Valuing long term investments

[couerdelion]

I have made some remarks in other threads that argues for a slightly more complicated approach to valuing long term streams of returns from certain civ investments. For example, a settled GS can not simply be valued as

[1h x (1+prod multiplier) + 6b x (1+science multiplier)] x No. of turns remaining.

The problem is that returns made far into the future are less important at the time that the decision is made than those returns made in the near future. To resolve this we need to introduce the concept of a discount rate to value these long term investments but this leaves open a very big question as to what is the rate we should use.

Let’s take a simple example of a GM. The choices are to settle it in Wall Street City for 1f + 18g each turn or to go on a trade mission to generate 2000 gold. For ease of calculation I’ve going to assume 1f=2c and in this city (2c is worth 5g). So I’ve got the equivalent of 23gpt from the settled GM

If we have a discount rate of 1%, then the value of the future stream of benefits from the settled GM is 23gpt / 1% = 2300 gold.

-> it’s worth more to settle GM

If we have a discount rate of 2%, then the value of the future stream of benefits from settling is 23gpt/2% = 1150 gold.

-> it’s worth a lot more to send on a trade mission


Now I’m not proposing here to argue what GPs should be used for: that’s for a completely different thread. The question here is how we determine a “discount rate” or it’s companion – the “multiplier” (above the multipliers are 100 and 50 respectively)

In the investment world, the discount rate you would use would be a risk free rate of return – the rate you could earn on investments without any risk. I think a natural place to start in the Civ4 world would be to look at the rate we are actually earning on our Civ assets.

Unfortunately we run into another problem here because there is no simple framework that even exists for valuing these assets. However, this is not too much of a problem since we are not so much concerned with the absolute value of the asset but about how much it has grown over a given period. What we can then do is look at our core output of gold, science and hammers from various information screens and see how these have grown over time. If we can assume a linear relation to the asset value then the increase in the amount of core output is the same as the increase in the total value of our assets.

The values I would suggest looking at are Mfg (Demographics screen), Science (Finance Manager) and Gold (Net of Expenses). Food, population and culture are unlikely to be meaningful since these are interim assets that are ultimately used to generate the core output.

The three values can be looked at separately although these will probably give different rates of growth so some method of combining them into one number would seem appropriate. I personally use 1h=1.5science=1.5gold. This is likely to be close to numbers that others use and I would not expect that a different choice of numbers would produce a different result for the calculated discount rate or multiplier. Let’s call this figure Core Output in turn t, or CO(t).

Having calculated CO(t) at different times, we then have to calculate a turn based rate. This is a relatively simple calculation and is as follows

1 + increase_rate(t1,t2) = [ CO(t2) / CO(t1) ] ^ [ 1 / (t2-t1) ]

The ^ symbol represent “to the power of”

It is likely that the rate of increase in your core output values will go through sudden stages of growth. For this reason, I would take long term period in which to perform the analysis and do this over each different age.

The “multiplier”, mentioned earlier, is simply the number which we can use to multiply any constant turn benefit we achieve from some “asset” to place a value on that asset. It is calculated simply as 1 / increase_rate

To give you some idea of the figures, an increase rate of 1% will be achieved if you can increase your output by 170% over a 100 turn period, nearly 7.3 times over a 200 turn period, nearly 3000 times over a 400 turn period . At 2%, you’ll be achieving this growth in around half the time.

Those sorts of figures suggest that 1% is probably too small but 2% is too high (at least over the course of the whole game). I plan to get some figures from a recent game to see what they look like.

Armed with a figure such as this, it should then be possible to decide between some general strategic alternatives (eg grassland workshop vs grassland plains + rep.specialist, towns vs rep specialists)

[Krill]

Now this is interesting

And I'm sorry to have to ask thi, but my maths skills are really sucky it seems. What is the basic meaning of increase_rate(t1,t2)?

[couerdelion]

Originally posted by Krill
Now this is interesting

And I'm sorry to have to ask thi, but my maths skills are really sucky it seems. What is the basic meaning of increase_rate(t1,t2)?

Sorry for taking a bit of a shortcut there.

It’s the rate of increase per turn, between turns t1 and turns t2.

[Krill]

ie the differential? That kinda thing?

[Diadem]

There's a few flaws in your math. A constant 170% increase over 100 turns means, indeed, a factor 7.3 growth over 200 turns, but it's 'only' a factor 53 growth over 400 turns.

But that's a small error, not really important in the overal argument.

I have more serious problems with your usage of percentages though. Your definition of discount rate seems to be rather arbitrary. Why do you define it like that, and why would it be equal to your growth rate, in this case?

[couerdelion]

Originally posted by Krill
ie the differential? That kinda thing?

