Camel archers and Numidian cavalry

[elitetroops]

Spread the spears across 100 cities and the same amount of spears will take no damage in the initial flanking attack, leaving the same amount of full health spears to be attacked in the second wave. A full health spear is a full health spear, doesn't matter if it has been attacked before or not.

In my experience stacks aren't that much larger on larger maps or slower speeds. Not in the timeframe you'd be using HAs. Maybe stacks get bigger later in the game but then you should probably use siege to deal with them, or at least avoid attacking massive stacks in heavily fortified hill cities at bad odds. Marathon also gives you such a massive advantage that at least on immortal or lower a huge pangaea can be conquered before that becomes a problem.

 

[dankok8]

Spread the spears across 100 cities and the same amount of spears will take no damage in the initial flanking attack, leaving the same amount of full health spears to be attacked in the second wave. A full health spear is a full health spear, doesn't matter if it has been attacked before or not.

A full health Spear is a full health Spear but it might make you overestimate the number of HA's needed to take the city if some of your second wave is actually the first wave i.e. attacking a virgin Spear. So 304 HA's for 100 Spears may not be the correct ratio in a more realistic situation.

In my experience stacks aren't that much larger on larger maps or slower speeds. Not in the timeframe you'd be using HAs. Maybe stacks get bigger later in the game but then you should probably use siege to deal with them, or at least avoid attacking massive stacks in heavily fortified hill cities at bad odds. Marathon also gives you such a massive advantage that at least on immortal or lower a huge pangaea can be conquered before that becomes a problem.

I have to disagree about stacks not being larger. I still recall a Marathon game from many years ago I played on the Earth map as Rome. In 1 AD I had 153 Praets. On Normal game speed that isn't possible.

 

[elitetroops]

You are still implying that there would be a difference if a full health spear has been attacked or not. These are individual events, whatever happened in the past is irrelevant. When a flanking HA fails to deal damage, there will be more full health spears left than flanking HAs. All of them exactly equal. It doesn't matter if the remaining flanking units attack "virgin spears" or not. Only thing that matters is that in both cases all flanking HAs are guaranteed to attack full health spears. If you spread it out over 100 cities with 3 attackers each, the only difference is that some attackers wouldn't get to attack at all while some other stacks of 3 fail to kill the spear. You wouldn't know how many units you need to take them all out until you edit in some more.

My experience with marathon is that I need less units because I can attack earlier and face weaker defenses. Haven't played the earth map but I believe it is quite a bit larger than a pangaea. Huge marathon pangaea should be over well before 1 AD.

 

[rah]

Nope, the war begins with Numidians and ends with Numidians if done right

Not on larger worlds, but I still generally agree.

 

[dankok8]

You are still implying that there would be a difference if a full health spear has been attacked or not. These are individual events, whatever happened in the past is irrelevant. When a flanking HA fails to deal damage, there will be more full health spears left than flanking HAs. All of them exactly equal. It doesn't matter if the remaining flanking units attack "virgin spears" or not. Only thing that matters is that in both cases all flanking HAs are guaranteed to attack full health spears. If you spread it out over 100 cities with 3 attackers each, the only difference is that some attackers wouldn't get to attack at all while some other stacks of 3 fail to kill the spear. You wouldn't know how many units you need to take them all out until you edit in some more.

Maybe I'm struggling to articulate what I'm trying to say. I don't think you understand me. I'm not saying it's different to attack a virgin Spear and a full health Spear that's already been attacked but didn't take damage. Of course there is no difference. However, attacking a virgin Spear has not been attacked before means that the second attacker is actually the first attacker. Thus all the numbers you got are probably slight overestimates of how many HA's are needed to kill a Spear. Honestly the combat HA's are probably even better than your numbers show in that case because some flanking HA's actually didn't even soften the defenders.

I realize that 3 attackers may fail to kill a Spear in my test proposal so maybe use 4 or 5 to ensure that every Spear in each city eventually dies and then average the results. It's very time consuming but results will be more accurate.

 

[elitetroops]

However, attacking a virgin Spear has not been attacked before means that the second attacker is actually the first attacker.

