Part 29: Mechanics: The true Civ 2 combat systemTM;DR
So after all that math, what was accomplished? Well I derived 4 formulae which tell us everything there is to know about the probability of an attacker winning a battle against a defender assuming we know their stats. Formula One is the odds of the attacker winning a round if Defense is greater than or equal to Attack. Formula Two is the odds of the attacker winning a round if Attack is greater than Defense. Then take the result P from Formula 1 or 2 and plug it into Formula Three and you can figure out the probability V that the attacker is victorious in the whole battle. Or with Formula Four I also figured out the amount of damage an attacker is likely to inflict before dying to a superior defender, which is critical for figuring out how many attackers you need to overwhelm the foe.
In all formulae, Attack refers to the post-modifier Attack of the attacker. Defense refers to the post-modifier Defense of the defender. Y is the number of rounds the attacker must win to kill the defender (Defender's HP / Attacker's Firepower). Z is the number of rounds the defender must win to kill the attacker (Attacker's HP / Defender's Firepower).
Probability that attacker wins a round of combat (P) = ( ( 8 x Attack) - 1 ) / ( 16 x Defense )
Probability that attacker wins a round of combat (P) = 1 - ( ( 8 x Defense) + 1 ) / ( 16 x Attack ) )
Probability that attacker wins the whole battle (V) = 1 - [ Sum over k from k = 0 to k = Y -1 of ( (k + Z - 1)! / ( k! x (Z - 1)! ) ) x ( P ^ k ) x ( ( 1 - P ) ^ Z ) ]
Average damage attacker will deal before dying (D) = (Z x P x Attacker's Firepower) / (1 - P ).
So now you can calculate your exact odds of winning a fight in Civ 2! And it will only take 10 minutes or so per fight if you have this page, paper, and a calculator handy! Figuring out roughly whether you have the numbers to overwhelm tough defenders is marginally easier and quicker at least.
To experiment around with possible stats and combat outcomes efficiently, I set up a little Excel spreadsheet that can instantly provide battle outcome %s given stats. Here it is on Google Sheets: https://docs.google.com/spreadsheet...dit?usp=sharing
Either copy/paste it into your own blank sheet or download it to get a copy you can actually edit. This is an extremely powerful and unique tool, and the first of its kind as far as I'm aware.
Comments & Implications & Interpretations:
In no particular order:
1) These formulae are extremely complicated
2) The instruction manual version is so clearly and dramatically inaccurate that it can't even be called an approximation of the truth. I conclude that the designers either deliberately lied about it or had a very poor grasp of their game's mechanics. It's not just them though; unless I misread something, the "simpler formula" on the page I linked to previously is even MORE clearly wrong. It doesn't even generate probabilities!
3) We don't have a good, simple approximation of the true formula. This makes it very hard or impossible to make informed choices about what to attack with what and therefore turns combat in Civ 2 from a strategy game to a guessing game. Even more than all the luck involved already does.
4) Even knowing the entire formula and having a program ready to crunch out the numbers -which no one in the universe did before I finished this calculator- instantly is not enough. You see, Formula 3 can only be applied if you know the current HP of the attacker and defender. But if they're injured, you don't! There is no way to find out the exact HP of an injured unit. You can make an educated guess based on what its max HP is and how full it's bar looks, but that's it. Which means that at the end of the day, no matter how complete your knowledge of math and the game mechanics, you do not have the information required to make strategic choices.
5) In fact, the information required to make strategic choices has been deliberately withheld. The designers seem to have purposefully tried to make combat in this game a black box in nearly every way they could, and then added massive amounts of luck on top of that. This is not good game design.
6) The secret "defender win ties" mechanic of who takes damage in each round is the worst thing ever. From this mechanic all weirdness with the combat system flows. I mean, intuitively you would think that 2 identical Warriors with 1 Attack, 1 Defense, 10 HP, and 1 Firepower fighting each other would have about a 50-50 shot of winning. But it's not even CLOSE to true! The attacker has roughly 25% odds to win!
