A Theory Contest

[Kramsib]

Originally posted by Peaster


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).

I cannot understand how can you refuse my theory so quickly.

It is totally clear that, 1s = 0,33g , implies that shields are cheaper than gold.

But I think that I explained that stock gold loses value in high speed so its better invest on shields.

Of course 2nd position is better than the first, but it doesn't in a standard game with scientific research.

I am talking about nominal value, and you are talking about value of "use".

Oh, oh, the marginalist revolution is near, as Estilpón said.


In other words, you are talking about the subjective shield value. As far as gold don't produce settlers, shields are more valuable. Of course.

But I am talking about objective shield value, shield value based on its cost of production.

I insist, I must end the theoretical concepts for all of you to understand the theory.

You should be patient with it.

[Peaster]

I cannot understand how can you refuse my theory so quickly.

Easy. I judge a tree by its fruit. I asked you a specific question which seems easy to understand - How much would you pay for a shield in Berlin ? [This is clearly about "value of use" as you call it]. Your answer was 0.33g !? If you didn't understand the question, feel free to revise your answer.

[Kramsib]

Ok, then we missunderstood each other.
In fact, you are making me think a lot, and my theory is growing in complexity and power.

If we are talking about "value of use", we are talking about how desired shields are. In that case, the highest price I would pay for a shield would be the rushbuy cost.

That's it , In Berlin on 4000 BC with 3 shields in box and producing a settler I would pay:

37 / 20 + 2 = 3,85 gold for the first additional shield.
36 / 20 + 2 = 3,8 gold for the second additional shield.

and so on, descending on 1/20 for each additional shield.

This is why, as rushbuying is the only way of obtaining more shields, it acts just like a blind monopolist stablishing a high price for shields.

An imaginary shield seller who offers shields on a higher price than the "rushbuy monopolist" would be refused from the market, as rushbuying is more efficient.

But if there was an imaginary shield seller who offers a lower price I would rather buy him his shields instead of rushbuying.

Now, let's imagine, there were several imaginary shield sellers who compete against them. Would they compete lowing shield price to nearly 0 ?.

Of course not, shields have an internal cost of production. That cost is my 0,33 gold per shield. In fact, when I produce 3 shields per turn I am employing 2 citizens, the same labour force as my trade arrows, If the imaginary shield seller were in the same situation, selling shields for a lower price than 0,33 is unneficient as he would rather employ his citizens to produce other more valuable goods. My 0,33 gold per shield is the value which makes equally worht producing trade arrows or shields. That means that, internally in my civ, I have an internal exchange rate between shields and gold of 0,33 gold per shield. This is an internal value.

In an imaginary shield market I would earn money if I sell shields over 0,33 gold (my internal cost).

Now, let's imagine, that there were more players with different internal costs.

Player 1 has 0,33 gold per shield internal rate.
Player 2 has 0,5 gold per shield.
Player 3 has 1 gold per shield.

Assuming rushbuy cost is the same in this turn and equal to 3,85, for every player, player 1 could sell shields to player 3 for 0,9 gold in exchange and buy shields to player 2 for 0,6 gold per shield, gaining a spread of 0,3 gold per shield traded without losing production.

Reproducing this behaviour, with all players competing among them, we would have a shield market with an equilibrium shield price which can be considered the real value of a shield in the civ play.

To sum up:

We have a range of shield value, where it can fluctuate:

from 0,33 gold to 3,85 gold per shield.

I will never buy shields over rushbuying price.
I will never sell shields lower than my internal exchange rate.

The shield trade, the number of shield buyers and shield offerers, their interanl exchange rates and the rushbuy cost, would determine the exchanged price.

In this simple position, I only trade with the IA at a fixed price, the rushbuy cost which varies depending on how full my production box is, and the kind of item I am producing.

[Peaster]

I think we are talking about the same thing now.

IMO, when most Civ2 players wonder about the "value" of something, they want to know how much it will help them win. So, that's what I want to focus on in this contest.

I don't see how you got "37 / 20 + 0,5 = 2,35" for the cost of the first shield, but this is much more reasonable than 0.33. I think the true cost is more like 6g, but that ignores the option of rush-buying just the first row, or disbanding the warrior (both are legal in this game). There is also the option to save your gold for the next city.

So, I don't think it is that simple. I expect my value will be about 3g, based on the best known playing/spending strategy.

Anyway, thanks for your patience.

[Peaster]

@Grigor - I think we are both interested in the question of when to rush the first setter from Berlin. I think we both start by rushing row 1 for 16g (84g left) and disbanding the warrior (delaying this has no benefit). So, we have 15 shields in the box. We can rush the settler in 4000BC, 3950BC or later on. Let's compare the first two options.

4000BC "pros": We build city #3 a turn earlier, gaining an extra 2 food + 3 shields +1 gold. Also "Mom" (Berlin) gets back to work a turn earlier, gaining 2 shields.

The "cons": We spend an extra 81g - 68g = 13g for this, worth about 6 shields (in city #3 for example).

IMO the pro's exceed the con's, so we should RB in 4000BC rather than 3950BC. The same reasoning shows that waiting for 3900BC is worse than 3950BC, so we can ignore that option. [In fact, the cons generally decrease each turn you wait].

I was careful with the pro's and con's, to only discuss shields in the "second generation cities" (city #3 and Berlin after her first birth). It is misleading to compare these shields with the 3 extra shields purchased in 4000BC Berlin. This is another example of the time-value of money (and shields) principle.

