This is good testing, etc...

However, it doesn't incorporate the dreaded pop cap. Especially if you are using a mixed army, it can be rather difficult to mass that many of the units early on.

You make an excellent point about not getting any upgrades w/o first having a sufficient amount of affected units first. But any edge is a good one -- in battle if your 7 hoplites and 6 toxites are pumped up a little more than your opponent's, then you'll win. Of course, this scenario assumes you have the bank to do this.

an early army will usually consist of only two unit types.
I would not find it hard to get 15 hoplites and 17 toxotes before heroic age. The point is, even if you have the resources to upgrade to medium, unless you already have the listed number of units (or are as you said at pop cap) then those resources would be better spent on more units.

another example then:

15 hoplites cost as much as 12 medium hoplites.

Total damage of 15 hoplites = 120 damage/second
Total damage of 12 med hops = 105.6 damage/second

So yes, if at pop cap this is irrelevant. However, if you're upgrading before you have the listed number of units and you're not at pop cap you're shooting yourself in the foot.

Calculate power as sqrt(Hp*Dam) [actually, as sqrt(EffectiveHp*DamPerTime) if you want to compare different units with different strike rates and different armor.]

Calculated this way, X units of power Y will fight to a standstill Y units of power X.

Power = sqrt(hp*dam) does end up making the upgrades seem very expensive. Trust me though, the equation works in simulation and in game testing.

I think ES might've slightly blown it here. WCIII upgrades seemed very cheap to me (less than a single unit to start with), until I did the math and realized that they were about right for the number of units you'd see in a game, with the final upgrade only being worthwhile for armies of 10+.

thewesson

thewesson... are you sure youre right? it seems to me that it should be a streight square relationship between hp and damage. For instance if you increase the attack of a unit it's alive for the same amount of time but does 10% more damage while alive. If you increase the hp of a unit by 10% it's allive for 10% longer but does 10% more damage because of the increased longevity. Thus if you increase both by 10% shouldn't the power of the unit be considered to have increased by 1.1*1.1= 1.21 or 21%?

[This message has been edited by The Devil (edited 11-20-2002 @ 07:26 PM).]

No problem TheDevil, I worked through all this for WCIII and I wanted to share the fruits of my struggle ...

Someone should look up the numbers for ranged units versus melee units.

Ranged units which focus 100% are about twice as effective as a melee army which focusses not at all, fighting entirely 1 on 1. (That is, an equal ranged army will defeat an equal melee army with 1/2 its hit points left.)

So being ranged should be equal to having about 1.4x as many units.

So the price for a given power in ranged units should be about 1.4x as much as the price for the same power in melee units.

I have the hunch that's about where Ensemble's numbers come out. Their ranged units seem to be about the same price but have about 2/3 the raw power, if memory serves.

If ranged units are significantly better than this, then you'll see them dominate the battlefield.

thewesson

thewesson, I've revised my above post, could you please read it again thanks for your help nutting this all out.

by the by, do ranged units still dominate WC3?

Clarification question -- do you calculations take into account how much damage they can receive as well as dish out? (it was mentioned in your other thread)

Also - not appropriate for ranged units - but melee units can only deal that much damage as long as they have a target and can reach them.

very nice work. keep it up.

No, Blizzard got that worked out and people don't mass 100% ranged at all, though they do use them. Well, people do mass Huntresses (short-ranged melee.)

Trying to work out all the factors on my spreadsheet, Huntresses do have about a 10% power advantage over other Tier1 units.

Estimating a 1.4x bonus for being ranged, Blizzard's units come out pretty close in cost-effectiveness in each tech tier otherwise, even across races.

I see you still don't quite take my word on sqrt(hp*dam) thing ... Intuitively, it seems as if 10% increase in HP should produce 10% more power all by itself, doesn't it?

Well, look at it this way. A 10% increase in HP produces 10% more lifetime damage, true. However, a 10% increase in numbers produces 10% more lifetime AND 10% more damage per time unit. So 10% more numbers is 1.21 times as much lifetime damage.