Not quite the differential. If you were increasing at 1% per turn (on average) then when your output is 10, it would increase to 11 in around 10 turns. When it is 1000, it would increase to 1010 in just one turn.

The differential measures the absolute “rate” of increase while the “rate of increase” definition I am looking at is a proportional increase figure

[Krill]

OK, thanks for that explanation Couerdelion.

[Diadem]

I think the assumption that the value of a certain output is reversely lineair with your total output makes sense.

If your total output is 10 beakers, then 10 beakers are worth 100% (= 1). But if your total output is 100 beakers then 10 beakers are only worth 10% of what they originally were.

So we first have to find out the growth rate of your empire. From my experience it seems to be more or less exponentional. This is of course what we would expect based on game mechanics as well. Your growth depends on your output, and your output depends on your growth. Mutually enhancing factors are a good recipe for exponentional growth.

So we can write our growth as: g(t) = g(0)*exp(a*t) with a some for now undetermined constant, and g(0) is of course your size at the point where we start measuring (which we definie as t=0).

So the *value* of a certain investment at a specific turn is: v(t) = p(t) / g(t) where p(t) is the profit of that investment at a specific turn. For great people added to a city this profit is of course constant - a super specialist makes the same amount of hammers / beakers / gold each turn.

The total value of a investment we now find my integration. Assuming that the number of remaining turns are much greater than 1/a (so that exp(a*t) >> 1 at t = last turn) we can savely integrate to infinity. Furthermore we'll assume that profit of an investment is constant, as in the case of a super specialist: p(t) = p.

We then find for the total value of an investment: v_T = Integrate[p/g(0)*exp(-a*t)] = p/(a*g(0))

A trade mission, for example, has a total profit that is equal to it's gold gain divided by g(0). So (as expected) the value of g(0) cancels out if we want to compare a super specialist great merchant and great merchant trade mission. We find that we should add the great merchant as a super specialist if:

p/a > TM

with:

p = profit each turn from your super specialist great merchant
TM = profit of a trade mission
a = some yet undetermined constant

Interestingly enough this is the formula that couerdelion began his post with. So we now know where that comes from All we now have to do is find this constant a and we're done.

[Diadem]

In a current game I'm playing over the last 200 turns I grew with about a factor 3. So a = 0.00549. With 175 turns remaining in the game that does not satisfy that the number of remaning turns is much greater than 1/a. In fact it's smaller. Still the error introduced by integrating to infinity won't be extremely big. So let's do that anyway.

A great merchant in my capital gives 18 gold a turn (200% bonus) and half a population point, which becomes half a specialist since the city is bigger than 20 already. The gpp don't matter much anymore and I'm not running representation. So that's really 9 gold per specialist, so 5 gold for half a specialist. So 23 gold per turn is a good estimation for the worth of a super specialist great merchant. This is rather low, normally it'd be more because your extra half a population point could work a town instead of be a specialist. Also you could run representation, or your gpps could still matter. Anyway, for me, right now, it's 23 gold a turn.

So a trade mission would have be bring in more than 4187 to be worth it.

Does anyone have any idea what trade missions on a normal sized map on epic speed bring in? Because I've never even done one, I think, so I haven't the faintest idea

[couerdelion]

I've had around 3500-4500gold on a large map/epic speed sometime into Renaissance period. But this was one of the larger examples. 2500-3500gold would seem nearer the mark for most trade missions at this time.

The growth rate figure looks a little low though it might simply be the “phase” of the game in which there are limited opportunities to grow. You may want to remove some large “one-off” factors that will mess up the calculations. Some that I can think of include Golden Ages, idle GPs, large unused treasury, start of war/end of war situations

Certainly at the beginning of the game, the opportunity to expand is high. Take a city with a floodplain and it will grow its science by 10% in 11 turns if it can work another commerce tile, or production by 100% if it can work a production tile. The early phase growth has to be one of the most significant and once resource techs are unlocked, this should sky-rocket.

[fed1943]

IMHO, any steadfast value of conversion food/gold is risky and

misleading.

If you care to look at SPDG - city Bananastary (Couerdelion knows the

game well) you will see a good example:

Two settled merchants speed the city'grow and said city has 11

cottages; so, the cottages will be worked, and later developed,

sooner.

The diference of commerce output with/without the +2 food can be

precisely determined until the turn that all would be towns in the

slower=without +2F version, so a global value, not a per turn one.

Obviously, the bonus gold (and here beakers,too) also present and

they are a per turn output.

Best regards,

[LDiCesare]

The asumption:

For great people added to a city this profit is of course constant - a super specialist makes the same amount of hammers / beakers / gold each turn

is false, as the amount does vary, depending on universities, markets and whatnot being built later in the game.
It's particularly obvious when going for cultural victories: You get a multiplier to your culture when you build cathedrals, broadcast towers and wonders, and depending on your civic. A typical late game multiplier for me is x3.5. When integrating that value for a cultural victory, the variables are quite simple and the choices easily computed, though the multiplier varies with time. In this case, a decrease of value with time is not a good idea, but I think it's the only case when there is no diminishing returns on the points granted by a specialist.

