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A theory of interest for Civ2

Discussion in 'Civ2 - Strategy & Tips' started by Peaster, May 5, 2006.

  1. Peaster

    Peaster Emperor

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    I have been asked to explain my Civ2 theory of interest by fellow GOTMer, AliArdavan. It is really just basic RL interest theory applied to Civ2, but it seems many players don't know about it. For simplicity, I will leave out most of the math unless people want to see that. Also (all modesty aside) I have been pretty successful with this in playing over 15 GOTMs, and I am pretty sure many strong players (eg SlowThinker) accept interest theory as common sense. Here are some of the main points:

    1) Obviously, it is better to get a free city (from a hut, for example) in 4000BC than in 2000AD, because a city can be used to make more cities. Likewise, a free unit, an extra 100g, whatever - they are all better in 4000BC. Thus, the "present value" of almost any item is greater than its "future value". Based mainly on experience, I estimate typical values "go down" about 5% per turn in most Civ2 situations.

    For example, if you hoard 1000g for 10 turns, instead of investing it, you just lost about 500g. This principle applies to temples and defensive units sitting idly in your cities - they are costing you! (of course they may also serve a valuable purpose, but you don't want to overdo them).

    2) The 5% estimate applies pretty well to settlers/cities too. You should be able to double the size of your civ approx every 14 turns (math omitted - but it's related to the 5%), assuming no major wars/etc get in the way. Eg, 1 city in 4000BC becomes 2 cities in 3300BC, 4 cities in 2600BC etc until you decide to level off. This assumes you invest in growth and not armies, etc.

    3) If you get paid 1 gold per turn, for example from the taxes of a small city, you are getting an "annuity". At 5% interest, it is worth the same as 20g (math omitted again) paid in advance. This "x20" factor is important in many Civ2 investment decisions.

    Example: You have a city making 10 shields per turn and want to build a factory there. The factory will increase production by 5s/t, which is an annuity worth 5x20 = 100s. However, the factory costs you 200s. Bad deal!
    And I didn't even add in the 4g per turn cost, which is worth 4x20 = 80g up front. I would not build a factory in a city unless it's making about 24s/t (or can do so in the very near future). And I almost never have such cities (see point 4 below).

    Example: The KRC wonder costs 300s, which doesn't make sense in a city of size less than 15 (15x20=300). You can evaluate Adam Smith and the Colossus in much the same way.

    4) If you can find a way to get more than 5% back per turn (eg thru trade bonuses) then DO IT! If you cannot do better than 5%, invest in growth (eg settlers), which usually returns about 5%.
     
  2. pi-r8

    pi-r8 Luddite

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    Um... what if there's no more room to build cities? Since, after all, that's what most of the game is like.
     
  3. Peaster

    Peaster Emperor

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    What do you mean by "no more room"? On a normal map, you can probably fit over 200 cities easily, though I don't really want that many.
     
  4. pi-r8

    pi-r8 Luddite

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    By the time you can build factories, all the good city sites have long since been taken. Are you suggesting building cities on Antarctica, or what? I guess you could build them in between your existing cities, but all that does is turn one good city in to two worthless cities.
     
  5. Peaster

    Peaster Emperor

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    That depends on your style, doesn't it? For example, in GOTM 63, where this question arose, I still had plenty of room to expand even after launching (in 570AD with about 60 cities). I built mostly vans in the last 40 turns of that game, since they were a better investment than factories or engineers.

    Having said that, I usually play for early conquest and finish long before getting the Ind'n tech. So, I actually use interest theory in lots of other ways, but the factory example is easier to explain.
     
  6. Ali Ardavan

    Ali Ardavan Mathematician Moderator Civ2 GOTM Staff

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    Very interesting. Thanks Peaster. I have to reflect on this.
     
  7. Ali Ardavan

    Ali Ardavan Mathematician Moderator Civ2 GOTM Staff

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    I have been thinking about this off and on today. The math is elementary, the approach is novel (to me anyways).

    Here is another angle using the same theory. From this angle the return ought to be 10% a turn:
    I build a settler for 40 shields and use it to build a new city. For simplicity I am ignoring the fact that I lose one citizen in the process.

    Choice 1: shield city: I build on a forest and put my citizen on a forest as well. This city will never grow. Nor will it produce any arrows. Just 4 shields a turn every turn. Thus I just turned 40 shields into a stream of 4 shields per turn or 10% return per turn.