Hmm, why are we calling Dam*Hp 'lifetime damage' and why is that our criterion?

I've decided it seems to be a good measure of unit worth to determine how much damage it will inflict in its life. Because Dam (actually dam per time unit) determines the rate at which the unit inflicts damage, and because Hp determines how long (how many time units) it stays alive, Hp x Dam reflects how much damage you expect it to do in its lifetime. You can see that a unit of 100 Hp inflicting 2 damage will be equivalent to a unit of 50 Hp inflicting 4 damage - if these two fight, then at the 25th time tic they will both be dead - at the same time.

Calling 'power' the square root of hp*dam is just a way of bringing 'power' in line with the fact that lifetime damage goes up as the square of the numbers. Deciding to call this quantity 'power' makes the power of a formation go up linearly. Then you can do things like add and multiply power by numbers and get accurate results.

Here's the math:
Formation one has N1 units that each have H1 hp and D1 dam.
Formation two has N2 units that each have H2 hp and D2 dam.

When are they equivalent? When lifetime damage is the same.
Each formation has N*H hitpoints and does N*D damage, so lifetime damage for each is (N*H) * (N*D)

(N1*H1) * (N1*D1) = (N2*H2) * (N2*D2)
N1^2 * (H1*D1) = N2^2 * (H2*D2)
square root of both sides:
N1 * sqrt(H1*D1) = N2 * sqrt(H2*D2)
N1 * power1 = N2 * power2

'power' becomes a convenient number that lets us decide what N1 and N2 have to be, to call two different formations composed of different numbers of different units equivalent.

If Unit A has twice the 'power' of Unit B, you are only saying that 10 A's will fight 20 B's to a standstill. And this in fact is the way it works out on the battlefield.

It is *not* true that 10 A's will do twice the damage that 10 B's will do. As a matter of fact, since lifetime damage is the *square* of the power, 10 A's will do ~4x the damage that 10 B's will do.

thewesson

Hey Gimli aren't you McLT_Gandalf?

then what percentage would armour be?

Thanks for your help thewesson, you were most definately correct

I hope the fact that I couldn't just accept your word for it didn't offend you. I'm a scientist (hence the bad spelling) and so need to understand things rather than just accept them. Your explanation helped greatly and my own calculations back you up.

Hopefully I can look forward to many more similar stimulating conversations and analysis... damn, I'm sick. I should have completed my degree in maths and computer science.

thewesson, would you care to work out when it's economical to get the hack armour upgrade for the hoplite? that seems a much more curly question than the medium upgrade.

That's a good point. You should definitely consider armor in these calculations. Some units for example do the same amount of damage to enemy archers and infantry, but are more vulnerable to pierce damage than to hack damage. Clearly in this case it's better to upgrade earlier if facing archers and later if facing infantry.

Also, it's important to know figure out how to use this information in a game. Do your numbers represent the number of units in your army or the total number of units you made in the game? That is, should you upgrade medium hoplite when you have 30 hoplites or when you are creating your 30th hoplite after some have fought and died? Or should you upgrade when you are thinking you will create at least 30 hoplites after the upgrade time, to give them more power? The key point here is that all hoplites who fight (or die) before the upgrade don't benefit from it. For example, if you create 15 hoplites in Classical age to rush with and you have enough resources for 3 hoplites or 1 upgrade left, do you go for the 3 hoplites or for the upgrade? Your rushers will probably die and be replaced by, say, 15 more hoplites. Should you upgrade after creating the 15 more hoplites, even though you still have 15 and not 30? Should you upgrade before rushing? I'd think that upgrading before rushing is the best choice since it adds power to the 15 hoplites that would otherwise have died and not benefited from the upgrade, in addition to helping any hoplites you make afterwards.

Also, later on in the game the pop limit gets important. When you reach the pop limit you should probably queue up some units for backup but then go for upgrades.

"Upgrade sooner. The key point here is that all hoplites who fight (or die) before the upgrade don't benefit from it."
or,
"If you are ever going to build 30 hoplites, then get the upgrade."