[Blake]

My first thought was also "how do you account for increasing multipliers?"
For example a 2000g cash bomb now, operating on +25% sliders (assuming all cash is turned into beakers via slider).
Vs a settled Great Merchant generating 10gpt (raw), but on progressively increasing sliders, starting at +0%, +25%, ending at +200%, ~+125%.

Of course sometimes you have an empire with ~+150% science multipliers (that assumes a large contribution from Oxfords capital) and 0% gold multipliers - in that case the cash bombs independence of gold multipliers works in it's favor.

But the other thing, is I disagree with the discounting sometimes. I don't believe that getting a tech earlier *always* results in more benefit than getting it later, there are some techs for which this is definitely not true, Philosophy being a brilliant example (while Taoism is still up for grabs), Civil Service being another good example (the sooner you get the +50%, the better). But there are also vast numbers of techs with no usable economic benefit (and in some cases, no applicable military benefit) and in some cases it's also entirely possible to research faster than you can get in infrastructure.

When considering lightbulb vs settle, you need to consider that a well timed philo lightbulb is priceless - that's what getting those 1000odd beakers worth faster is - priceless.

We could also look at the space race, I say that it really doesn't matter how quickly you research each particular tech, the only thing which matters is how quickly you research *all* of them, granted a few of these techs bring along additional benefits: Computers, robotics, genetics but a lot of them don't or are very situational benefits.

A lot of the time I find myself asking "How much will this option speed up the space race" : including when thinking what to do about a 1AD great engineer - settled in the Ironworks city he will indeed accelerate the space race. Indeed, a Great Engineer is often a question of Plop THIS wonder faster, or get EVERY future wonder faster, and here I feel that if you generate a lot of GE's it's best to go for the latter - getting every wonder in the game faster, rather than just the first wonder to come after the GE is born.

Then it gets even more complicated - in some cases I build a wonder (ie Angkor Wat) and I'm nearly completely neutral as to whether I get the wonder or refund, as such there's no way that the lump sum from an engineer is in some way more valuable than the "discounted" future yield of the settled GE - because in terms of hammers, the city is just twiddling it's thumbs and hoping to get the hammers converted to cash - which is not to say that I don't want more hammers there for future use, settling GE's can increase "burst" hammers (ie need something done asap) at multiple times in the game (obviously using hammer bomb is one single very large burst).

So hmmm....
What can be said is that the discount factor obviously varies greatly, and in some cases may even be negative (ie a cash bomb before Education may be worth less than a cash bomb after education and the round of university+oxfords building).

[couerdelion]

Lot’s of comments here so I’ll try to go through them individually.

Fed’s comment about food/commerce values is correct. I simply used the figure to enable me to come up with a consistent value rather than have to say 1f+21g/turn is worth x gold + whatever. The same can be said for the food/production comparative values and we could extend it also to comparative values for gold, science, culture and GPP. One thing that is clear to me is that food is relatively more valuable in small cities and cities with a granary. But I don’t think my assumption is necessarily wrong. You could always use a different value for food:gold conversion but I would suggest that the more important assumption in the whole calculation is the discount rate.

LDiCesare makes the valid point that further multipliers can be added to the city. This actually does not undermine the discount method approach. It simply means that you have a slightly more complicated formula which needs to allow for increases at future dates and you cannot simply apply a multiplier to the current turn-based gain from the specialist. Such considerations get even more complicate with early shrines which will later gain both additive and multiplicative bonuses.

Blake’s comments I will have to think about because I find many of the points contradict the framework – that is to acquire as many “assets” as quickly as possible. Part of the problem may be that my measure of valuing the existing infrastructure is to add together the three fundamental measures of output (gold, beakers and production). There will be times within a game when you might experience low (or negative growth) as part of a longer-term strategy. But these short-term dips should simply be just that, a temporary blip in the longer-term trend towards greater productivity, technological advances and wealth. In fact, what is happening in all these situation is that there are other “assets” that you are investing in which will give a better long term return than one that adopts a more basic, turn by turn maximisation of growth.

One obvious example is an early rush strategy. During the military build-up, your costs will be mounting without a corresponding increase in wealth generation. Since I don’t have any comprehensive framework for valuing military units, these are simply not taken into account but basic inspection of the strategy will reveal that those units have quite significant value. Quite apart from any other strategic gains they will give you, the units will be converted into new cities which will ultimately grow faster than you would do simply growing organically – at least that is the hope of the rush strategy.

