A Theory Contest

[Peaster]

Originally posted by Grigor

The optimal result for the practice position should be 48 cities.

Grigor - Your idea is tantalizing. I guess you are splitting the 60 turns into 4 cycles involving movement and production: m-p-m-p-m-p-m-p-m. Each "m" takes at least 2 turns of movement plus 1 turn to build a city, so there are 59-15 = 44 turns left for 4 p's, an average of 11t per cycle.

I estimate that a grass-city needs 33g for rushing 14s, to birth that fast. Cities produce 11-15g in taxes per cycle, so each new grass-city needs support from 2 neighbors. Maybe from its mother and a forest city, for example. But the mother and the forest city will need some gold, too, to maintain an 11t pace.

I don't think we have enough resources to make this all work perfectly, so unless someone has a new idea, I don't think 48 is possible. My max is 41. Have you done any more trials lately?

[Kramsib]

Originally posted by 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.

Let me explain:

This weekend I have more time for posting, but only this weekend (because monday is holiday).

I started the other thread because I can follow it , from a weekend to other, and as it is only theory (by now) it is easier for me to follow.

After that, in june when I am on holiday I will start putting my theory in practise.

[Kramsib]

Originally posted by Peaster
I am thinking seriously about giving you Fundy at the start, to eliminate unhappiness completely. This would make warriors useless (except for disbanding purposes). Any objections?

Here are my first thoughts about solving this position:

1) IMO the optimal growth strategy is to send a few Settlers from Berlin to the major nearby forests (no warriors, no roads). Then, the new cities mutliply in clusters around the forests. With Fundy, I got about 115 s/t in 1000BC this way.

2) A formula like "1 shield = 2 gold" means both sides of the equation are equally effective in pushing this strategy. It might be possible to deduce such a formula either from reasoning or from testing.

3) From reasoning, I figure these are about equally useful:

25 gold = 10 shields (2 NON-warriors) = 15 sheaves (1 food van)

4) I briefly playtested the effect of adding each of these 3 options into the 4000BC save. The 15 food seems a bit better than 10 shields, which is better than 20 gold (maybe because IRB'ing is hard to do). So, I do not have a final answer yet, but feel I am getting closer.

Disagreement is encouraged, of course.

I do not think we will get the correct shield value in such a simplificated game, nor in a game where the goal is maximising shields. In fact, by maximising shields, the value of shields will decrease as they become abundant against gold.

In such a map, where bonus grasslands is preferred than simple grasslands, food is abundant so its value is reduced, shields are quite abundant so its value is also reduced, but gold is dificult to obtain without roads.

In a well developed country, average tile will have, 2 food / 1 shield / 1 trade arrow, so values will be aproximatelly.

1 food = 0,5 gold
1 shield = 1 gold
1 trade arrow = 1 gold.

As there are some forests these values will be distorted.

Let's see a better aproximation:

Cities won't develop more than 3 citizens as in ICS we are continuosly building settlers. So production in each city woul be

1 forest, 3 bonus grassland,

so production by city would be

7 food / 5 shield / 3 trade arrows maximum

So values are:

1 food = 0,43 gold
1 shield = 0,6 gold
1 trade arrow = 1 gold.

The less roads, the cheaper shields.

[Kramsib]

According to these values:

Irrigation, has no sense.
Mining grasslands changes 1 food for 1 shield, so we are changing a 0,43 gold unit for 0,6 gold units (food by shields), so we are increasing our value. Mining grasslands is profitable.
Building roads is even more profitable, as they give 1 extra gold per tile (except on forests).

Consequently, the tile improvement strategy implies mining simple grasslands and building roads on bonuss grasslands.

[Peaster]

Kramsib - I don't think anyone has a complete answer to how much gold a shield is worth in Civ2. If there is a meaningful answer at all, it must depend on whether the player is going for conquest or OCC, which city the shield is in, etc.

This contest is not intended to directly answer that question. But as part of your theory, you should tell us how much a shield in 4000BC Berlin is worth. That is, how much would you pay for one shield? If someone can find a clear and correct answer to that, maybe we can talk about shields in other cities and settings.

I do not believe your reasoning about values posted above is correct. When you wrote "1 shield = 0,6 gold" do you mean that you would only pay 0.6 gold for one shield? In other words, you would never rush-buy ? I also think your conclusions about mining (etc) are wrong, but if you can do more with this position than I can, I will have to change my mind.
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@ Grigor - I tried for 48, but gave up. I don't see how the average new city can produce a settler in just 11 turns. Maybe if every city had its own forest square, or if we could play to 950BC....

