A Theory Contest

[Peaster]

Originally posted by rjmatsleepers

There is only a hidden cost of buying shields in the first row if there is a "time value of money" - ie if 1 gold now is worth more than 1 gold in 10 turns. AFAIK, it has always been difficult to establish this in Civ II.

I am sure there is a time value of gold in this contest, at least in the 3500-1800BC growth period when there are many good chances to spend it. Gold is good only for getting settlers out faster, and settler-value decreases each turn. I am pretty sure the correct interest rate is about 5 per cent (same as the growth rate).

I believe this also applies to most situations in real Civ2 I have seen (mostly ICS/conquest, with barbs rarely making much difference). But let's discuss real Civ2 in more detail later.

I believe the question as to whether you should rush buy the final row is different. If the city produces 1, 2, 3 or 4 shields, then the next settler will be produced anything from 8 turns to 1 turn earlier by rushing. The question is one of cost and benefit.

I have been thinking mainly about the case of a city producing 4 shields per turn. which seems fairly typical in this contest. For now, let's ignore the size of the city (assume it is size 2) and the option of moving workers around (but IMO they should be on forests). I have tried to find the best ways to spend gold. Here are the lowest costs of saving one Settler-turn (Row 1 figures were adjusted upwards using a 5 per cent interest rate. Also, I assumed 1 saved-turn = 4 shields there):

In Row 1:
With 6-9 shields in the box: 12 g/t
With 4-5 shields in the box: 13 g/t

In the last row(s):
With 27, 31 or 35 shields: 11 g/t
With 23, 26 or 30 shields: 12 to 12.5 g/t
With 34 shields: 13 g/t

Most other possiblilties were over 13 g/t.

For example, with 31 shields, you pay 22 gold and save 2 turns. I am not sure whether my previous estimates for a Settler-turn are consistent with this new info (but notice there is also some benefit to the mother city in birthing early). Also, I have not thought much about the size 1 vs size 2 issue in this context, or about cities that don't make exactly 4 s/t.

But if this is all approx correct so far, the question remains - how to spend your gold? Until I have time to play-test, I expect you can spend most of your gold on 11-12 g/t deals and avoid paying more. I would keep my gold only if I could save over 5 per cent per turn that way. IMO only this kind of free-market thinking can define the value of gold. It may be very hard to find, justify and use formulas like "2.3g = 1s" (though I am still trying).

....The underlying theory has to place a very high value on shields to make that worth while - a lot more than your 2.3 gold per shield.

I am thinking that "real" shields in the final row are worth almost 4g each, though it seems better to talk about turns saved. As you said, you always have to pay for 1-4 extra shields in a final row rush-buy.

...I find it difficult to believe that any benefit from having a settler 1 turn earlier will be greater than the cost of rushing.

I assume you refer to real Civ2? [In this contest, there is no other use for gold]. I would prefer to postpone that, but I think most conquest players IRB settlers pretty often in real Civ2.

So, it seems to me that the value of thinking in terms of turns rather than money is a result of the underlying ICS strategy. The quicker you can build a settler the sooner you can establish a new city. When you have finished the current test, it may be worth repeating the exercise, but with a different terminal condition - perhaps one that favoured an OCC strategy.

I agree. There are many options for what to do next. We could simply tweak this one a little (remove fundy, or adjust the food box, etc) to see whether a small change has a big effect. Or we could make a big change and switch to OCC. I just hope it gets easier with practice!

Sorry to post at such length, but I believe this exercise is sufficiently important that it is worth exploring all the nooks and crannies that surround it.

Sounds good to me. Post as much as you want.

[Peaster]

Originally posted by Grigor

Grandma also notices that rushing during 2-shield production is more valuable than during 3-shield production.

At the risk of doubting Grandma, I am not sure about this. Please assume for a moment this is a game where you can rush each row separately and that we can ignore interest rates and food boxes etc.

Example 1: City A produces 5 s/t and City B produces 10 s/t. Player X rushes rows 2 and 3 of City A for 50g. So, he has 2 settlers after 4 turns. He repeats this, for 4 settlers in 8 turns.

