The economics of food velocity in a Despotism

[absimiliard]

Okay, so we've all realized that rush-building during your early-game Despotism is a good thing. We've even gone so far as to believe that it's a very powerful thing. But no one yet, that I've read has tried to quantify this tactic. I will tackle it, but I'll need some help since it's not quite all straight in my head.

First I will establish some assumptions. A point of population, when used to rush is worth 28-40 shields in my experiments, I will venture an average of 35 shields therefore as the value of that point of population. There is a cost to rushing however, the cost is the amount of lost resources the pop would have built during the time you take to rebuild that point of pop.

That opportunity cost (economics term here) varies dependant on how long it takes to build your point of pop. For a city with two food to spare, the norm, that time is 10 turns. During those 10 turns we can reasonably assume the point of pop would yield 20 food, 10 shields, and 10 trade, this will obviously vary widely but an average isn't unreasonable I think. So for 40 mixed resources you are netting 35 shields. In and of itself this isn't that impressive.

It gets better as your food-velocity, the speed with which a given city grows, gets higher. If you have a velocity of 4 food/turn you are only looking at a 5 turn growth rate. Your exchange rate then is much better, 35 shields for a loss of 20 mixed resources. A velocity of 5 food/turn brings your loss down to 16 mixed resources. Now you're really cooking with gas!

My conclusion is that it's only worth pop-rushing if you make 3 extra food/turn or more or are in an emergency situation. At 3/turn you're losing 28 resources to get 35, that's a good trade, whereas at 2/turn you're losing 40 resources for your 35 which I find less appealing.

Now on to transfer mechanics.

Pop can be transferred by means of Workers or Settlers. Settlers obviously carry larger chunks of pop than Workers do, but cost proportionately more. The math has Workers costing 10 and Settlers costing 15 per point of pop. For a 35 shield return on that pop we get efficiencies of 70% and 60% respectively. (That's a rough figure, 71% & 58% are more accurate). Clearly population transfer is best done with Workers, not Settlers. But when is it worth it to transfer that pop to build something, that's the important question.

Using Workers adds 10 resources to your opportunity cost when you pop-rush. So a 2/turn food situation now costs 50 resources instead of 40, clearly a poor bargain for 35 shields. 3/turn is better, now we're talking 38 lost for 35 gained. I might use population transfer for rushing in this circumstance if I really needed to, like to build a Temple to expand my borders.

4/turn and 5/turn is where this changes though. At 4/turn you're actually making resources by transferring that pop to another city where it can be used to rush, you're not making much, only 5 shields, but it's still a positive figure. 5 food/turn is even more compelling, I cost in expenditure & opportunity cost only 26 resources but I can gain 35 by pop-rushing. That's cooking with gas!

My conclusions are as follows. At a food velocity of 2, 2 extra food/turn, pop-rushing is feasible but not practical, use it only for essentials. At a food velocity of 3 become efficient at pop-rushing in that city and should do so as much as you can given the value of shields early in the game. At a velocity of 4 or 5 you become efficient as a source of export, in one of these cities you should build just Workers and use them to pop-rush in other cities.

Now I need your help. I'm pretty sure I've missed some factors, but I'm not sure what. As it stands I'm pretty pleased that my thoughts give us a good yardstick to guage pop-rushing by, but I'm afraid of those missing variables.

So . . . . if you can think of how to better express this relationship, please speak up. I'm convinced pop-rushing is an essential early-game tactic and I'm certain it can be quantified. I'm just not absolutely sure I've gotten it done right.

[mharmless]

To help you with your math, the first person killed in pop rushing is worth 40 sheilds. Second and all beyond (up to eight at least) are worth 20 each.

The rush cost of an item is doubled if it has no work into it yet.

Finaly, in the early game your cities are going to be capped at 6 pop points (for the most part), and it will be a long time before you unlock the 12 point cap. Once you get to 12, your city will collect food untill it would grow the next turn, and then halt there. Thus if you rushbuild an item at that point you will replace the worker immediately the next turn and suffer only one turns production loss for 40 sheilds of production value (roughly).

[absimiliard]

Thanks. That doubling of cost & the lower value of extra pop used in rushing must have been what was making getting an exact figure difficult for me. I think that working with an average figure is acceptable though, I never get the full 40 from my rushes, normally I pull in like 35 though it does vary a bit from time to time.