BTW - When I estimated that doubling occurs every 14t, I meant it in an average sense (some cities may take 16t, some 12t, etc). Also, the actual average might be more like 13.9 or 14.2. For your hope of 48 cities, it seems we need 14.0 uniformly and exactly. I don't see how to do that.

[Grigor]

48 Cities and 141 shields/turn

A very productive evening. By focusing on making sure I got a complete doubling each round, I achieved 48 Cities and 141 shields per turn in 1000BC starting with 100g. In 4000BC, i rushed the first row, then waited one turn before disbanding the warrior and buying the third settler. Both the second (NONE) and third settler walked an extra turn to get to a forest square. The second settler from Berlin walked two extra turns to get closer to the SE forest.

I can probably do even better with some careful thinking - I could have squeezed a fifth doubling out of both Berlin and Leipzig, and there was no reason for any of the fourth round cities to found between 1400 and 1000BC. It seems that rushing the first row whenever possible saves enough turns and enough money to finish the last doubling in 1150 BC.

I apologize for the crummy numbering in the E-class cities.

[Peaster]

Grigor - Congratulations!! I also found a way to get 48 cities tonight, but you did it first, and I only got 136 points from mine. Our openings were very similar. I think I waited one extra turn to RB in Berlin. When one settler appeared a little earlier than needed, I used it to build two road squares (for the trade arrows). I also felt afterwards that I could have squeezed out one more city if I had tried.

Sorry for doubting your idea. I am still kind of surprised that 48 is possible.

[rjmatsleepers]

Originally posted by Peaster

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 ?

My my, what a lot of posts in the last 24 hours. It will take me some time to absorb them.

The value theory I'm trying to articulate has a number of elements.

The "intrinsic" value of a shield is low - perhaps 0.5 geu (the geu is an accounting unit arbitarily set to equal 1 gold).
food is slightly more valuable than shields - perhaps 0.8 geu.
shields (and perhaps food) may have an additional value based on their ability to buy time. This is perhaps 3 - 6 geu per extra turn. (I separate the intrinsic value of a shield from the time enhancement value because the extra turns bought per shield can vary from 1 downwards. Combining the two would require using an average that wouldn't apply in some of the interesting cases.)
The value of trade arrows depends on the type of government and the city development. In despotism and without a market, a trade arrow is 0.6 geu.

I would like to use these values to "justify" some of the decisions I feel are "correct". For example in a size 1 city placing your worker on a shielded grass tile, but in a size 2 city placing both your workers on forest tiles.

At the moment, it feels like a theory with some interesting implications, but which doesn't quite work.

RJM at Sleeper's

[Kramsib]

Originally posted by Peaster
Kramsib - I don't think anyone has a complete answer to how much gold a shield is worth in Civ2. If there is a meaningful answer at all, it must depend on whether the player is going for conquest or OCC, which city the shield is in, etc.

I think my theory can give a complete answer to that question.

This contest is not intended to directly answer that question. But as part of your theory, you should tell us how much a shield in 4000BC Berlin is worth. That is, how much would you pay for one shield? If someone can find a clear and correct answer to that, maybe we can talk about shields in other cities and settings.

According to my theroy, shield value has not a constant value in the game, it varies depending on its abundance during the play.

A little example, when you produce less shields, they become more scarce, and more "expensive", they have a great value for you and you will pay more for them, in such a situation you would rushbuy more often (change more abundant/cheap gold for scarce/expensive shields). When shields become abundant, they become cheaper and rushbuying is less worthwhile (less worthwhile but it can still be useful depending on the interest rate, intrinsec gold value).

I do not believe your reasoning about values posted above is correct. When you wrote "1 shield = 0,6 gold" do you mean that you would only pay 0.6 gold for one shield? In other words, you would never rush-buy ? I also think your conclusions about mining (etc) are wrong, but if you can do more with this position than I can, I will have to change my mind.

1 shield = 0,6 gold, does not necessarily mean that rushbuying is not worthwhile. It depends on intrinsec gold value, in fact, in this simple position, stock of gold has no value at all (except for rushbuying).

This game is a competition against time, and time affects on gold value.

Let me explain it more clearly.

First of all, we need to discern between "flood of gold" and "stock of gold".

Flood of gold can be identified with the amount of trade arrows, as they "flood" every turn.
Stock of gold are gold accumulated in your treasure.

Flood of gold is employed on science investigation, return it to population in luxuries form, or keep it in your treasure.