    Choice 2: coastal city: I find a plain or grassland by the ocean and build a city there. My first citizen, and all subsequent ones, work on water.
    Depending on the form of my government and wether I started on plains or grass, I max out at size 2 or 3. This city produces 1 shield and 5-10 arrows. If we convert arrows to shields at the rate of 2 per 1 we get 3.5 to 5.5 shields for our investment of 40 which is roughly 10% again.

    Let us see how other investments do in comparison:

    A library, at 80 shields, should return 8 shields or 16 arrows per turn. Add its cost of 1 and you get 17. The science out put of your city needs to be 34 before the library to get this kind of return!

    A barracks, at 40 shields, should return 4.5 shields per turn (the .5 for its cost of 1). Assuming the city only makes miliary units it should be at 9 shields output for the barracks to make that return.

    A temple, at 40 shields and maintenance of 1, should return 9 arrows per turn. It can make 2 citizens content which is worth only 4 arrows.

    A road takes a settler 2 turns to build. This causes the loss of 2 turns of revenue or 8 shields. Thus the road costs 8 shields or 16 arrows. It should return 1.6 arrows per turn. It returns 1 in commerce. But it also speeds movement. Evey time a unit passes through it saves 2/3 movement cost.
    For a 40-shield unit that is 4*2/3 shields or 16/3 arrows in saving. If the road is used even once every 8 turns (by a 40s unit) its total return is above 10%.
     
  8. Ali Ardavan

    Ali Ardavan Mathematician Moderator Civ2 GOTM Staff

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    If you apply this theory widely you will reach the conclusion that the only worthwhile thing to build in Civ2 are settlers and caravans (I am ignoring combat for simplicity). And in fact that is what the ICS strategy is all about. No wonder it is so powerful.

    This reminds me of a note I once saw in this forum. It said there are only 2 rules for being a good Civ2 player.
    1. When in doubt, make a settler
    2. When not in doubt, admit the possibility that you may be wrong and go back to step 1.

    However, there are caveats. Besides the obvious ones there is the issue of compounding. Best example of this is science city. Its individual components may not be worthwhile by themselves but when put together they compound and enhance one another's effects. Typical scince city, at its max, can produce 800-1000 science (not to mention a decent amount of gold and shields). Going with the low figure of 800, and using the 10% return theory, any investment below 8000 arrows or 4000 shields is worthwhile. Yet the infrastructure in the city is worth about 2000. Add to that the cost of land improvement (average of one road and one irrigation per tile) 20x(5+2)x4=560 and it is still a great deal. Spread the infrastructure among several cities and it will be an awful deal.
     
  9. Peaster

    Peaster Emperor

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    You really caught on to this idea quickly! It took me a pretty long time to work it out, but these are the kinds of thoughts I now have in every game. You may be right that 5% is a tiny bit low, but you are ignoring some factors. The lost citizen represents one food box - rather hard to evaluate I admit. But we did a very detailed study of this at Apolyton last year, and IIRC it's worth about 25 shields (unless you are celebrating in a republic/demo govt, for example). There is also a delay of approx 3-4 turns from a settler's birth to the time it can found a new city, which reduces the value of the investment a bit. The settler also requires support during this period.... so, with all that included, I think 5% is not far off.

    BTW: I usually consider 1 shield = 2.5 arrows, but this is also debatable.
    Improvements are usually rush-bought (no?), so to me a library costs 160g = 160 beakers, and I would want 160/20 = 8 new beakers per turn. So, a city already making 16 beakers could get a library.

    But as you know, I mainly play for conquest and don't usually care much about beakers. Also I don't grow very large cities, so basically I never build libraries.

    Interesting... I never thought of barracks using interest theory. Maybe because I usually RB or IRB units in those cities rather than wait for the shields. So, my rule of thumb for barracks is that it pays for itself as soon as it produces 2 (needed) vet elephants or the equivalent [because it raises the value of those 80 shields to 120 shields]. On second thought, maybe it does so even sooner [since the barracks can be RB'd cheaper than units can, and it can always be sold]. I like a barracks city to produce 5 shields per turn, to make IRBs work out nicely.
    I almost never build em [mainly because Wonders like HG and MC are more cost-effective], and I agree with you. Except that with a 5% rate, you'd only need to get about 5 arrows back, which you could probably do in a well-developed city currently needing Elvises.
    I mostly agree, and I probably build roads sooner than most ICS players. But roads are tricky to evaluate. For example "It returns 1 in commerce." is true IF some nearby city constantly uses that tile. In my games, that is likely if the tile is a special or shielded grass, but not too likely otherwise. Also the cost of the road is paid up front, while the movement benefits may not kick in until much later.
     