This would happen also if you spread the attack out over 100 cities. If you first attack all cities with one flanking HA each, the expected amount of full health spears the second wave has to deal with is exactly the same as if they all were in one city. Out of 100 flanking HAs, a certain percentage will fail to deal any damage to a full health spear. When one fails to deal damage, it makes no difference if the next attack targets that same spear or some other full health spear. Once the initial 100 attacks are over, the amount of remaining full health spears is the same.

 

[dankok8]

This would happen also if you spread the attack out over 100 cities. If you first attack all cities with one flanking HA each, the expected amount of full health spears the second wave has to deal with is exactly the same as if they all were in one city. Out of 100 flanking HAs, a certain percentage will fail to deal any damage to a full health spear. When one fails to deal damage, it makes no difference if the next attack targets that same spear or some other full health spear. Once the initial 100 attacks are over, the amount of remaining full health spears is the same.

It's different in that when you have 100 defenders in one city, you don't know if the first wave is 100 units. To attack all the defenders at least once, you may need more than 100 attackers.

 

[elitetroops]

This can't be that difficult.

Ok. Say you have them spread out in 100 cities. When an attacker fails to deal damage, the AI cheats and has the surviving full strength defender switch places with a defender from another city that hasn't been attacked yet, so that after the initial 100 attacks, some defenders have not been attacked yet. Can we both agree that this makes no difference, as long as a full strength defender is swapped with another full strength defender? Then realize that this is exactly what is going on when they are all in one city and some may defend multiple times in the first 100 attacks if they don't take any damage the first time.

 

[dankok8]

This can't be that difficult.

Ok. Say you have them spread out in 100 cities. When an attacker fails to deal damage, the AI cheats and has the surviving full strength defender switch places with a defender from another city that hasn't been attacked yet, so that after the initial 100 attacks, some defenders have not been attacked yet. Can we both agree that this makes no difference, as long as a full strength defender is swapped with another full strength defender? Then realize that this is exactly what is going on when they are all in one city and some may defend multiple times in the first 100 attacks if they don't take any damage the first time.

Yes I agree it makes no difference but you're missing my point.

Basically if you have 100 Spears in one city and use just 100 flanking HA's the flanking HA's aren't enough to be the first wave. Some combat HA's end up making up the first wave. If I attack multiple cities (more realistic scenario) I will attack every city with one flanker first before using combat so then my flankers are truly the first wave of attack. That's all I'm saying. I'm not saying it's somehow a different combat outcome if you swap one fully healthy Spear for another.

 

[elitetroops]

I can see what you think is your point perfectly well. And I can also see that it's complete nonsense because it is based on the false assumption that there's a difference between a full strength unit that has been attacked and another that hasn't. The fact that you even use terms like "virgin spear" or "truly the first attack" to make a distinction between units that have been attacked and units that haven't shows how flawed your reasoning is. Both of those are completely meaningless. If you really do understand that there is no difference between full strength units that have been attacked and full strength units that haven't, you should stop referring to them by how many times they have been attacked and instead only refer to them as full strength units. Then you will see why your logic fails.

 

[dankok8]

I can see what you think is your point perfectly well. And I can also see that it's complete nonsense because it is based on the false assumption that there's a difference between a full strength unit that has been attacked and another that hasn't. The fact that you even use terms like "virgin spear" or "truly the first attack" to make a distinction between units that have been attacked and units that haven't shows how flawed your reasoning is. Both of those are completely meaningless. If you really do understand that there is no difference between full strength units that have been attacked and full strength units that haven't, you should stop referring to them by how many times they have been attacked and instead only refer to them as full strength units. Then you will see why your logic fails.

I'm not making the point that you think I'm making that attacking a virgin vs non-virgin Spear is somehow different. I'm saying you ran an experiment where flanking HA's were supposed to be the first wave but didn't even attack all the defenders.

This argument is kind of becoming fruitless. We both agree and understand how the combat engine works and that's not even what the discussion was about so let's stop.

 

[drewisfat]

A true believer in flanking would continue promoting flanking against full HP spears, possibly even spears that have only taken one or two hits, but the full HP part is assumed in even bothering to do the test. So yes, ideally we would want to see some additional # of HAs promoted with flanking after the first 100 have gone.

But for how immature this discussion has gotten, I'm surprised we're missing such a golden opportunity to giggle about the term "virgin spear".