That's an intuitively attractive argument. But I believe that if you are going to fight with *this* army (which is why you are building it) then you need to maximize the power of *this* army in whatever way is best. That minimizes your casualties, maximizes enemy casualties, and keeps units alive to overpower the diminished enemy in future battles.

TheDevil, I am certainly not offended by skepticism and I enjoy these discussions, that's why my posts are so lengthy.

As for armor. Suppose you had an upgrade that took hack armor from 30% to 40% (I suppose that's what ES is calling a 10% increase.)

Then damage taken goes from 70% to 60%.

Before upgrade, effectiveHP is Hp* 1/.7 (1.43), and afterwards, effectiveHP is Hp * 1/.6 (1.66).

Unupgraded unit power is sqrt(1.43 * Hp * Dam) and upgraded power is sqrt(1.66 * Hp * Dam)

The armor factor on power is sqrt(1.43) in one case and sqrt(1.66) in the upgraded case. That is, 1.2 and 1.3 approx. The upgraded unit has 1.3/1.2, 1.08 times as much power. 8% more power.

How much does 10% more hack armor cost? I don't know, so I'm going to say as much as 3 units.

Break even point:
X+3 = X * 1.08
1 + 3/X = 1.08
3/X = .08
X = 37.5

So if hack armor costs as much as 3 units, and your units are only facing hack damage, then you should pay for the hack armor when you have 38 units already that can benefit from it.

We can see that Ensemble Studios expects you to be fighting with pretty massive armies.

Note that if you had a hack armor upgrade that took you from 80% to 90%, that would *double* your eHp, and it would make your units 44% more powerful versus hack-ers. A good deal!

However, Ensemble has been careful not to allow hack armor to get past about 50% or so, so extreme effects like this don't exist. I think.

thewesson

Very interesting. Small note though:

Armor upgrades your dmg vulnerability, not your %armor. For example, Copper Mail gives you -10% hack vulnerability. Which means if you had 30% hack armor, you'd have 70% vulnerability, meaning armor would increase 7%.

For example, a unit with 10 attack would be giving you 7 dmg before, and after it would be 6.3.

This makes the armor upgrades even less attractive.

Also, I tried setting up 30 medium hoplites versus 33 hoplites in the editor. I ran it a few times with different starting formations, but never microing them, and the results varied pretty widely. Sometimes the medium hoplites won with 5 guys left, two at full hp, and sometimes the hoplites one with 7-8 left at half hp. So, basically, it seems about right. And of course with good microing you could probably win more often than not.

[This message has been edited by Blotter (edited 11-21-2002 @ 04:05 PM).]

Oh, so ES is doing multiplicative upgrades? So (to take an extreme example) when they say -10% hack damage, and you already had only 10% hack vulnerability, that would take you to 9% hack vulnerability?

You're right, that does make it even less attractive.

It is the sensible way to do it however, since it avoids weird effects as you approach the limits of 0% or 100%.

In any case, the general way to evaluate armor would be sqrt(1/vulnerability). An upgrade would make a unit X times as powerful, where X = sqrt(VulBefore/VulAfter).

If the upgrades are multiplicative as you say, then every '10%' armor upgrade is the same power boost of sqrt(1.0/0.9) = a 5.4% power boost.

21 non upgraded guys would be the same as 20 upgraded guys.

If damage is increased by 10% (multiplied by 1.1) by every weapons upgrade, that is even slightly less nice.

That would be sqrt(1.1/1.0) = a 4.8% power boost.

I hope these upgrades don't cost a lot. If they cost as much as three guys, you'd have to have 60 guys in the field to make it mathematically worthwhile.