A similar thing can be said for CS (or other slingshots) where early growth is put to one side while attempts are made to acquire some “state-of-the-art” technologies. These in turn will be used to generate the production, gold, science that was left as a secondary objective during the slingshot.

That said, there may be features within the game which naturally attract higher or lower growth phases. Early game growth might be rapid and then there could be a small stumbling block as matching civs start reaching the limits of their expansion. I took a closer look at my current game at different stages and identified periods of high and low growth. While the general trend rate of increase was a little under 1.5% per turn, there was a period of 1720-100BC in which the rate was 1.17%, another of 455-995AD with 1.04% and a third of 995-1172AD with a rate of 0.79%. If I look at the situation at the end of each period there is a common theme – war. The first period ended with my capture of two barbarian cities so the military build up over that period clearly impacted on growth while the return on those investments was yet to arrive. The second period ended while I was still fighting my second (and final) war against China and had started a first war against Japan. The third period was right at the end of the first Japanese war so seems to suggest that this was not a very successful period for me. If I recall, the one city that I captured had 12 units that needed dislodging and for a long time, those stacks were just to staring each other out – not my idea of an effective strategy

But what I have also found is that there is a reasonably close trend line which follows a smooth pattern of 1.45% growth in core output per turn. At it’s most extreme point, the actual output and the trend line differed by 16% and this was the point above after I had sorted out the mess with Japan. I rather think that the message here was that I was simply punished for bad play rather than this being any natural dip in the game at this stage of development – though I wonder also if things like cultural battles are also things that will dampen growth.

Higher later game growth might be indicative of civics and techs that unlock the greater potential of improvements. Or it might be the fact that I am running Golden Ages now and these are being used to accelerate growth – I was careful not to measure output during a GA since this would mess up the general trend-line.

What I don’t really agree with is the idea that there are some techs that do nothing. They all do something, even if it is just to unlock other techs thereby getting you there earlier. Scientific Method is a very good example here. Perhaps you can give an example of a tech where there is no advantage in getting it sooner rather than later. I can agree there may be better techs but often it makes sense to get most of the as soon as possible within the framework of your strategy (which can include avoid a tech so that you can trade for it).

But I don’t quite understand why 2000gold is more important after Education (and I presume Writing too) than before. Perhaps this may your way of explaining the changing gold:science exchange rates that arise as the multipliers are built. It certainly makes sense to emphasise (via sliders) on one particular asset prior to the building of multipliers for the other – and then reversing this after they are built - but I don’t think this necessarily changes the value of gold itself – or not if everything is priced in gold - universities just make science cheaper. It’s quite possible that I have missed something here so I will ponder this matter over the next day or so.

[EDIT - IGNORE THIS RUBBISH. CASE OF WRITING WITHOUT ENGAGING BRAIN FIRST] One question about negative rates is what this implies. If gold today is worth less than gold tomorrow then spend it NOW. This implies maximising the science rate because that gold doesn’t gain interest and the more you keep, the more you will loses as it devalues. This, however, seems to contradict the idea of storing gold in periods of low science multipliers to fund science in periods where there is a step change up in the science multipliers. [EDIT - /END IGNORE RUBBISH]

Please note that the discount rates I have applied are to an aggregate measure of science, gold and production and, in a more general sense, could include other assets which I have, as yet, not factored in. I see no reason why individual gold, science or production discount rates might fall in a certain period, but think it unlikely that all three would do so unless there is some other “unquantified” asset being acquired which we have not taken into account in our equations. (note that the acquisition of population for whipping is another example here)

As a general rule, any time I find myself disagreeing with what Blake says, is also a time when I should look at these things a lot more closely.

Finally, if I were to return to the worker calculations I made some weeks ago and use the 1.5% discount rate. If I assume that the worker builds three floodplain cottages and then spend eternity chopping, the value of that worker at the time it is completed is almost 1,350g !! And in the scheme of things, I’ve chosen some pretty menial jobs for the worker here. I’ve used Blake’s conversion rates of 1f=1.8h=2.5c and in this reckoning, a floodplains cottage is worth about the same (per worker turn) as a plains farm.

The case for worker first is getting stronger.

[zeace]

Great analysis guys,

I also think that some games have much higher rates than others. I played a game a while ago where Mansa, Gandhi and I were all on a single continent which was nicely divided in three by choke points. None of us ever built much of an army, and we were all great friends and trading partners.

This game flew along way faster than any other I've played. Our combined tech rate was amazing. As such, I never had a shortage of things to build. In this game, the rate of return was probably twice normal. Knowing that the game was going so fast and would be over sooner, the immediate gain things (lightbulbs, wonders and trade missions) were far more valuable than planting a GP in a city.

On the other hand there have been games where no one reasearches anything while they build massive armys that destroy one another... A slower rate should mean that planting a GP will be more valuable long term.