 

[Qactus]

you mean virgin spear as opposed to battle-hardened spear?

On topic, I just recently had the chance to compare F2 vs C2 promos on Cataphracts during a looong campaign against 4 AI in a row.
Initially I favored F2, because it grants FS immunity (which Cataphracts lack) on top of the withdrawal chance. Ultimately, combat promos still turned out to be clearly the better choice.

 

[elitetroops]

the full HP part is assumed in even bothering to do the test.So yes, ideally we would want to see some additional # of HAs promoted with flanking after the first 100 have gone.

That was never an assumed. In that case a multi-city test would also need multiple flanking at each city just in case. Danko wanted to spread them out only to make sure the first unit to hit each one is flanking.

Danko, I'm gonna ask you one last question.

Make it more simple. One city, two spears, you have a stack with two flanking HAs and a bunch of combat. The first flanking HA attacks spear A and retreats after dealing no damage at all. Now we still have two full health spears and one flanking HA left. You get to choose which spear you attack with that remaining flanking dude. My argument is simply that it makes no difference which one you choose. Whether you pick spear B to make sure both spears are attacked in the "true first wave", or if you pick spear A again, leaving B a "virgin spear" has absolutely no effect on the overall outcome of the this little war. Do you disagree with this? Do you expect a more reliable result if the second HA attacks spear B?

 

[dankok8]

That was never an assumed. In that case a multi-city test would also need multiple flanking at each city just in case. Danko wanted to spread them out only to make sure the first unit to hit each one is flanking.

Danko, I'm gonna ask you one last question.

Make it more simple. One city, two spears, you have a stack with two flanking HAs and a bunch of combat. The first flanking HA attacks spear A and retreats after dealing no damage at all. Now we still have two full health spears and one flanking HA left. You get to choose which spear you attack with that remaining flanking dude. My argument is simply that it makes no difference which one you choose. Whether you pick spear B to make sure both spears are attacked in the "true first wave", or if you pick spear A again, leaving B a "virgin spear" has absolutely no effect on the overall outcome of the this little war. Do you disagree with this? Do you expect a more reliable result if the second HA attacks spear B?

It's a fact that it makes no difference. But I don't understand why you're asking me this question?

Fighting 100 Spears is 100 independent events so of course it doesn't matter in what order they are done. My point is simply that testing 100 attackers attacking 100 defenders in general gives us no information about variance of outcome. Whereas doing attacks spread over 100 cities gives us more information because now we can make a distribution. For example, in 12 cases, we needed 1 HA, in 47 cases, we needed 2 HA's, in 22 cases, we need 3 HA's, in 8 cases we needed 4 HA's..... and in 1 case we need 9 HA's.

It's going to be a left skewed distribution of some sort.

When I said fewer attackers are required I meant that the median (middle value) is in fact smaller than the mean. So when you say 3.23 HA's per Spearman is the average number of flanking HA's needed you're actually overestimating the number of HA's that are needed in a typical situation. Kind of like how if you're earning an average salary you're actually making really good money because you're making more than maybe 80% of the population. The distribution of salaries is left-skewed just like combat outcomes here.

Of course my point here applies to combat HA's as well. Those numbers are probably overestimates as well. Sorry I had to go all ham with statistics but I do know what I'm talking about here.

 

[elitetroops]

I asked you the question because it is the 100 spear example scaled down to only two. All this time you insisted that it does make a difference in the case of 100. Can we at least agree that 1 city vs 100 cities makes no difference for the mean?

Median vs mean is a completely different issue, but I wouldn't expect the median to be significantly lower than the mean (I can immediately tell that the median in your example cases is way too low). By "significantly lower" I mean that the difference is not big enough to affect our decisions. For practical decision making median is kind of useless anyway. The mean can at least be used to estimate how many units you need to take out a large stack or estimate what you need to win a war. Median only tells you how many units you'd need to get some job done 50% of the time. You'd probably rather know how many you need to have 90% success rate. This can of course also be tested and tests require multiple cities, but those are different test than the one we're discussing here.

When testing the effectiveness of a promotion, all I'm interested in is what good it does me overall. Would frequent use of this promotion help me win my wars faster or with fewer units. For this the mean is the measure I'm interested in.