On the other hand, as many have already pointed out, you don't have to house or micromanage improved weapons and armor.

thewesson

thanks again thewesson. I checked the cost of the copper mail upgrade (150f 150g) so yes, it costs about as much as 3 hoplites. As Blotter said it doesn't increase hack armour by 10% but rather reduces vulnerability by 10%. Hoplite hack armour is already 35%, thus their hack vulnerability is 65%. Hence the copper mail upgrade increases hack armour by 6.5% to 41.5%.

eHP before upgrade = 1/.65 * HP * Dam or 1.54*(HP*Dam)

eHP after upgrade = 1/.585 * HP * Dam or 1.71*(HP*Dam)

thus sqrt both is 1.24 and 1.31 respectively. The upgraded unit has 5.6% more power.

so break even point is
x+3 = x*1.056
thus 3/x = 0.056
x = 53.6

so it becomes economical to do the first upgrade when you have 54 hoplites.... which will almost never happen. Of course if a unit started with very low armour the upgrade would be more economical, but not much.

What this all means?

Only upgrade once you're at pop cap (unless for some reason you manage to actually get 53 hoplites first!)

Thx TheDevil.

I would maintain that for a multiplicative upgrade it wouldn't matter what the unit started with; sqrt(Before/After) will be the same.

To throw a couple of confounding factors into the mix here ...

You can do some additional math to try to take account of the fact that units aren't just a big blob of HP+Dam but instead come in chunks. A unit doesn't stop inflicting damage until it is dead; a half dead unit inflicts full damage.

Fragmentation example:
Unit A may be twice as powerful as Unit B. However, due to the difference in fragmentation, one of unit A will do better fighting two of unit B than you expect. Halfway through the battle, one B will be dead and A will be half dead but still inflicting full damage. The B force is now at half damage but the damage of the A force isn't be reduced at all. It turns out that A wins with 25% HP left.

There are some nice formulas for fragmentation but I can't recall them off the top of my head. You can often ignore fragmentation because with large numbers of units (that can focus) like 10 A vs 20 B the fragmentation factor is trivial. However, you should take fragmentation into account where a battle breaks up into a bunch of small battles, 1 on 1 or 2 on 1.

The fragmentation factor would make upgrades look slightly better - more power in a smaller number of units, essentially less fragmented.

It's interesting to note that in WC3 in a same-cost battle, because of the fragmentation factor, Grunts should win a low-cost battle and the Footmen should win a high-cost battle. (Grunts are big and bulky.)

Also, the extra units aren't just the ostensible cost. Each unit also costs 5+ wood (for housing and barracks), and whatever villager time it costs to build more houses and more barracks. Add in all those and maybe you should get medium hoplites when you have 25 or 27 existing and not 30.

Anyhow ... despite all that, I think the priorities for the planner of the military economy should be:
1) raw unit numbers
2) more houses/settlements
3) upgrades

But I often upgrade as soon as I think of it because I am a poor player and I often have lots of food/wood/gold sitting around doing nothing. Also, I feel sorry for my poor primitive units getting killed and I want to give them the best equipment possible. Even though giving them more buddies would probably work better.

thewesson

Ok so what's the bottome line here? You need roughly 20 odd units on the battlefield before upgrading before a pop cap? I do like the idea of future expectations though.

The bottom line would be 25+ units before medium/heavy/champion upgrade, something like that.

The armory upgrades would almost never win out vs more units. Something like 50+ units would have to be affected.

(The math works out such that really expensive/powerful units make the upgrade worthwhile at a smaller number of units.)

However, sometimes spending X resources on more units is not really an option, because you don't have the barracks capacity, or you are housed, or you are at pop limit. If you don't think you can spend the resources on units before you start to fight, then the upgrade may well be worth it.

In terms of future expectations ... let's take an extreme case ...

Say you had one hoplite and the enemy had one hoplite. For some reason the enemy has decided he will always send exactly one hoplite at you for every battle. He will do this indefinitely. (Thus, the future expectation is infinite.)

Your choice is to upgrade now so that every new hoplite is worth 1.1 old hoplites, or to add three hoplites to your army. After that you will only have enough resources coming in to add one hoplite to your army for each battle.

If you upgrade now, your one hoplite will survive this battle with about 10% Hp left. On the other hand, your four hoplites will emerge with a nick - one guy will have been reduced to 70% and that's all.

