. However, these special features that would allow us to do the calculations more easily, could also cause us to miss the "bigger picture" as MariOne put it. That's a valid objection that I cannot refute as long as I can do the calculations only for battles involving heavily damaged units.

As far as testing is concerned, it would be interesting to use your 94-units save-file to repeat the Impact Rover vs Synthmetal Garrison test (the one I did 100 times at your suggestion) on a larger scale.

[Vev]

State searching, is searching all the states the health of both units could be. Probably use a recursive depth-first search.

My concern is amount of raw data. During my repeats of 94 (which I will explain that number) I have found the values varies +/-6 from the mean. So mean is 83 then the ranges are around 77 to 88. Keep that in mind. As for 94 units, that is because I used Tiny map and ran out of space to put units in.

I think I need to do tests with hits 2-1, 3-1, 4-1, 5-1, all at fixed combat strengths to see what sort of pattern emerging.

[Vev]

Noticed that my previous data could be flawed due to something I did not take into account. Rest assured this is correct.

Huge Map with 500 units vs 500 units
5 vs 4 with both fission
1398 wins in 1750 trials giving 79.88%

The reason why it is 1750 not 2000 is because occassionally for some reason, some units refused to fight. The F7 security nexus would show this discrepancy of equal number of extra units on both sides. Just deduct them from the total. Or if you are lazy like I was, just drop the batch of 250 completely and redo another round.

This is still a long way from my goal of calculating odds with varying hits on both units.

[Verrucosus]

Happy New Year, Everyone!

I think I came up with a scenario to test the hypothesis of a variable combat modifier reflecting the combatants' hitpoint ratio.
Among the models we have been discussing, the variable modifier is unique in the sense that it is relevant even when the combatants have exactly the same strength at the start of the combat.
In order to make it as simple as possible, let's consider a combat between two scout patrols (both damaged 80%) in flat terrain.

If there is no hidden combat modifier (Sikander) the probabilities are as follows:
2:0 victory for attacker: 0.5 * 0.5 = 0.25
2:1 victory for attacker: 0.5 * 0.5 * 0.5 * 2 = 0.25
1:2 defeat for attacker: 0.5 * 0.5 * 0.5 * 2 = 0.25
0:2 defeat for attacker: 0.5 * 0.5 = 0.25
(The probabilities for three-round-sequences are doubled because there are two possible sequences leading to the same result.)

The same probabilities apply if there is a fixed modifier (MariOne) as in our scenario, there is no stronger unit at the start of combat.

With a variable modifier, the odds change as a result of the first round's outcome. Because the winning unit has now twice as many hitpoints as the loser, its chance of winning the second round is 67 % while the loser's chance is reduced to 33 %. The probabilities for the different event sequences are as follows:
2:0 victory for attacker: 0.5 * 0.67 = 0.33
2:1 victory for attacker: 0.5 * 0.33 * 0.5 * 2 = 0.17
1:2 defeat for attacker: 0.5 * 0.33 * 0.5 * 2 = 0.17
0:2 defeat for attacker: 0.5 * 0.67 = 0.33

If there is a variable modifier the chances of straight victory are twice as high as those of a three-round-victory while they are the same if there is no modifier or a fixed one that does not come into play for units of equal strength. This is a clear difference that lends itself to being tested.

My tests consisted of 300 combats that were resolved "manually", i.e. by reloading the same scenario time and time again. The results were as follows:
2:0 victory for attacker: 69 (23 %)
2:1 victory for attacker: 80 (27 %)
1:2 defeat for attacker: 73 (24 %)
0:2 defeat for attacker: 78 (26 %)

As you can see there is not even a hint of straight wins being twice as likely as three-round-victories.

I'd be obliged if you could comment on this. Unless there are any flaws in my calculations and, as always, subject to different data from more comprehensive tests, I believe we have to abandon the idea of a variable modifier reflecting hitpoint strength.

With best wishes for the coming year,
Verrucosus

[Vev]

Yay you're back. I kinda stopped working since I work better with feedback. Because this problem is so annoying. I respect your patience in repeating these tests.

Those are strange results. Hopefully they will reveal more light. Meanwhile, my savefile doesn't work so well if two units with different reactor size fights. I have to manually engage fighting.