  10. Ali Ardavan

    Ali Ardavan Mathematician Moderator Civ2 GOTM Staff

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    Attribute that to a good teacher ;)
    Good point. Here is a revised version:

    If I produce my settler in a city of size n it will go down to size n-1. Assuming the food box is not overfilled at this point, I need 10n bushels to get back to where I was food wise. Since typical small city produces 2 extra foods this takes 5n turns. Add to that 4 turns of support for the settler while it finds a new spot to build and (assuming 1 bushel per turn) you need extra 4 bushels or 2 turns. If the settler did not reduce my size and did not need food support I could have switched two citizens from shielded grass (2F1S) to forest (1F2S) in a city producing two extra foods and not increase my food storage. Thus the 5n+2 turns that took me to restore my food storage cost 10n+4 shields if I immediately up my size after the reduction (and more if not).

    Now the cost of the settler is at least 10n+44 shields not 40. So the return on my investment is 4/(10n+44). For n=2 to 6 this is 6.3%, 5.4%, 4.8%, 4.3%, 3.8% which averages to 4.9%, the value you reached empirically. My earlier estimate of 10% was obviously too much because of ignoring reduced size.
     
  11. Ali Ardavan

    Ali Ardavan Mathematician Moderator Civ2 GOTM Staff

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    Peaster,

    Another issue I have been thinking about, and I am interested to know if it is part of your theory, is early expansion.

    When I compare my early results (-1000) to yours, I see you have 2-3 times as many cities as I do. In this part of the game I do not make infrastructure; just settlers and the occasional unit when I have to. Thus, I suppose the only difference between us is that you place your cities much closer than I do. You probably put cities 2-3 tiles apart. I put them 4-5 tiles apart. The problem is I do not think this by itself explains the final outcome. Extra 2 turns in placing a settler that takes over 10 turns to build cannot mushroom into a ratio of 2 to 1 in 60 turns.

    Your theory of interest does not explain this either. Using 5% a settler is worth 2 shileds per turn. Since I take an extra couple of turns to make my cities I lose about 4 shields. This should work out to 40 shields per 10 cities which (ignoring size reduction) is 1 extra settler! Certainly this does not explain the difference. Even using a 10% rate would not.
     
  12. TimTheEnchanter

    TimTheEnchanter I...am...an Enchanter!

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    A couple thoughts on this:
    1) Compounding is a powerful thing. Peaster talks about a 5% payoff. Applying this on a compound basis, he should double his investment just about every 14 turns which, probably not coincidentally, was his goal for doubling his number cities. If instead of getting 5% return you get 10/13 of that (3.84%), you double about every 18 turns. After 60 turns at these rates from an initial "investment" of 1 unit, the 5% gives you a payout of 18.7 units, while the lower rate gives you 9.6 units. This is very close to a 2:1 ratio.

    2) Perhaps you're underestimating the cost of moving those extra couple squares. Are you building roads between them? if so, that adds to the delays. If not, your travel times can compound quickly as well. One thing to consider is the free road a city provides. A classic ICS grid only requires one road to be built between cities to allow free movement to a city 2 squares away. If you put the cities 4 squares away, you need to build three roads per city to allow the same rate of movement for future settlers. The 2 square distance lends itself to the 14 turn doubling rate: 10 to build and move 1, 2 to build one road, 1 to move to city location, 1 and start a new city. With cities 4 apart, it's almost impossible to sustain this rate of expansion even for the first few cities. You would have to build in 10 and spend the next three turns moving to the city location to build on the 14th turn. There's no time for ANY roads without slowing down the rate of payback. Even then, you quickly start having to cover more ground for your capital city settlers and there are no roads in place to speed you out to the outer rim of cities.

    3) Extra travel time you spend slows down your ability to fill the food box a second time as well, meaning you are less likely to maintain production as fast. As the travel times get worse, so does the impact on restoring your cities to full production.

    4) another factor is how quickly you rush your production and how youbalance food/sheilds/arrows. In the early game it's tough to get a food box full in less than 10 turns. You would likely have to do some rushbuying and focus on sheild production at the expense of arrows to have a settler ready to go when that food box is full. If Peaster is more aggressive with this, it might make an additional turn or two difference on the "doubling" rate.