Example 2: Same situation, new player. Player Y rushes two rows in City B instead, and repeats this for a second City B settler. After 8 turns, City A has made 1 settler and City B has made 3 settlers. So, Player X and Y come out even.

I think there is a slight error here (cities lose a turn of production when they give birth) and the change of city size might throw it off a little too. But overall, I think production level will be less important than location in deciding where to spend gold.

Is the issue in this contest when to wait and when to rush?

I guess each player can choose their own issues (I am just trying to solve the position, and don't know which decisions will turn out to be the most important). IMO it is a mistake to "let the city grow" under these conditions, so waiting is not really an issue for me. I think ST suggested a rule change [smaller food box] to make it more of one.

At the moment, the most interesting issue for me is that a simple gold formula like "2.3g = 1s" doesn't seem feasible anymore. I hope we can at least find a good rule of thumb for spending gold wisely.

BTW - Is anyone ready to get started? If not, I don't mind postponing the start again.

[rjmatsleepers]

Originally posted by rjmatsleepers

I find it difficult to believe that any benefit from having a settler 1 turn earlier will be greater than the cost of rushing.

RJM at Sleeper's

Originally posted by Peaster

I assume you refer to real Civ2? [In this contest, there is no other use for gold]. I would prefer to postpone that, but I think most conquest players IRB settlers pretty often in real Civ2.

I was indeed refering to "real Civ 2", but as soon as I sat down in front of a Civ 2 screen I found it much easier to believe that the benefit from having a settler 1 turn earlier was greater than the cost of rushing. At a reasonably good site, a new city will produce at least 1 trade arrow and 2 shields in this hypothetical extra turn (not to mention some food). If the shields are worth 2.3 gold each and the arrow worth 1 gold, there is a benefit of 5.6 gold (or perhaps we should call them gold equivalent units ). If this is the case, rushing the final 6 shields in a city producing 4 shields per turn gains you the value of the 2 extra shields (hypothetically 4.6 geu) and the value of the extra turn (5.6 geu). Total value 10.2 geu at a cost of 13 gold. So if you can micro manage the city to produce an extra 2.9 geu in the final turn, you have made a profit - this would certainly be possuble if you can switch 2 workers from forest to ocean.

Similar ideas can be used for the value of rushing that first row. In a 5 shield city, rushing the first row would give you 11.5 geu from the shields themselves and 5.6 geu from the extra turn - 17.6 geu for a cost of 10 gold. On the face of it this is a real bargain. In a 1 shield city, the benefit is even bigger. The 9 shields are worth 20.7 geu and the 8 extra turns, 44.8 geu - 65.5 geu at a cost of 23 gold.

All of this left me wondering if I was overvaluing the shield. Perhaps by valuing it at 2.3 geu I was double counting the value of the extra turn. With this in mind, I experimented with a lower intinsic value for the shield. A hypothetical value of 1.3 geu per shield gives a value of the extra turn as 3.6 geu. At that valuation, rushing the first row in a 5 shield city gives you 6.5+3.6=10.1 geu at a cost of 10 gold - a profit, but not an irresistable bargain.

This may help a little with the value of food. If using your workers for food in a size 1 city builds the settler earlier, it may be possible to value the food by looking at the sacrificed shields and the value of the earlier build. (The details are left to the reader as an exercise. )

The problem still remains that these pesky values insist on changing, even in your controlled environment. Anyway, I am at least groping towards a theory of value that I can use in the contest.

RJM at Sleeper's

[rjmatsleepers]

Time value of gold

Originally posted by Peaster

I am sure there is a time value of gold in this contest, at least in the 3500-1800BC growth period when there are many good chances to spend it. Gold is good only for getting settlers out faster, and settler-value decreases each turn. I am pretty sure the correct interest rate is about 5 per cent (same as the growth rate).

I believe this also applies to most situations in real Civ2 I have seen (mostly ICS/conquest, with barbs rarely making much difference). But let's discuss real Civ2 in more detail later.