 

[dankok8]

I asked you the question because it is the 100 spear example scaled down to only two. All this time you insisted that it does make a difference in the case of 100. Can we at least agree that 1 city vs 100 cities makes no difference for the mean?

Median vs mean is a completely different issue, but I wouldn't expect the median to be significantly lower than the mean (I can immediately tell that the median in your example cases is way too low). By "significantly lower" I mean that the difference is not big enough to affect our decisions. For practical decision making median is kind of useless anyway. The mean can at least be used to estimate how many units you need to take out a large stack or estimate what you need to win a war. Median only tells you how many units you'd need to get some job done 50% of the time. You'd probably rather know how many you need to have 90% success rate. This can of course also be tested and tests require multiple cities, but those are different test than the one we're discussing here.

When testing the effectiveness of a promotion, all I'm interested in is what good it does me overall. Would frequent use of this promotion help me win my wars faster or with fewer units. For this the mean is the measure I'm interested in.

Mean is the same in both tests.

Median is actually the measure that tells you what the middle of the road situation is like (not too lucky or unlucky). How much lower it is than the mean it's hard to tell without extensive testing but it will be lower. It's the point I was trying to make from the start but I didn't want to get all into statistics. In my opinion, using 100 HA's to attack 100 Spears in a single city is flawed because the average number of HA's gives us no idea about the distribution of outcomes. Let's say your apartment building has 20 apartments. One guy earn $10 million/year and the other 19 people earn $20,000/year. Guess what the average income in the building is $519,000 a year. The average paints a very flawed picture of the kind of people who live in that building. The median is critical to know in this situation because it represents the "typical tenant" in this building better than the average does. In civ 4 combat situations we are obviously not getting the same kind of skewed distribution as here but extreme bad luck does happen and in 100 rolls you will get a few ridiculously bad rolls that will skew the mean.

I think having the median and generally the distribution of outcomes is important if we are analyzing this thing at all.

 

[elitetroops]

It's the point I was trying to make from the start but I didn't want to get all into statistics.

You should have said median right away to clarify you're talking about something else. I know my statistics, I just don't think median is a very useful measure here. Variance is a part of the game and should be considered in decision making. Again if we use some more simple numbers, say you do a bunch of attacks at 51% odds. The median should be 1 and mean slightly above 1.5. I believe this is about the largest difference between median and mean you could achieve (except I don't think civ4 combat would ever give 51% odds, but let's ignore that for now). Knowing that the median is 1 serves no practical purpose. I can't base any of my decisions on this number. If I plan to fight some units at these odds, I know from the mean I need to bring more than 1.5 units/defender to expect a positive outcome.

I wouldn't say 100 rolls must include a few ridiculously bad rolls. You can expect to see events that have 1% chance of occurring and those I consider very much to be a part of normal gameplay. Common enough to happen every now and then. In the case of combat a 1% event is not a huge outlier that greatly skews the mean. In this case my guess is that it is at most a +1 compared to a 10% event. They will also happen both ways. In 100 rolls you can expect on average one flanking dude to take out the defender in one shot (they have 1.03% victory odds in this scenario) bringing the mean down.

Oh and just to clarify, when I do tests like this I always run them a few times to make sure the mean is somewhat reliable and I don't draw conclusions from one exceptionally lucky/unlucky run.

 

[dankok8]

You should have said median right away to clarify you're talking about something else. I know my statistics, I just don't think median is a very useful measure here. Variance is a part of the game and should be considered in decision making. Again if we use some more simple numbers, say you do a bunch of attacks at 51% odds. The median should be 1 and mean slightly above 1.5. I believe this is about the largest difference between median and mean you could achieve (except I don't think civ4 combat would ever give 51% odds, but let's ignore that for now). Knowing that the median is 1 serves no practical purpose. I can't base any of my decisions on this number. If I plan to fight some units at these odds, I know from the mean I need to bring more than 1.5 units/defender to expect a positive outcome.