7) As Attack and Defense values grow, the "defender wins ties" mechanic matters less and less. This makes mathematical sense because the "defender wins ties" rule only comes into play when the attacker and defender roll the exact same number. And rolling the same number is much less likely when you roll dice with more sides, which you do as your Attack and Defense rise. For example, assume HP and Firepower for both units in a fight stay fixed at 10 and 1 and see what happens to the odds of victory as the Attack and Defense go up together. At Attack = Defense = 1, the attacker wins the battle only about 25% of the time. At Attack = Defense = 2, the attacker wins about 40%. At 5 and 5 it's about 45%. 18 and 18 is about the highest one can normally encounter in Civ 2 and the attacker's odds to win the battle have reached 49%. Remember of course that the "defender wins ties" rule comes into effect in all battles, not just ones where Attack and Defense are equal. I brought up only Attack = Defense matchups because in those it's clear that the outcome should be 50-50, so you can see the size of the effect clearly.
8) Attack and Defense values grow over the course of the game. Therefore, the defender wins ties mechanic has the biggest effect on battle outcomes early in the game. Which makes it even WORSE than it seems. See, the early game is when the outcome of individual battles matters the most. Having combat work even less like the player is told it does at the part of the game where it matters most is the worst possible way to make things go.
9) This defender wins ties mechanic is why Warriors and Triremes are unusable in combat and part of why Settlers are invincible early and several other weird phenomena that people are sure to notice- and that the designers certainly should have in testing.
9.5) It's also part of why there's no good way to protect yourself from barbarians early. Due to their high-tech units and (secret) Attack boost, their Attack stats are already so huge at the start of the game that they roll big dice and thus the "defender wins ties" rule rarely hurts them. But if you try to hit their small Defense with your small Attack, the "defender wins ties" rule slams you hard.
10) Even if for some reason you wanted to keep the rest of the pointlessly complicated and poorly thought out combat system as is, there are several ways the stupid 'defender wins ties" rule could have been prevented from spoiling everything. For example, remember how one is always rolling a random number up to EIGHT times Attack or Defense to find out who wins a round? That eight is an entirely arbitrary choice of constant. If it was higher, the dice would have been bigger and the tie mechanic would therefore matter less. Let's look at the Warrior vs Warrior case if the 8s are replaced with 20s. Guess what? It's better than 41% odds instead of 28% that the attacker wins with nothing else changed at all. What if it's, say, 100? We go up to 49%. Changing this one arbitrary constant (while leaving the rest of the messy combat system intact) instantly solves the most terrible problem that pointlessly screws over people trying to play strategically. There are so many obvious ways one could leave the combat system basically the same but solve all problems with minor tweaks that I lost count before finishing this sentence.
11) Why the heck is it secret? All of the above stuff might have been acceptable if we were told defenders had a hidden advantage. I mean, it would still be a bad idea overall in many ways. Like it's what makes combat odds so hard to understand intuitively. And it makes Triremes unusable on offense when they're clearly meant to be capable of attacking. And it cripples player efforts to deal with barbarians even more than their secret attack boost already did, essentially making them unbeatable in the early phases unless they do something amazingly stupid. And there are many other problems. But all of that would be forgivable if the player just knew it was going on and could therefore know to plan accordingly. But no, the instruction manual formula very clearly states that combatants with equal Attack and Defense should have even odds of victory. Maybe the designers honestly didn't know? Maybe they never even took two seconds while writing out these enormously long mechanics for combat to think about how they'd actually work? It's disturbingly plausible.
12) No seriously, why is it secret? Just one sentence mentioning that defenders have a slight advantage would have been enough.
13) Fire Emblem also told some major lies and kept major secrets about the supposedly transparent combat rules. The clearest case is the "True Hit" mechanic. But critically, True Hit actually almost always benefited the player. Secretly benefiting the player is less bad than secretly undermining the player as "Defender Wins Ties" tends to do.