I had forgotten the cap on pop. That would imply that rushing is economical for every instance where you would have a city sitting still in pop-growth. I suspect you're still better not doing much rushing at under 2 food/turn though, so if a size 4 city had no more 2 food squares to work I'd rush it then, before my food velocity dropped. But that's just a hunch.

The largest factor I've missed though is the presence of a Granary. Having a granary nearly doubles your food velocity, you have the same amount of excess food per turn but only need to go half as far. In fact I'd go so far as to venture a hunch that rushing is always the right idea with a granary a velocity of 2+.

[David Weldon]

I would argue that the opportunity cost is even less than presented. The reason is that you are not capped only at 6 in the early game, at emperor and deity you are often practically capped at 2 or 3, depending on how many MPs you wish to waste sitting around in your cities. At levels where you can have larger populations more easily, I don't rush-build things very much but at higher levels I find it very important.

The biggest missing factor in your math is that the first thing that should be rushed is a granary. This halves all of the opportunity costs you discuss, as well as greatly accelerating the point at which you hit the natural population cap (due to happiness or lack of aquaducts).

In my experience on difficult levels, if I ever rush anything from a city then my max population is 1, even with a temple and multiple luxuries. The opression unhappiness is a large factor when you're rush building every two turns thanks to some irrigated cows or wheat. This means that I can't grow my cities to two unless I have MPs or I immediately rush another military unit. I bet you can guess which I normally choose...

The opression unhappiness eventually wears off, but I normally don't worry about growth until I've conquered or subjugated at least two civs.

[Dale]

Pity ya can't pop-rush the pyramids. Then you'd become some sort of Power Pop-rush Empire.

[pchang]

In addition, the resource trade off doesn't include the benefit of getting the shields NOW instead of waiting X number of turns for them to build up slowly. TO do this properly, you would have to include some sort of net present value calculation.

[absimiliard]

Hey pchang. I think that my analysis is only valid over a short run of turns. Over that time period the depreciation will not be too much. We should be able to ignore it, or if need be account for it by introducing a variable.

Mr. Weldon's comments are more damaging. He is correct that I am not accounting for granaries adequately in my math. I think the solution is to redefine quite a few things.

First I define an average value for a square, call it A. Food is F, shields are S, and commerce is C. A = F + S +C. This is a little simple, but equally valuing the resource types can be dealt with later. (I think a typical square is like 2F + 1S + 1C, though it's easy enough to get higher-value squares as well) This value, A, can also be simplified to the sum of the values in shields by claiming that food, shields, and commerce are all equally valuable. The example square then has a value of 4S.

We'll switch to growth (G) instead of velocity. Using time as turns (T) and a unit of population (P) grown, we get G = T / P. Growth is defined as Turns per unit of Population grown. It's odd to see time on top of a fraction, anyone who can elegantly get it into the denominator where it belongs gets kudos.

O = G * A. If your opportunity cost is under your gain of 40S, the value of a unit of population in a rush, then you should rush, hypothetically. So if (G * A) < 40S you should rush. For a typical value square you thus rush if (4S * G) < 40S, or G<10. This says you can hypothetically gain value from rushing if you can grow a unit of pop in under 10 turns. Given some inevitable loss in your shields from rushing I'd say you're more practically looking at 35S in returns making a rush at 8 turns practical.

Given that a special can make a city have a G of like 2 to 4 the effects on the game are spectacular. A city with a G of 2 nets a gain in shields (G) equal to say 35 minus the opportunity cost. 35S - (G * A) = 35S - (2 * 4S) = 35S - 8S = 27S. So every two turns a G2 city that rushes gains like 27 shields in production value relative to itself not rushing.

That goes somewhere too. Let me think about it.

[i]As a brief aside, may I note that I've been rushing brutally and have found that it leaves me horribly poor and ignorant. I'm gonna try a more mixed game next, some rushing cities, others not.

[Easy Rhino]

abs, I just want to say I'm really impressed with this topic.

Uses all the concepts that I never learned in ECON

ER

[Grim Legacy]

Some more thoughts:

1) The population gets unhappy from rush jobs. In a size 3 city, you need 2 MP already. Rush a job and you may end up with a size 2 city needing 2 MP... as soon as it grows to 3 again you have a disorder. If you need an entertainer to compensate, you'll lose the revenue of a laborer.
So you'll need a temple to help and to make them forget sooner in addition to a granary (without a granary, the food revenue of a Despotism city is typically only 2 unless special tiles are near).