Treasure is employed for paying maintenance, rushbuying or international transfers.

In your simple savegame, flood gold cannot be expended on science, so it can only be used for treasure or luxuries. In fact, in the first turns, when luxuries aren't needed to keep population happy, the best distribution is asign the maximum percentage on taxes, making your treasure grow.

As there is no rivals, there is no need to employ gold for international transactions, and as there is no buildings to build (only barracks but they are useless without enemies) treasure is only useful for rushbuying.

In other words, liquid gold (stock gold / treasure ) has no further use, and it is only useful for rushbuying, that is to say, bringing today, future production.

Changing, stock gold, which is gold with low "value of use", for shields which have a greater "value of use" is "profitable".

This reasoning is perfectly compatible with Adam Smith's thoughts who discovered that countries' wealth depends more on a great "flood" of gold than on a great "stock" of gold.

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I have prepared a practical example.

Let's assume a city with 4 shield production per turn which is building a Settler (40 shields - 10 turns). And let's assume there is no change in production during the following 10 turns. Let's assume there shield value is constand during this period too and equal to 0,6 gold per turn.

Remember that Rushbuying cost for units is:

N*N/20 + 2N

And twice when the box is empty.

Here is the meaning of the Columns in this excel file.

A: Here is the shield production of the city, 4 shields per turn.
B: The turns left until the settler is finished.
C: Shields left until the settler is finished (A * B)
D: Cost of rushbuying according to the formula mentioned before.
(C*C/20 + 2 * C). Twice for the first cell of the column.
E: Real shield value in gold terms using my theory, assuming it keeps constant during the 10 turns to simplify the analysis.
F: Simple interest rate for the period remaining.

C1 = ( 1+i*t ) * C0

Considering rushbuying as a way to bring today future production.

That is to say, if production remaining has a real value of C0 (E column) and rushbuy cost C1, what is the aditional cost in terms of an interest rate?

G: Simple interest rate in turn terms. By dividing by the remaining turns, we obtain the interest turn "payed" per one turn.

OBSERVATIONS.

When shields are cheap (0,6 gold per shield), rushbuying is less worthwhile, as we are paying more gold for rushbought shields. In fact the interest rate is positive. But you can notice that, as turns go by the interest rate is growing. Rushbuying at the second turn is cheaper than any other turn.

Interest rate is also a measure of how much value your stock gold is losing and reflects the cost of opportunity of not rushbuying.

But on the other hand, the second case where shields are expensive (3 gold per shield) there is a point when rushbuying become worthwhile (interest rates become 0 in the 6th turn = 5 turns remaining) and interest rates become negative.

CONCLUSIONS.

In your savegame, rushbuying is the only way to make stock gold useful, and the sooner you expend it the less value it loses.

When shields are cheap, rushbuying is less worthwhile except we have better uses for our money (science, international transfers, maintenance).

When shields are expensive, there is a point when rushbuying is always profitable.

[Peaster]

RJM - That looks like a good start. I am not sure what you mean by the "intrinsic" value of a shield. To me, a shield has value only for making the next settler faster, which (I think) is what you mean by "an additional value".

Kramsib - If your theory can give a complete answer to shield values, we would all be grateful. But so far I have not learned anything from it. I have almost no idea what you are talking about, and judging from readers' responses in the other threads, they do not get it either. I urge you to focus on the specific questions at hand -

1) How to play this position correctly ?

2) How much would you pay for an extra shield in 4000BC Berlin? (also similar trade-offs with food, etc)

I suspect that part of the confusion is over what you mean by formulas like "1s = 0.6g". To me, it means 1s and 0.6g are equally useful in achieving my goal (which means your numbers are horribly wrong). Apparently, you have a different meaning in mind ?? Can you explain this clearly, please?

And if it does have a different meaning, how do you use it to make decisions?

[Grigor]

I think I see where Kramsib is going, and I look forward to the rest of it.

So far, it seems to not be of practical use. We all know (from OCC among other practical trials) that rushing in the second turn is the cheapest strategy, but we usually do not have that option. In Civ II the most powerful practical tool for the human player is partial rushbuying, (which I think is perilously close to cheating but that's a different thread) and I look forward to any discussion of when and what kind is optimal.

In Peaster's current test position, there is only one partial rushbuying option, and after much playtesting the intuitive wisdom is to rush the first row as soon as the gold is available. I hope Kramsib's full theory will shed some light on this eminently practical situation.

