Rough payoff times and exchange rates

SevenSpirits at Realms Beyond said:

Civ is a game of investing. Citizens are investments which earn interest over time by working tiles. A university, to take a fairly simple building, is an investment which earns back its cost by producing beakers, and it has a better rate of return if the city's base beaker output is higher. Techs, units, and cities are all investments, albeit ones that are very difficult to analyze.

What makes Civ interesting is that it has an irregular and difficult-to-calculate set of available investments. How much return do you get from having Machinery? You wouldn't even know where to begin calculating that! (There are several quite complex things about it. Mostly, (1) it's required to research other techs in an incredibly long and interconnected chain and (2) it enables a military unit which has incredibly hard-to-judge value.) But even the straightforward investments (citizens, say) are not as simple as they look because investment costs are often large compared to per-turn output. For example, perhaps you have a city producing 5h per turn into a Library. It takes a dozen turns before the hammers you invested start compounding. You might as well have produced nothing for 11 turns and then 60h on the 12th. Compound interest is a powerful force and it's actually a pretty big deal to have an increasingly large pile of hammers sitting there in an incomplete building and not returning anything. The farther in the future the completion time, the more potential value you are forgoing.

So this is hard. We know that, and this is why we keep playing. It's deep. Also, trying to catch a glimpse of its true form, trying to gain little pieces of understanding, is pretty fun. Here is a tool I have been using to chip away at the mystery a little. It's some math and some estimates. They are wrong, but they've helped me wrap my mind around some things and their values.

1. Conversion of food, hammers, and beakers
1 food = 8 points
1 hammer = 5 points
1 beaker = 3 points

Notes: 1h into a worker/settler is worth 6 points. Food is assumed to be granary-enhanced food at a city of approximately size 7 growing onto early-game grassland tiles. Food is obviously stronger at lower city sizes since growth is cheaper there, and is also affected by other factors such as available health/happiness and the strength of the tiles being grown onto (including bonuses from buildings). Also note that a population point costs 2f and 1b every turn (citizens tend to add about $1 each to total maintenance) i.e. 19 points.

Some examples:
grass river farm: net +1f -> 8 points
grass lakeside farm: net +1f -1b -> 5 points
lighthouse coast: net 1b -> 3 points
grass hill river mine: net -1f +3h -> 7 points
grass pastured pig: net +4f -1b -> 29 points

Does anything strike you? There are gigantic differences in tile value which you don't really notice until you subtract out the 2f and 1b. Rivers are a big deal. Resources are a humongous deal. You knew they were important but seeing that a 6/0/0 tile nets you almost six times as much as a 3/0/0 tile might be a surprise.

How did I arrive at these numbers? I looked at tile choices I and other tend to make, such as slightly preferring a lake to a forest. I also looked at the actual conversion rate you can get between food and hammers by whipping.

You may wonder how to measure a cottage. It's trickier, because cottages get better over time. But it's not THAT tricky...


2. The interest rate
1 per turn = 20

Notes: That is 1 per turn forever. This is for quick speed. And, it corresponds to a per-turn interest rate of 3.53%. (1.0353 ^ 20 is approximately equal to 2. Take $20 and invest it at that rate and after 20t you will have $40. Alternately Trade the $20 for $1/t, wait 20t collecting $20, and then trade the $1/turn back for $20 - you get $40 that way too. So they are equivalent. Note that this is based off the assumption that in the actual Civ game where you make $1/turn, you will not be able to continuously reinvest your interest for effect - see the intro paragraphs. All in all it's very hand-wavy but the important thing is that it gives plausible results.)

Examples:
You might pay 20 food to grow onto a river grass farm which nets 1f/turn.
You might whip away a 2/1/1 tile (netting -1h/turn) to gain 20h.

Got it? Now to apply this to cottages. A cottage is worth
1b/t +
1b/t 6 turns from now +
1b/t 19 turns from now +
1b/t 45 turns from now

How much is 1b/t 6t from now worth? We can just discount its value according to the interest rate we've decided on. So a cottage is worth:

(1 + 1 * 1.0353^-6 + 1 * 1.0353^-19 + 1 * 1.0353^-45) b/t

which comes to just over 2.5 b/t. Seem reasonable?

A printing press cottage is of course worth more:

(1 + 1 * 1.0353^-6 + 2 * 1.0353^-19 + 1 * 1.0353^-45) b/t

which comes to just over 3 b/t. And naturally the civics help out even more, but you can see how it's worth about as much as a farm. What factors make farms better? 1) Production modifiers + whipping. 2) Small city size. 3) They become Bio farms, or other good late-game improvements later, while the increased strength of cottages is already accounted for in that 2.5-3b figure. 4) If you are not confident in being able to maintain control of that land or keep the tile from being pillaged, or you don't plan to work the tile as frequently. And what factors make cottages better? 1) Commerce modifiers e.g. Bureaucracy or libraries or banks. 2) Cottage civics late-game. 3) Faster build times. 4) No freshwater requirement. 5) Improved by Financial and/or golden ages.

How did I arrive at 1/t = 20? A bit of estimation based on whipping, and a bit of trial and error. You can see some of that trial and error in the way I played pbem 23. For various reasons (neglecting to consider that citizens cost about $1 in maintenance, and failing to account for the lack of continuous compounding), I was estimating then that 1/t was about 13 or 14. Which meant I overvalued lump sums and undervalued per-turn stuff. That's not to say that Novice and I didn't play that game really well, but you can see that at the end we were some giant number of beakers ahead but had a pretty poor tech rate, and it was costing us. And part of that was that I advocated not using a single great person for a permanent benefit, and another part was that I overvalued citizens working coast compared to buildings they could have whipped out.

... OK, I think that's all I've got that has crystallized into communicable thoughts. I'm not at all confident in these equations. But even though they are very inaccurate, and ignore so much context, I find them useful for getting a rough idea of how useful things are relative to one another. Thoughts?

P.S. Before anyone thinks that I play like a robot, in fact I play very intuitively, and that is the joy of the game for me. I love games that are deep enough that you simply can't play them by calculation. But I also enjoy analyzing what my intuition tells me - that's this stuff.

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