I'd like to believe that there is a time value of money, but I'm not sure where your value of 5% came from. Anyway, let's take a hypothetical 5 shield city and a hypothetical 10 gold (and a hypothetical tax rate that gives you a zero net income). If you can spend your 10 gold to rush the 1st, 2nd or 3rd row and in each case get a 1 turn improvement in the time to build a settler, I can't see very much difference. In practice, I'd rush the 3rd row because that leaves me the flexibility to use the gold for something else in an emergency. This seems to imply that there is a time value, but it is very small.

If you have 25 gold to spend, rushing 5 shlelds in the 1st row and 5 more in the second seems to be a much better bet than rushing 10 shields in the first row. However, this is because of the higher cost of buying in 10 shield units compared with 5 shields twice, not because of any time value.

I would be delighted if someone can post an example where the benefit of rushing later rather than earlier is clear and measurable.

BTW, on the question of the 5% value - in this example, if you have 20 gold to spend you can rush rows 1 and 2 or rush rows 2 and 3. With a time value of 5% you should be 1 gold (or its equivalent) better off by rushing rows 2 and 3. I can't see where this extra gold or equivalent is coming from. My tentative conclusion is that the value is a lot less than 5%.

RJM at Sleeper's

[rjmatsleepers]

Originally posted by Peaster


At the risk of doubting Grandma, I am not sure about this. Please assume for a moment this is a game where you can rush each row separately and that we can ignore interest rates and food boxes etc.

Example 1: City A produces 5 s/t and City B produces 10 s/t. Player X rushes rows 2 and 3 of City A for 50g. So, he has 2 settlers after 4 turns. He repeats this, for 4 settlers in 8 turns.

Example 2: Same situation, new player. Player Y rushes two rows in City B instead, and repeats this for a second City B settler. After 8 turns, City A has made 1 settler and City B has made 3 settlers. So, Player X and Y come out even.

I think I'm with Grandma on this. Your examples are using 5 and 10 shield cities whereas Grigor was talking about 2 and 3 shield cities. The benefit of an earlier turn seems to be about the same, but the cost of gaining the earlier turn seems a lot lower for 2 and 3 shield cities. Putting it another way, rushing a large number of shields in the first row produces more "extra" turns for a 2 or 3 shield city. This seems to make rushing more valuable in this case.

RJM at Sleeper's

[Grigor]

Very exciting, guys, if still very confusing.

I like RJM's attempt to quantify the value of the turns saved. Whether he has it right or not, it is certainly correct to include turn-saving into the value equation.

It seems that one of the main issues here is availability of money. With apologies to Slow Thinker, my mathematical mind consistently looks for singularities; if gold were at 20,000, there would be no time value at all. So whatever time value there is must be dependent on the amount of gold one has.

Also, with gold at 20K, the optimum strategy would be to maximize food, not shields, because that would let the settler be rushed at the earliest time. So we are looking for a theory which tells us when it is good to empty our treasury. Sustainability is part of it.

I still am looking for an answer as to how to get the 4th doubling with cities on grass.

[Peaster]

I am uncertain about many things, but am quite sure about this time-value of money principle. If you have played the position a few times, you realize that your final score (about 135, for example) is roughly proportional to your number of cities (about 40, for example). Your number of cities doubles every 14 turns or so. That's an average 5% growth rate.

This means each game turn is worth 5% of 135 points = 7 points. Obviously, this applies to turn 1 as well as turn 60 [even though your score changes faster towards the end]. Now, the value of a game turn is mainly for production of stock (shields/food) for new settlers. If you are producing 7 stock on turn 5 then at that time 7 stock = 7 points. If you are producing 70 stock on turn 45 then at that time 70 stock = 7 points. So, the value of stock (and therefore gold) decreases exponentially with time.