2) When you have a granary and temple already, you might consider Monarchy instead of looking to get Rush benefits: under Monarchy food production balloons, yielding a very tangible benefit over rushing even worthy city improvements like libraries -as these offer limited gains in a small despotism city. In addition to the extra food and thus laborers, you get extra happiness from a 3rd MP as well. One more laborer.

3) Taking into account the pop cap, a factor that is often overlooked is the possibility to re-allocate production tiles: switch the grassland laborer to forest or mined hills (+roads of course) and you'll receive a nice production bonus, maybe even a trade bonus if you have gold or dyes etc near (these special resources typically appear in low-food squares that can be used in the situation I described).

[absimiliard]

Thanks ER, I appreciate the compliment.

Grim, I'll try to fit the unhappiness factor into my equations. I'm aware of it's existence as a population cap to be sure. My practical limit in a rushing city is 4 pop, that's being dealt with by 2 MP & 1 Temple. A luxury can help out here of course, but 4 is a viable limit since it allows a rush of a Settler unit.

Perhaps the proper approach is to view a long-term opportunity cost, O(lt), as being different from a short-term opportunity cost, O(st). In the short-term unhappiness isn't much of an issue, since I'll deal with it by rushing my pop down to the point of happy again. In the long-term however you're quite correct that I'm losing the use of a pop-unit for 20 turns. (Until the city calms down again)

That invokes a cost of either 20A or the cost of an MP. I value an MP at 5S + 20C in this circumstance. (the cost of production plus the cost of upkeep) Clearly the MP is better, cost-wise, than 20A. Thus Unhappiness (U) as a factor is valued at 5S + 20C, or 25S for simplicity's sake. (someone really needs to figure out a decent conversion ratio for F, S, & C.)

So O(lt) = G * A + U. Or for a normal value of A, O(lt) = 4S *G + 25S. Plugging in our value of 40S for a rushed pop and doing a bit of math gets us a hypothetically profitable long-term rush when G < 3.75. So even in the long-term rushing can still be profitable, it's just much harder. This is also a purely theoretical break-point, lots of inefficiencies can get into the equation.

If you can't use an MP to keep your pop happy then O(lt) is even worse. Then O(lt) = G * A + 20A. Basic assumptions in place yield O(lt) = G * 4S + 80S. Since a rush yields 40S, a long-term rush without MPs to quiet people will never be profitable. Can't be since O(lt) will never be under 40S. If my math's right, that's brutal.

Regarding your third point, I fully agree. If I'm in a non-rushing city and am bouncing on a pop-cap, either hard (aqueduct) or soft (unhappiness), then I firmly believe in reallocating if I'm not focussing on pop-growth. The trick is knowing when & where to rush, not just rushing blindly everything in sight, like I did when I first started trying this technique.

Grim's second point is actually the most easily addressed. All this is pre-Monarchy or Republic. I'm of the opinion that this tactic is mostly useful in the earliest of early games. By the time I'm getting close to one of the better governments my cities are fully developed & just need pop. At that point I'm looking more at the long-term and am view O(lt) as more important in my judgements than O(st)

[Aeson]

Pop rushing and corruption

One thing no one has mentioned in this thread, is how pop rushing affects highly corrupt cities. In those cities its trade off is practically nil, as no matter how large the city gets, they will only be able to produce food. A completely corrupt city should always be pop rushed at the level where it is producing the most excess food. If a city has three 3+ food spaces it can use, then it should be pop rushed whenever it hits a pop of 4. This doesn't account for civil disorder, which would limit the max pop differently depending on luxuries available and difficulty level. All that said, I usually keep each city that is being pop rushed down to 1, as its just easier to spot which cities need to be rushed each turn.

Another thing, pop rushing can be worth 39 shields for each pop point if you are willing to switch production after each step up in the unit/improvement cost scale. Though if an improvement costs 60 shields, it's going to take 2 pop points either way. One benefit of using production steps is that on higher difficulty levels it alows you to actually build things that cost more than one point without having to worry about going into disorder by growing to size 3 or 4 or however large a population you would need to finish the unit or improvement. Instead you just keep knocking your pop back down to 1 at each step.

[rah]

I'm surprised everyone has overlooked one aspect of the long term benefits here.

Suppose you're rushing archers. What are those archers gonna do? TAKE OTHER CITIES. So you can spend those pop point on building settlers or TAKING AI cities. Depending on circumstances and territory will determine the best path. A few extra cities won early have a tremendous impact in long term benefit calculations. Especially if taking those cities helps you create the Vassel AIs that you can rape/milk for centuries using diplomacy. Done right, they will give you every tech and all the gold they have. That's worth a few pop point. Too much of the dimplomacy seems to be based on the size of you peni..... I mean army. So you better have a strong one.