14) There's no good way to talk about the relative usefulness of Attack, Defense, HP, and Firepower. More is always better, but how much another point of any of them affects combat outcomes depends on the exact value of all stats of both combatants. This makes it really hard to know whether, say, a 6 Attack, 10 HP veteran Elephant attacking a 3 Defense, 20 HP Musketeer is at an advantage or not. The answer is actually yes, a huge advantage. But if it was a 4 Defense, 20 HP Rifleman on defense then suddenly the Elephants chances of winning are actually cut in half to 40%. It's the interactions of the stats with each other than make combat outcomes hard to understand and make it impossible to give a good summary of how much increasing a given stat matters in a vacuum.
15) The value of every point of advantage in Attack or Defense increases MASSIVELY as HP goes up for both combatants. So for example, a 6 Attack guy fighting a 5 Defense guy when both are badly injured with 1 HP and 1 Firepower has a modest advantage overall; 57% odds to win. Which is roughly the kind of advantage you'd think 6 attack vs 5 defense would have. But if they're both at 10 HP? That 6 Attack vs 5 Defense difference suddenly means 75% odds for the attacker to win! If they both had 40 HP, the attacker would have 90% odds to win! Heck, even the advantage of "defender wins ties" is massively magnified by higher HP. If everyone has just 1 Attack and 1 Defense and 1 firepower and also 1 HP, the attacker has 43.75% odds to win. If HP goes up to 10 for both, his chance of victory drops to just 28% or so! If HP was 40 for both, the attacker would have only about 10% odds to win despite all their stats being the same!
16) The above makes mathematical sense of course. More HP means more "trials", which means the results are more and more normalized and a few outliers where the disadvantaged guy gets lucky mean less. However, it doesn't make game balance sense. And it is the other half of why true combat odds are so hard to calculate and so counterintuitive and contrary to everything we're told.
17) HP was added to the game in the first place to solve the "Spearman beats Tank" problem. In Civ 1 there was no HP, so all combats were decided in 1 round. But the Attack and Defense stats were much the same as in Civ 2. So in Civ 1 a 2 Defense phalanx had a 10% chance to kill an attacking 10 Attack Tank. In Civ 2, even if the Tank was badly injured and thus had only 10 HP, the Tank would have 99.999993187% odds to win. Just on the basis of both units having 10 HP, not even the fact that the Tank is supposed to have 30. So clearly the addition of HP really did solve the "Spearman beats Tank" problem.
18) However, as outlined in 15, HP also turned even the smallest Attack and Defense advantages into absolutely massive differences. In Civ 1 a 3 Attack Archer attacking a 2 Defense Phalanx would have about 65% odds to win. In Civ 2, the archer has 90% odds to win. The archer is not higher tech than the Phalanx. It's not some kind of dedicated Attack monster. Yet it absolutely flattens the Phalanx, the game's first Defense specialist, if the Phalanx isn't fortified. But if the Phalanx IS fortified, well then HP inflates the defender wins ties advantage for the Phalanx into nearly 60% odds to win all of a sudden despite having the same Defense as the enemy's Attack. And City Walls! City Walls. 3 Attack Archer vs 6 defense Phalanx in City Walls in Civ 1 would have had a 25% chance to win. In Civ 2? Its less than 1%!
19) As I mentioned in my discussion of Formula 4, swarming can overcome these monstrous advantages fairly quickly. See, fascinatingly, the NUMBER of Archers needed to probably beat that walled Phalanx is 4 in both games despite the massively different odds of victory for a single one fighting. So in Civ 2, it's incredibly important to have overwhelmingly numerous and concentrated attack forces to take any kind of defended city or deal with any other foe who has even slightly more Defense than you have Attack.
20) Whether the above was good or not, different maximum HP was totally unnecessary. Just having HP in the game already "normalized" things and made it so the Spearman was never beating the Tank. Heck, it wasn't even necessary for it to be HP with people taking lasting damage, that's a separate issue. Just having combat be a race to 10 round wins would have solved the putative problem. Making it so that modern units have more HP than older units was a very, very bad idea. It didn't solve a problem, but it did massively complicate things. I don't think the designers actually understood this. Passages in the game talking about HP seem to imply that it is modern units having MORE HP than ancient units which prevents ancient units from beating them. That's not really true. Having HP at all already did that. Having more HP is the same as having more Attack or Defense except harder to understand and compare.