This can be confusing because there might be some turns in which nothing happens (no new cities, or spending opportunities, etc) and maybe the value of gold doesn't change then. But after 3500BC, or so, growth is fairly continuous and this model works well.
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Dear Grandma and friends,

I am not quite as sure about this, but I think production levels are not too important for RB decisions. I used 5 and 10 in my example just for divisibility reasons. The main point is that in 8 turns Cities A and B will produce 8x15 = 120 shields, plus the 40 bought shields, which makes 160 shields = 4 settlers. It doesn't matter in which city the shields were bought. (Am I missing something?)

Think of the 10 s/t city as the "strong mother" and the 5 s/t city as the "weak mother". By rushing the strong mother, we get her back to work faster. We shouldn't focus only on the settlers.

I have ignored food, city size, etc, but I don't think those factors will change the conclusion much.
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I am still working on the first row vs fourth row question. If we could rush the third row, I believe that would be the optimal way to spend gold.

[rjmatsleepers]

Originally posted by Peaster

Dear Grandma and friends,

I am not quite as sure about this, but I think production levels are not too important for RB decisions. I used 5 and 10 in my example just for divisibility reasons. The main point is that in 8 turns Cities A and B will produce 8x15 = 120 shields, plus the 40 bought shields, which makes 160 shields = 4 settlers. It doesn't matter in which city the shields were bought. (Am I missing something?)

Think of the 10 s/t city as the "strong mother" and the 5 s/t city as the "weak mother". By rushing the strong mother, we get her back to work faster. We shouldn't focus only on the settlers.

I have ignored food, city size, etc, but I don't think those factors will change the conclusion much.

That argument is attractive. At the moment I can't see any flaw in it, although I wonder if there is a time factor. I'll have to think about it.

RJM at Sleeper's

[rjmatsleepers]

Originally posted by Peaster
Dear Grandma and friends,

I am not quite as sure about this, but I think production levels are not too important for RB decisions. I used 5 and 10 in my example just for divisibility reasons. The main point is that in 8 turns Cities A and B will produce 8x15 = 120 shields, plus the 40 bought shields, which makes 160 shields = 4 settlers. It doesn't matter in which city the shields were bought. (Am I missing something?)

Think of the 10 s/t city as the "strong mother" and the 5 s/t city as the "weak mother". By rushing the strong mother, we get her back to work faster. We shouldn't focus only on the settlers.

I have ignored food, city size, etc, but I don't think those factors will change the conclusion much.

OK, I've given it some thought. Try this example:

Think about 2 cities, both of which have already built 25 shields towards a settler. One produces a single shield, the other produces 5 shields. We can use 10 gold to part rush in either city. After 15 turns one city will have added 15 shields and the other 75 shields. It doesn't seem to matter where we buy our 5 shields. But ...

If we add them in the one shield city, it will produce its settler after 10 turns and after 15 turns will have added 5 shields to its next settler. The 5 shield city will build a settler on turn 3, another on turn 11 and will have added 20 shields to its next settler.

If we add the 5 shields to the 5 shield city, it will produce its first settler on turn 2, its second on turn 10 and will have added 25 shields to its next settler. The one shield city will produce its settler on turn 15 and have no additional shields.

The first option gives you 5 additional turns with your settler. The second option gives you one additional turn with each of the 2 settlers. This is an apparent net benefit of 3 turns. Of course if you add shields to the 5 turn city it will continue to produce settlers one turn earlier, but the benefit of this comes in turns 18, 26, 34 and so on.

It comes down to the value of future benefits. Rush buying in the 5 shield city gives the earliest benefit, but this is relatively small. Further benefit happens every 8 turns. Rush buying in the 1 shield city gives the initial benefit rather later, but it is much larger. Even in this case there is a stream of future benefits. The 1 shield city goes on producing a settler 5 turns earlier with rush buying than it would have done without.

It would be possible to apply a DCF style calculation to this (using your 5% rule). I might try this later.