[Kramsib]

Originally posted by Peaster

Kramsib - If your theory can give a complete answer to shield values, we would all be grateful. But so far I have not learned anything from it. I have almost no idea what you are talking about, and judging from readers' responses in the other threads, they do not get it either.

That's why I started the other thread to show the whole theory from the beginning to the end, statement by statement, discussing them separately and with quite time between them. It is quite revolutionary as it is based on different ideas from Economics, but I believe it applies perfectly to the game. If all of you are patient I will be able to explain it completely.

In "Hurry Production Thread" I stated it very quickly, so I was missunderstood quite often.

I urge you to focus on the specific questions at hand -

1) How to play this position correctly ?

2) How much would you pay for an extra shield in 4000BC Berlin? (also similar trade-offs with food, etc)

Unfortunately, my theory does not apply for city placement or unit moving, but it is useful to determine which squares work in a specific city (it determines the more efficient tiles), it also helps to determine when rushbuying is worthwhile, as I have said on my previous post. And the most important application, it allows to give value to every item in civilization, in gold terms obviously. Units, buildings or even techs can be translated into gold thanks to my theory.

I suspect that part of the confusion is over what you mean by formulas like "1s = 0.6g". To me, it means 1s and 0.6g are equally useful in achieving my goal (which means your numbers are horribly wrong). Apparently, you have a different meaning in mind ?? Can you explain this clearly, please?

And if it does have a different meaning, how do you use it to make decisions?

When I say, 1s=0,6g I mean a exchange ratio, like 1 $ = 1,3 €, that is to say, If there were a shield market, I could buy 1 shield for 0,6 gold.

Of course, there is no such a market in civ (excepting rushbuying), but in a standard game, I can exchange techs, gold, or units.

This theory is specially useful in Multiplayer games, moreover when using Civ2Dip by Yaroslav (a great friend of mine), where human players are really interested in the real value of things.

How much gold does a settler cost?, a settler cost 40 shields, but how much gold does a shield cost?, this is the question.

My theory states that the whole shield production in a city has the same value as the whole food production and the trade arrows. That is to say.

food production value = shield production value = trade arrows production value

As trade arrows are directly convertible in gold (gold returned into luxuries, gold invested on science or gold in your stock of treasury), we have the following expression.

trade arrows = flood of gold = shield production value = food production value.

The ratio, trade arrows / shields, give me the value of 1 shield in gold terms.

Using your savegame as an example.

Berlin in 4000 BC, produces:

7 food / 3 shields / 1 trade arrow = 1 gold

So, 1 shield = 0,33 gold (1trade arrow / 3 shields = 0,33)

Now, if you ask me, how much gold I will pay for an aditional shield as a stock in my production box?, my answer will be 0,33 gold per shield.

[Peaster]

Originally posted by Kramsib

Berlin in 4000 BC, produces:

7 food / 3 shields / 1 trade arrow = 1 gold

So, 1 shield = 0,33 gold (1trade arrow / 3 shields = 0,33)

Now, if you ask me, how much gold I will pay for an aditional shield as a stock in my production box?, my answer will be 0,33 gold per shield.

I am glad you cleared that up, because it is finally completely obvious to me that your theory doesn't work. I suggest that you play the position two ways (use cheat mode to make these changes, and govt = Fundy):

1) Start with 110 gold + 3 shields in the box.

2) Start with 100 gold + 13 shields in the box.

You will have much better results with 2).


@Grigor - You are a very nice man, but let's be honest. We both know Kramsib is off by at least 600%. I would not be so harsh with him except that he pretends to know divine secrets which are really just untested ideas.

@All - I probably will be ready to start in a few days. How are your theories coming along?

[Grigor]

Peaster -

Yes, I am nice

But in our playtesting we cannot even decide whether the play is more efficient if we wait to rush the settler in Berlin 0, 1, or 2 turns. Having enough gold to rush the first row early in the 2nd gneration settler builds can take off significant turns.

I will try tonight, but I expect that waiting 2 turns with 110 gold will give enough gold left over to rush all 3 first rows on the first available turn.

[Peaster]

Grigor - I was suggesting the exercise with the 110g to Kramsib as a way to test his theory. You already know that 10s is worth much more than 10g.

I hope you will write up your understanding of this position, including a statement about values to use in the auction. Your idea of four doubling cycles is wonderful IMO, but I am not sure it will work the same way from a slightly different starting position. So, I hope you will form a theory that doesn't assume exactly 4 cycles.

I will probably revise my theory a little in the next few days and put some more thought into how to run the auction. I have been thinking it might be easier and fairer to create a few different starts and let each player choose one. Any opinions on this?