    Like I said, these differences compound themselves very quickly. The rate I used was less than 25% off the 5% rate Peaster strives for. the lower rate resulted in doubling every 18 turns instead of every 14 and within 60 turns, a 2:1 ratio is to be expected.
     
  13. ElephantU

    ElephantU Deity

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    Perhaps we should have an "ICS Comparison Match" with posted saves every 10 turns...
     
  14. Ali Ardavan

    Ali Ardavan Mathematician Moderator Civ2 GOTM Staff

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    Thank you Tim for a very informative response; as always.

    I reexamined my analysis and also looked at data from recent GOTMs and here is what I found out:

    1. During the initial phase when I am building only settlers it takes me on average 20 turns to double my number of cities; it takes Peaster 15. (Hence, in 60 turns at -1000 he has about double the number of my cities.)

    2. As I said before, the fact that my cities are about 2 tiles further away (4-5 versus 2-3), by itself does not explain the difference of 5 turns; it only explains two. However,

    3. I neglected to account for the food support of the settler. Thanks for pointing it out. Since cities in that era typically produce 2 extra foods per turn, the 1 food per turn support is equivalent to a half turn delay. Thus, by moving 2 tiles further away I lose 3 not 2 turns. This still leaves about 2 more turns to account for though which brings us to the next point.

    4. As you correctly guessed, I neglected to account for road building. I do build some roads in this period though I do not necessarily connect all my cities. Each road takes 2 turns which accounting for food support turns into a 3 turn delay. If I build 2 roads for every one that Peaster builds that more than explains the difference. I do not know for a fact how much road building he does, but with cities close by he needs fewer of them than I do.
     
  15. Peaster

    Peaster Emperor

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    Tim hit the main points of early expansion pretty well. I have been thinking pretty hard about these things ever since I read DaveV's guide to ICS, and I still do not have it down to a science. But Grigor and I (and others) did a pretty detailed study of early growth at Apolyton last year - something like Elephant's proposal - but it was on a highly simplified map (only grass and forests - no specials). IIRC we decided that doubling every 12 or 13 turns was the ideal. That required raising the tax bar to the max and using the gold for IRBing settlers only - not even a warrior for defense.

    http://apolyton.net/forums/showthread.php?s=&threadid=132147

    IIRC we didn't find the correct "rule" about roads. In my games, I tend not to build any roads until I have about 3 cities, and then only single-tile roads that connect 2 nearby cities. I prefer roads on shielded grass since those tiles are likely to be worked a lot, and then the road adds an arrow per turn. By the time I have about 8 cities, I want to connect them all, because I usually try to defend the whole network with just 1-2 dips or horses. And by then I am starting to build vans/etc that need to move.

    Other factors: An early nomad from a hut can instantly double the size of your civ (in an extreme case). I don't usually build many units for exploration, but if I have any spare ones I send them out. Also, I favor food specials (or plain old grass) in the early game over arrows or shields, because a food shortage is more likely to slow you down. I like to get monarchy quickly, but it is not so high a priority for me as growth, so I do not raise the science bar until I have a few cities and a decent cash flow. Finally, watch for a good chance to use the size-1 trick.
     
  16. TimTheEnchanter

    TimTheEnchanter I...am...an Enchanter!

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    This is what I was getting at with my fourth point. I would not underestimate the effects of this on the differences between people's games (e.g., Ali v. Peaster).

    So many of us have been conditioned to "Max science" and "Get to Monarchy ASAP", we have less gold to RB. I wouldn't be surprised if one of the turns difference between Ali and Peaster is due to the ability to rushbuy more quickly. In the very early game, without food specials it takes 10 turns to fill a food box. however settler production will lag behind and requires IPRBs to keep pace with the food box.

    Those of us focusing on science will not be able to keep up on the shields for the settlers because we will not have enough gold, but I guess Peaster can. Getting to monarchy will usually give a food surplus of 3 instead of 2, thus cutting the food-time in a "standard" grassland city from 10 to 7 thanks to the auto-irrigated city square (I'm assuming Settler-Irrigation is not worth the 5 turn investment in this analysis) and perhaps a bit more in food-special cities. It would be interesting to see the math comparing the "cost" of stressing science (in delays to earliest expansion through limited rush buying) vs. the "cost" of delaying monarchy (delaying the accelerated food growth) a bit later because science was not stressed. Based on what you guys are saying, my guess is that once you compound the value of the early RB benefits, it probably outweighs the later benefits to earlier monarchy. Then again, maybe it's counterproductive, and the roads/travel differences are even more of a factor than we realize. :crazyeye:

    Either way, this is really interesting. It's amazing that you guys can get me questioning the well established, almost universally accepted, opening strategy nine years after the game has been released. :goodjob:
     
  17. Peaster

    Peaster Emperor

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    For me, it is hard to evaluate the benefits of monarchy precisely, but as a ballpark figure, I guess it boosts production by about 20% (?). My style probably delays the advance approx one oedo cycle on average, or 4 turns [not much more because my extra city(s) eventually boost my science rate]. If that happens around turn 40, I've lost the 20% for only 10% of the game, for an overall loss of only 2%.