RJM at Sleeper's

[Peaster]

RJM - Good example. We can calculate the value of each city's production using the same mathematics as for the present value of an annuity (I assume that's what "DCF" means?). I was an actuary long long ago, so this is pretty easy for me. I am not sure what interest rate to use (My 5% figure was based on a 14 turn doubling cycle, but your example takes around 16 turns to double). With a 5% rate the values are:

Value (Rushing in Slow City) = Value of Slow City + Value of Fast City = 0.715 + 2.67 = 3.385

Value (Rushing in Fast City) = Value of Slow + Value of Fast = 0.560 + 2.806 = 3.366

IMO the difference is negligible, and probably due mostly to using the wrong interest rate. In a nutshell, Fast City production is worth 5 times as much as Slow City, but the rush buy has 5 times greater effect in changing the value of Slow City production. This reasoning is probably not quite right because compund interest is not linear, but it's pretty close when the interest rate is small.

[rjmatsleepers]

Originally posted by Peaster
RJM - Good example. We can calculate the value of each city's production using the same mathematics as for the present value of an annuity (I assume that's what "DCF" means?). I was an actuary long long ago, so this is pretty easy for me. I am not sure what interest rate to use (My 5% figure was based on a 14 turn doubling cycle, but your example takes around 16 turns to double). With a 5% rate the values are:

Value (Rushing in Slow City) = Value of Slow City + Value of Fast City = 0.715 + 2.67 = 3.385

Value (Rushing in Fast City) = Value of Slow + Value of Fast = 0.560 + 2.806 = 3.366

IMO the difference is negligible, and probably due mostly to using the wrong interest rate. In a nutshell, Fast City production is worth 5 times as much as Slow City, but the rush buy has 5 times greater effect in changing the value of Slow City production. This reasoning is probably not quite right because compund interest is not linear, but it's pretty close when the interest rate is small.

DCF stands for discounted cash flow and is pretty much the same as your annuity calculation.

Thanks for doing the sums. I'm surprised the results come out so close. We get back to the question of what is the appropriate rate to apply.

RJM at Sleeper's

[Peaster]

If your civ takes T turns to double in size, and your growth rate is G, then (1+G)^T=2, and G = exp((ln 2)/T)-1, which is approximately 0.7/T. For example, in our contest, T = 14 (approx) and G = 0.05.

In your example, we don't know the terrain, or the production in future cities, so we don't know T.
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I have given up on solving the row 1 vs row 4 problem. I think rushing in row 1 is simpler and not bad. There may be chances for better deals in row 4, but they are hard to plan for.

So, my theory hasn't really changed much since I posted it. When I say "2.3g = 1s", this refers to a shield in row 1 of a typical new city. I think value formulas like this will almost always need such qualifications. Algorithms seem easier to work with than value formulas. But we have seen that values can help us find algorithms (ex how far to walk for a forest).
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I am still wondering whether my theory will hold up. Has anyone found an improvement or a major flaw ? Has anyone beaten 134 s/t in 1000BC? (I only got 132 in a 2nd try). And what's happened to ST and Kramsib and LaF ?

[Kramsib]

Originally posted by Peaster

I am still wondering whether my theory will hold up. Has anyone found an improvement or a major flaw ? Has anyone beaten 134 s/t in 1000BC? (I only got 132 in a 2nd try). And what's happened to ST and Kramsib and LaF ?

Sorry, I cannot follow the thread because lack of time so I have opened a new thread to defend my theory in a "step by step" way.

http://www.apolyton.net/forums/showthread.php?s=&threadid=132874

There, we can discuse every statement, mainly the theoretical aspects on which the whole theory is based. After the theroy is correctly understood I will pass to practice.

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[Peaster]

Sorry, Kramsib, but I don't believe that. If you have time to develop a general theory of Civ2 and then refine it down to a set of practical playing formulas, then you have time for this simplified position.

[Grigor]

Peaster and rjm -

I am still interested in singularities and counterexamples. There needs to be a factor involving current cash on hand and anticipated cash x turns later in order to decide whether rushnuying is an optimal strategy.

And in practical terms, I don't have the answer to the optimal strategy, especially for cities without forest access. The optimal result for the practice position should be 48 cities. Why is my best result only 41? Where could I have spent money or placed workers more efficiently to get that fourth doubling in the remaining 7 cities?

Help.