Here's a tip if you haven't figured already, Ctrl-X to zoom out all the way helps reduce animation.

[Verrucosus]

Ah, "patience" is a nice way of putting it. I suppose there are more fulfilling ways to spend half an hour than to press the exact same keys 300 times.

Anyway, I'm afraid my patience for testing depends on a specific hypothesis to verify or disprove. This is where I am at a loss at the moment. There is still MariOne's suggestion that there is a fixed combat bonus for the stronger unit, but so far we have no theory about how such a modifer should be calculated. MariOne's test results indicate that the attacker enjoys an additional bonus of 50 % in the 2 v 1 combat irrespective of the defender's reactor strength, but the bonus was lower (30 % and 20 %) in the two other combat situations he tested.

Unless we can come up with an explanation for this that can be transformed into a formula for calculating the elusive modifier, all we can do is collect data to test Sikander's model without having a specific alternative in mind, but to me that is a far less appealing task.

Verrucosus

[Vev]

Yeah, at the moment, I am just cranking so much data, it is more like statistics than figuring out the magic equation.

[Verrucosus]

Maybe it is time for a bit of comparative combat theory. SMAC has borrowed a lot of its game mechanics from Civilization II. As can be seen from the Combat thread in the Civ2 Strategy Forum (easily accessible through the Great Library Index), modern Civologists have encountered similar problems.

What is presented under the heading "On to the real calculation ..." is too puzzling for me at the moment, so I'd like to stick to the simplified formula suggested in that thread. It is as follows:

Odds = (S + (S - W)) * Shp * Sfp / (W * Whp * Wfp)

where:
S = Stronger unit's modified attack or defense value
W = Weaker unit's modified attack or defense value
Shp = Stronger unit's hit points
Sfp = Stronger unit's firepower
Whp = Weaker unit's hit points
Wfp = Weaker unit's firepower

Now, we know that the firepower (i.e. the amount of damage inflicted on the loser of a round of combat) in SMAC is always "1". The hitpoint factor seems to correspond to the hitpoint modifier that we had been discussing earlier, so there isn't anything new in that respect. What is intriguing however, is that the difference between the combatants' attack strengths is added to the stronger unit's strength. I didn't have the time to check yet, but I have a distinct feeling that such a modification might provide an explanation for MariOne's test results.

[Vev]

Nice find.
My brain is too frazzled to think properly.

[Verrucosus]

Let's have another look at MariOne's test results. The calculations and data are taken from his "SMAC combat test.xls" file.

1st scenario: 2 v 1r2
According to the manual, the odds of winning a single round of combat should be 2 : 1. MariOne's test suggests that they are about 3 : 1. This is exactly the result we get if we increase the stronger unit's strength (2) by the difference between the two strengths (1).

2nd scenario: 2 v 1
see above

3rd scenario: hard2 v El1r2
I'm not sure what is happening here because I don't know what "El" represents. But if the predicted odds are presented in column D of MariOne's sheet, they are 3 : 2. Adding the strength difference to the stronger unit's strength we get 4 : 2 or 2 : 1 which is more or less the result of MariOne's test.

4th scenario: 1.25 v 1
This could be a veteran scout patrol attacking a green scout patrol. Predicted odds are 5 : 4, test results indicate odds of 3 : 2 which again reflects a bonus corresponding to the strength difference.

So the formula

Odds = S + (S - W) / W

Where
S = stronger unit's modified strength
W = weaker unit's modified strength

would explain MariOne's test results.

The bonus is applied to the stronger unit rather than the attacker because that is what the Civ2 researchers have found out, but of course we'd have to check with tests where the defender is stronger.
The two other factors in the "simplified" Civ2 formula(hitpoints, firepower) should be disregarded for the time being, because previous testing here indicates that hitpoints are irrelevant to the odds of winning a single round and because there is no firepower statistic in SMAC. I believe the simple modification that is left provides us with a hypothesis worth exploring.

However, as we are looking for the real formula, I'd appreciate comments on the relevant section in the Civ2 guide. I do understand what is said about calculating battle odds, but the two formulas for calculating the chance of either the attacker or defender winning a single round are still puzzling to me.