Just as you would have kept the benefits of upgrading for all future battles, you get to keep the soldiers that survive this battle.

Your five hoplites (one scratched) will fight the next battle and emerge even better. And so on. Your win/loss ratio will always be better than the person who took the upgrade instead.

Ultimately in this scenario you will reach the unit count at which an upgrade would be preferable to another three hoplites. At that point you should not create any hoplites for three battles and get the upgrade.

You can see here that the person who took the upgrade early is much worse off. He lost almost one hoplite the first battle, most of a hoplite the second battle, and so on ... he'll always be lagging behind you, because your army dominates each battle better than his did.

In a different scenario, where you add .5 hoplite to your army for each battle, and the enemy comes at you with 1 fresh hoplite every time, you with your dominating army will win every fight and Mr Upgrader will lose every fight past the first ...

In summary, even if you expect a large number of small battles continuing into the indefinite future, your criterion for upgrading should be the same: numbers on the field right now.

*IF* you did not get to keep soldiers that survived a battle, then upgrading would be better. But since you do get to keep the survivors, a bigger army has the same persistent effect of affecting future battles that an upgrade has.

If you're like me and you let your soldiers wander randomly into TC fire so that they're unlikely to survive whether or not they win the battle, then the upgrade is probably a better choice for you.

thewesson

What hasn't been considered is this, and is KEY to this discussion.

Assuming that each battle consists of the same units fighting the same units, and that AFTER each battle, both players sit and wait for the next army, then the answers to this discussion thus far hold true.

However, rarely do players (good ones anyway) win a small battle, and then leave their units standing till the next battle. Considering you DO upgrade earlier, the second battle, you will come out on top, because:

Battle 1: Each player uses the same amount of resources (lets say 50r per unit)

PA: pays for 10units and upgrade (Costing 650r)
PB: pays for 13units (Costing 650r)

Each player even kills the other

Battle 2: Each player has same resources at his disposal

PA: pays for 13units-- Now heavy (650r)
PB: pays for 13units-- still medium (650r)

Now, not only is the battle outweighed in PA\s favor, but those few units left actually have EVEN a LONGER "power", because they can attack Player B unhindered (destroying villagers, etc.) this becomes exponential as Player B now won't have as many resources at his disposal, and it landslides from there. Now also, Player B is on the defensive, so his first units must destroy Player A's remaining units before amassing for the forth coming small battle.

Battle 3:

PA: pays for 13units, still heavy (650r)
PB: Even if he can manage to still output 650r for this battle, his first units will either be dead, or atleast damaged from fighting the "spill-over" units from Battle 2, which swings the favor toward Player A even further!

Now Player A can do even MORE unhindered damage to Player B, AND therefore exponentially increasing his economic advantage over Player B.

[This message has been edited by NobleKain (edited 11-22-2002 @ 03:33 PM).]

I think you're right that at an even power where A and B merely destroy each other, it's better to have the upgrade in your pocket for the future.

However, there is some other point, somewhat below the break even point I suppose, where you would be better off with more units.

Is that point very far from the power equality point?

I'm running these numbers in my simulator, which I can make available to people on request. I'm simulating Hoplites as having 10 Hp and doing 1 dam/sec non upgraded, and having 11 HP and doing 1.1 dam/sec upgraded.

Suppose A upgraded (10% W & A) when he had 20 Hoplites and B didn't upgrade and got 23 Hoplites instead.

A's doesn't seem like a very bad decision. But fighting it down to the ground, B emerges with 5.6 units left over and A emerges with none.

Next battle, they each make 20 more Hoplites; B still doesn't upgrade. It's 25.6 'normal' vs 20 medium. Now B emerges victorious with 12 units left over. And so on.

Okay, let's try 25 medium upgrade vs 28 non upgraded. This is getting quite close to the break-even point.

Closer this time ... B still comes out ahead with 3.14 units left over. Both parties make 25 more. Next battle he has 28.14 versus 25 upgraded Hoplites ... he wins better this time. And so on. B is going to gradually crush A.