21) Firepower was actually counterproductive! The reason that HP served its intended purpose was that it meant combat took more rounds. This made it much less probable that a disadvantaged combatant would just get lucky and win; it would need to get lucky and win 10 times instead of just once. But Firepower cuts the number of rounds back down again! Which means it undoes some of the progress made by HP existing even when both units have the same amount of Firepower greater than 1. Once again, it makes things far more difficult to understand, different amounts of it among different units just do the same kind of thing that boosting their Attack and Defense (or HP if you insist on having different HP) would, but now it actually undermines stated design goals.
22) Also, the game can't handle multi-round combat. When lots of units are fighting on enemy turns, it freezes up and glitches and you can't see what's happening. If your game's engine can't actually handle a mechanic, maybe the mechanic shouldn't be included.
23) All these crazy complexities in the combat system don't actually encourage different or more interesting play. You still just want to have the units with the highest stats- and to have more of those units. What's different is that you don't have a clear understanding of what the highest stats necessarily are or of how much they matter.
24) The most important and useful of my 4 results is Formula 4, the one you can use to figure out how many attacking troops you need to take a city. It needs to be applied carefully though since it's just an approximation. First of all, there is no penalty for overshooting and bringing along too many guys; the extra survivors can go on to attack the next enemy. But launching an attempted conquest that fails because you run out of people will cost you hundreds or even thousands of shields of Production. The enemy will build a new defender or two and will heal up within a couple of turns, so you'll have gained nothing. So plan for worst-case scenarios. Assume all enemy troops are veterans and if you don't know their exact number, then assume it's a high number of the most problematic type available to the enemy. Double check that you've included all Defense multipliers before figuring out the numbers. And you may wish to bring one more attacker per defender than the formula says you need. Oh and don't forget that when all the defenders are dead, you need to have 1 unit (even if it's a weak one) that can walk into the empty city and claim it!
25) If you look at a lot of the combat rules, one reason they're full of weird special abilities which don't actually add much for gameplay purposes and complicate things needlessly is simulationism creeping in where it doesn't belong. For example, the instruction manual Combat section has a sub-section called "Pearl Harbor" talking about how ships attacked in port have their Firepower reduced to 1 and attackers against them have their Firepower doubled. The combat section is riddled with weird little exceptions and special abilities and boosts with no real rhyme or reason to them but always with some note about how it's supposed to represent X or Y real thing. This is just not a good idea. Civ 2 is not a realistic game in any way and it makes less sense the more you think about it. Especially unit movement and combat. Like with planes being able to stay aloft for exactly 2 years if the game is post 1850 and not a second longer, but able to fly for 50 years or more without refueling if you somehow unlock them early. It's true in reality that planes must be refueled, but the attempt to include this in the game made less than no sense. It only attracted attention to other things that made no sense, like how Tanks and ships and such DON'T need to be refueled and how it can take decades to sail a boat across a small lake because that's how long a turn is at that point in the game. Simulationism is the enemy of good game design, but it's seductive and often hard to spot while you're working.
26) Of course, many of those weird special abilities are not mentioned. Or they're described falsely. For example, the bombarding ship in shore bombardments actually takes a firepower penalty! I have no idea what the reason for that is, but it's not explained anywhere and it's a serious issue that can get your ships killed fighting weak units. Another issue is that there's no good reason why some special combat bonuses take the form of, say, doubled Defense vs missiles and others take the form of doubled Firepower or whatever.
So with all of that established, I can now state Melth's Theorem: Civilization 2 suffers from game mechanics which are overly complicated and luck-based while being insufficiently thought through and explained; this is especially true of the combat mechanics.
The proof is left as an exercise for the reader.