It probably muddies the water enough that it will be hard to quantify. But please continue, because the discussion highlights the factors that you should be considering when you make that decision, even if it's all theory.


RAH

[absimiliard]

Corruption is actually easily dealt with. Basically it's just an alteration in the local value of A. If A = 2F + 1S + 1C ,then in a totally corrupt city you reset it to A = 2F. This doesn't change the equations, but it does effect the value of O(lt) and O(st). I think the effects on O(lt) are much higher since it includes a lot of A.

So here's an example city, now using 2S as the simplified value of A. (A nasty thing to do, but I'm too lazy to figure out complex math quite this early in the morning.) O(st) = G * 2S. This means short-term rushing is practical at values of G < 20. For O(lt) it's just as striking, if not moreso. O(lt) = G * 2S + 25S. This means long-term rushing, with an MP, is practical in a corrupt city if G < 7.5. Long-term rushing in a non-MP city remains impractical economically.

Excellent suggestion on step-rushing by the way. I hadn't thought of that yet. Very good idea indeed.

Rah, while I don't dispute that you're correct, 'cause I think you are, I'm not sure that's germane. I'm only talking about the point of profit in a rush economy. You're talking strategy based on the use of your rushed units. I'm totally trying to stay away from strategy because I don't think it's easily quantifiable. For my equations purposes it doesn't matter what you do with your profit they only exist to tell you if there is profit.

Not sure how I could change that. Perhaps strategy will always remain unquantifiable.

[rah]

Originally posted by absimiliard

Rah, while I don't dispute that you're correct, 'cause I think you are, I'm not sure that's germane. I'm only talking about the point of profit in a rush economy. You're talking strategy based on the use of your rushed units. I'm totally trying to stay away from strategy because I don't think it's easily quantifiable. For my equations purposes it doesn't matter what you do with your profit they only exist to tell you if there is profit.

Not sure how I could change that. Perhaps strategy will always remain unquantifiable.

That's the problem with all the QUANTIFICATION that is being done in this forum. Money Opportunity is critical in any economic equation concerning PROFIT. It's a big difference if you invest your money at a low interest rate or do a more high risk investment. (and personally I don't think SOME early conquest is that high risk). It can't be ignored.

I have seen the same problem with people quantifying civ characteristics or anything else. To look at long term benifits, leaving out conquest/diplomacy profits eliminates the most lucrative investment.

Having said that, please continue what you're doing. Even military inclined leaders need to know the cost of the conquest to determine if the cost/risk/reward ratio is worth it.





RAH

[absimiliard]

I think quantification in general is a good thing, kind of obvious since I'm goind it here, but I would agree that the quantification of the civ-traits is flawed. The analysis we've seen so far has failed to take into account inflation, which is why the early game low-value stuff like expansionist isn't total suck. The problem as I see it is that no one yet has a good value for inflation in Civ III.

I see my work as providing input for strategic decisions, not as deciding them. Heck, even a highly unprofitable rush may be great if the strategic or tactical situation requires it.

As an aside, I view early-game conquest as a fairly low-risk investment as well. I'm generally fairly sure I can take the AI so basically it comes down to are the units worth the cities gained, and that's a trivially easy question to answer "yes" to.

--------------------------------

I've been thinking about population transfer mechanics again. I'm an old Stars! player, and in Stars! your pop is really all that matters. It's all about the exponential growth curve. Civ III is looking fairly different though since I've been refining my economic equations.

Basically I thought early on that building a Worker in a high-food city and using it to rush something in another city might be profitable. To be sure there's some wastage in the 10S cost of the Worker, but I thought it might work in general.

I still think pop-transfer can be profitable if you are considering O(st). However in the long-term the high value of A makes up most of O(lt). This means that it's not the cost of the lost pop that's your main loss, it's the unhappiness, and that won't change even if the pop is from another city originally. So I think pop-transfer is only practical for short-term returns, or for corrupt cities where the value of A is low.

---------------------------------

As a final aside. I think I miscalculated the cost of an MP. I said 5S to build and 20C to support. That should be 10S + 20C. I think Warriors cost 10, and they're the cheapest MP I can think of. This should only slightly change the equations though. The actual math is left as an exercise for the student.