How close do you have to be? Well, B still wins when he has 30 unupgraded hoplites and A has 27 upgraded, but only by a bit (he has 3.05 hoplites remaining). But B's dominance will *still* become more prominent over time, if both parties continue feeding chunks of 27 hoplites into the fray.

So you just about have to be at the break-even point to justify your decision to upgrade.

As you said, too, when B wins the battle his 3 remaining non-upgraded hoplites can be causing some trouble while A is rearming.

thewesson

PS I understand of course that real life is much more complicated than the simulator. I don't claim to know exactly what it is that makes AoM players sometimes prefer upgrades more than they "should"; it might be a perfectly good reason.

[This message has been edited by thewesson (edited 11-22-2002 @ 06:53 PM).]

Thanks yet a gain thewesson, your help has been invaluable.

I'd love a copy of your simulator if you don't mind. If it's not too big you could send a copy to me at kenclair@senet.com.au

I tend to suck at this game at the moment too, and my instinct is to upgrade as soon as I have enough resources. However I was faced by a poseidon player who insisted on persistently attacking me with hippikon. He defeated my first lot of spearmen (only 7 I'm sad to say...vs 8 hippikon) and wen't on to kill villies. I moved all my exposed villies away from my forward and my tc to out of the way locations where he wasn't likely to scout, just left a few farming and mining near my tc (to give the impression that he'd slowed down my econ). Then build three baraks under my towers, and build spear till I was housed. He persisted in sending hippikon, but slowly added toxotes to the mix. In the meantime I continued building houses and villies and added slingers to the mix. I kept winning the skirmishes and spending less each time to reach pop limit, even when he used bronze. Eventually I was at pop with res to spare so got the medium upgrades, then the blacksmith upgrades and was advancing to the next age when my ally finished with the other enemy and moved on to help me. All over bar the shouting.

The point of all this is that I had the units on the ground and because I had more units it cost me less to re-build after each skirmish. Eventually my econ and military were in a better state than my enemy.

The Devil

P.S. my friends have done some ingame unit testing on the medium upgrade stuff... results pending.

I think you have to take into account that even if the armor upgrade or medium upgrade costs as much as three other hoplites (or whatever unit) it makes the player WAIT until 3 units have been built before he can take advantage of the so called "bonus".

It is much better to leave with an army early and get the upgrade on the way then wait until all three hoplites are done then leave. By waiting for the build times you are essentially giving the opponent time to prepare & get more units. By leaving early you are applying pressure early and taking the fight to him.

You can't accurately represent a battle with a spreadsheet. Tactics, strategies, and chance are all big factors. You also have to consider that your comparison is in some ideal situation that never happens when you use these formulas. You sort of treat two armies like two massive units fighting 1v1. That leaves room for a lot of error.

21 base units might be the same as 20 upgraded if they walk up to eachother in a line and start hitting eachother at the same time. They are going to fight in a much less organized way, most likely. That extra man will probably have to walk around or wait its turn in the crowd to reach a target. The upgraded units aren't going to fall as fast, and so their numbers are going to thin slower. An unupgraded unit dies 1 or 2 hits before it would have killed its target. That upgraded unit that defeated it with 10 hp's remaining has nothing hitting it, but is dealing out full damage unless something else eventually finishes its target and locks onto it next.

This means that your spreadsheet representation of power of a group is invalid as soon as the first guy falls, or someone does something other than march his army to meet another in an open field. It only works accurately for 1v1 battles between units and no one ever does that.

Do your upgrades as early as you can without slowing your production of units. If you only have 6 military units, but the resources to keep 3 - 4 barracks (or whatever you use) constantly flowing AND buy the upgrade, do it. If you do not, wait until you do. You should not need to wait until your pop cap to do unit upgrades if you have a decent economy going.

To answer this before it's said, no you shouldn't build an extra military building and pump more units instead if you can afford the upgrade. It's not worth a whole new structure for the 1 or 2 units an upgrade minus the cost of the building can buy you.