I wouldn't say 100 rolls must include a few ridiculously bad rolls. You can expect to see events that have 1% chance of occurring and those I consider very much to be a part of normal gameplay. Common enough to happen every now and then. In the case of combat a 1% event is not a huge outlier that greatly skews the mean. In this case my guess is that it is at most a +1 compared to a 10% event. They will also happen both ways. In 100 rolls you can expect on average one flanking dude to take out the defender in one shot (they have 1.03% victory odds in this scenario) bringing the mean down.

Oh and just to clarify, when I do tests like this I always run them a few times to make sure the mean is somewhat reliable and I don't draw conclusions from one exceptionally lucky/unlucky run.

Good post.

Here's the thing. In the example you gave with an estimated mean of just over 1.5 and a median of 1 (numbers I agree with by the way), you can't rely on the mean of 1.5 units/defender to tell you much unless you know the distribution of outcomes. Bringing the mean number of units could cover you in 60% of cases or 90% of cases which is quite a significant difference and you'd want to know that. For example if you're attacking 2 defenders at 51% odds, you will need as many as 4+ attackers just under 25% of the time. This is a case where if you bring the mean number of units which is roughly (just over) 3, you'll get smacked pretty badly pretty often.

It's combination of measures (ideally the entire distribution) that gives you the best tool for decision making. That's why the Advanced Combat Mod was created too to give the player more data to base their decision from (health remaining, XP gained etc.) rather than just W/R/D percentages. Unfortunately no mod has been created that lets you calculate overall odds from multiple combat rounds to answer questions like "What are the odds of my 10 axeman taking out these 4 archers?".

IIRC the civ 4 engine can give odds very close to 50% if the modified strengths of the units are equal but FS are involved. It's very rare in practice.

 

[dankok8]

I ran a test of HA vs. Spear in a city with 50% tile defense 100 times. It didn't even take that long maybe 30 minutes. Both the HA and Spear were unpromoted but I got a bunch of interesting data and I'm sure the distribution of outcomes is mostly the same.

HA's Needed to Kill Spear:

1 HA -- 5 times (3.81% odds so a bit higher than expected)
2 HA -- 51 times
3 HA -- 30 times
4 HA -- 12 times
5 HA -- 1 times
6 HA -- 1 times

Total: 256 HA
Mean: 2.56 HA/Spear
Median: 2 HA/Spear

HA's Died to Kill Spear:

0 HA -- 12 times
1 HA -- 56 times
2 HA -- 24 times
3 HA -- 7 times
4 HA -- 1 times

Total: 129 HA
Mean: 1.29 HA/Spear
Median: 1 HA/Spear

I tracked withdrawals as well. A total of 27 HA retreated. The death to withdrawal ratio was 129/27 (4.78:1) which is a bit higher than the expected 4:1 ratio with 20% withdrawal odds. The sample size seems large enough to get a good idea of outcomes though.

Of those 27 HA's who withdrew, 18 of them were withdrawals of the first attacker. In 10/18 cases (55.6%), the retreated unit failed to even scratch the defender leaving him at full health and the most damage that the retreated unit did was to bring the Spear down to 2.6 (healthy Spear is 4) which still doesn't give awfully good odds for the next attacker..

Anyways how good would two flanking promos be? It turns out that it's easy to analyze from this data because flanking promos don't change the modified strengths of either the attacker or the defender and so shouldn't influence the distribution of HA's needed, just replace some deaths with withdrawals. Given that 156 HA's of 256 HA's used either died or withdrew, unpromoted HA's should withdraw 31 times (20% of the time) whereas FI/FII HA's (50%) should withdraw 78 times. This means 47 extra HA's would survive. That's the net result of flanking promos to this data.

How about two combat promos? A C1/C2 HA has outright odds of 9.7% of killing the Spear in my test. The odds of winning are much higher so the number of needed HA's and number of deaths should be much lower. It's easy to invision this distribution looking a lot more favorable.

Verdict: Given that more than half of the time the withdrawing HA leaves a full strength defender and a virtual certainty of leaving the defender with at least 2.6 health, the flanking promos just don't bring much bang for the buck. Assuming on average a war involves killing 5 Spears, flanking could save you 2 HA's total compared to using no promos at all (saves you fewer vs. combat HA's because those win outright) while needing more attackers total, at a cost of fewer XP gained, having to wait for healing etc. It just isn't worth it to get flanking. I'm pretty much totally convinced at this point.