    On the other hand, I've had approx 30g extra for IRBs, which might translate into an extra city (because of compounding), which might be an overall boost of perhaps 10% or more.

    I am NOT sure these numbers are very accurate, but I think a small delay in getting to monarchy is not bad.
     
  18. TimTheEnchanter

    TimTheEnchanter I...am...an Enchanter!

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    That's along the lines of what I was thinking. although instead of "an extra city" it might be more accurate to think of it as "having had each of several cities a bit longer". Basically the compounded value of the early gold investment (which manifests itself in more cities, sooner, and thus an ability to start catching up on science at a lower sci/tax rate) probably offsets the potential food boost that monarchy may give you for some period of time.
     
  19. ElephantU

    ElephantU Deity

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    Some further thoughts on the "revise early tech priorities" theme: with no starting techs you can only IPRB two rows of a Settler, the first and the last. If you set your first tech to Bronze (Phalanx) or Horseback (Horse) you add the second row in 10-12 turns, half that if you start with two Settlers. If you make your second tech WarCode (Archer) or Wheel (Chariot) you will add the third row.

    Unfortunately, all of this is off-path for Monarchy, so the tradeoff needs to be quantified - which "pays" better, getting to Monarchy or adding 2nd and 3rd IPRB rows?
     
  20. TimTheEnchanter

    TimTheEnchanter I...am...an Enchanter!

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    This would obviously need to be confirmed, but I think this is a case where the benefits will be hard to justify. Getting the 2nd row in 10-12 turns really doesn't help much because by then you're beyond 20s in your first settler. It will help on the 2nd settlers, but the help is not likely to be meaningful. Similarly your third tech probably doesn't start to help you until the 3rd round.

    The "standard path" for us science/trade fiends would probably have you go Alpha-Bronze-CerBur-Code-Monarch with HBR being another possible candidate for the 2nd tech). The 10-15 turn delay in getting bronze available first instead of 2nd probably only affects the two '2nd generation settlers" and even if you were able to buy 4s/row the difference is very small. With the 2nd row available, you can buy 4s to 20s and 4s to 40s for 16g total. Without the 2nd row available, that same 16g will buy you 7s to 40s. Depending on how many shields you're making, that one extra shield may not even save a single turn getting those 2nd generation settlers out. Either tech path should give you the 2nd row by the 3rd generation of settlers so the choices don't matter.

    The interesting choice is whether to get the 3rd row as well. There, if you are looking at the cost of rushing to 20s, 30s and 40s if you're buying 4 per row the cost for those 12 sheilds goes from 24 to 31. The 24g would not quite buy you the last row, so it is likely to save you one and maybe in some cases two turns, and if you do both techs right away, it might make a measurable difference. The 2nd row would impact the 2nd gen (cities 3&4)and the 3rd row should help you on the 3rd generation (cities 5-8) and beyond (I could be off on the timing of these). However, doing this will delay monarchy by two full techs, and as you get up to the 6th and 7th techs this could be a significant delay. I would think the delay in food relief from monarchy would outweigh the rushbuying advantages.

    I think the first question is: Do the gold savings of the IPRB allow you to complete the settler a turn or more closer to when the food box is full? If you can already complete the settler the same turn the food box is full, having the IPRB is of little marginal value. If you can't, then for this to be of value, the IPRB savings must be enough to actually shave a turn or two off the settler production time. If the gold savings only shave a turn off every other settler, it's almost assuredly not enough to justify the delays.

    By the way, I don't think you could start with Bronze because you wouldn't have the prereq for wheel or writing and War code would be suppressed as the 2nd tech. to get 20s and 30s units on your first two techs you would have to go Horse-Wheel. Assuming you still went for Monarchy as soon as possible after that (tech 7), it would delay trade to at least the 10th tech. I'm not sure what the "rate of return" is for MPE or Trade vans, but they would be delayed as well.
     

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