As usual, I must qualify my calculations a little. The 6 shields mentioned in the "cons" might be worth more than they seem, because you have some choice in where they go (eg, maybe you need them in city #2).

It is very easy to miss something in this kind of comparison. What do you think?

[rjmatsleepers]

Originally posted by 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".

If a shield "has value only for making the next settler faster" then its intrinsic value is 0; I wouldn't rule this out, but for the time being I prefer to use a small but positive figure.

The computational difficulty with the "value for making the next settler faster" is that it does not directly relate to the value of a shield. You have to know how many shields will make the next settler faster, and how much faster. Once you know how many shields are required and how many turns you save, you can multiply the value of making a settler 1 turn faster by the number of turns saved and then divide by the number of shields to get a value per shield. It's all horribly complicated, which gets us back to your idea of using turns as the accounting unit. The part that makes me uncomfortable is that the value extra turns is apparently relative. I would like to be able to establish a fixed point so that I could always do the same comparison - how many extra turns does x give you compared with the standard?

At the moment I'm concentrating on a theory that can cope with an even simpler situation than your game - that is where do I place my workers in typical size 1 and size 2 cities? For example, I can usually choose between a shielded grassland tile and a forest. If I know that 1 food is more valuable than 1 shield, I put my worker on the grassland. But if I don't have a shielded grassland and the value of a shield is more than half the value of 1 food, I assume I would choose a forest rather than an unshielded grassland.

BTW I'm following the exchanges between you and Kramsib with interest, but a high degree of incomprehension.

RJM at Sleeper's

[rjmatsleepers]

Originally posted by Kramsib
In an imaginary shield market I would earn money if I sell shields over 0,33 gold (my internal cost).

After 1 turn in Peaster's game you can have 3 shields in the box. You could use the cheat function to remove those shields and increase your gold by 2. (That seems to be the equivalent of selling the shields for 0.66 gold which you say would earn you money.)

I don't think that there are many of us that would prefer 3 less shields and 2 more gold.

So let's restate Peaster's question: Assume you can start the game with 10 shields and 100 gold or with 0 shields, but more gold. Which is it to be? Your theory seem to be saying that it is better to start with 0 shields and 105 gold. That's the only only construction I can put on the above quote.

Assuming you wouldn't really settle for 5 extra gold, how much would you settle for? My answer by the way is anything over 50 - the cost of rushing the first row from scratch. If I start with no shields and 151 gold and rush that first row straight away, I'm better off than starting with 10 shields and 100 gold. (I might settle for something less than 50, but more than 50 is certainly worth taking.)

RJM at Sleeper's

[Peaster]

Originally posted by rjmatsleepers

It's all horribly complicated, which gets us back to your idea of using turns as the accounting unit. The part that makes me uncomfortable is that the value extra turns is apparently relative. I would like to be able to establish a fixed point so that I could always do the same comparison - how many extra turns does x give you compared with the standard?

The ultimate value is your final score. Each turn adds about 7 points to that, so we know 1 turn = 7 points. If you have 14 cities, then each city-turn is worth about 7/14 = 0.5 points at that time. And so on - in principle, we can evaluate any commodity at any time in terms of points (though it is usually easier to compare 2 other commodities directly).

But IMO evaluation has to be done in the context of a strategy for maximizing your score. Whenever Grigor finds an improvement in strategy, I have to re-think my values. Fortunately, they don't seem to change too much, which gives me some hope that we don't need to know the absolutely best-possible strategy before claiming to know the approx values of things.

For example, I can usually choose between a shielded grassland tile and a forest. If I know that 1 food is more valuable than 1 shield, I put my worker on the grassland. But if I don't have a shielded grassland and the value of a shield is more than half the value of 1 food, I assume I would choose a forest rather than an unshielded grassland.

Yes. The hope is that a theory of value will answer that question for you. But I think some questions, like this one, can be answered more simply with a rule than with values. For this contest, my current rule is:

Keep the worker on forest, unless this leads to obvious problems, such as disbanding the city.

This is part of my strategy, that each city should make a settler ASAP (rather than build up its food box, etc). It is a little different from what I said in my first draft of values, so will have to revise that statement.

BTW I'm following the exchanges between you and Kramsib with interest, but a high degree of incomprehension.
RJM at Sleeper's

I think Kramsib was writing formulas, and using the word "value", with some odd economic meaning, which I still don't follow. Maybe he will explain it more in his other thread. His latest post here seems fairly clear, and I regard that as a good starting point for more discussion. He needs to include the objective of the contest into his theory though.

It is hard to believe that I would say anything incomprehensible, but if I did, please tell me, and I'll try to explain.

[Kramsib]

Originally posted by Peaster
I think we are talking about the same thing now.

I don't see how you got "37 / 20 + 0,5 = 2,35" for the cost of the first shield, but this is much more reasonable than 0.33. I think the true cost is more like 6g, but that ignores the option of rush-buying just the first row, or disbanding the warrior (both are legal in this game). There is also the option to save your gold for the next city.

So, I don't think it is that simple. I expect my value will be about 3g, based on the best known playing/spending strategy.

Anyway, thanks for your patience.

I was wrong, the correct expresion is:

37 / 20 + 2 = 3,85

That is to say, the rushbuying cost of a unit is:

N*N/20 + 2N

Where N are the left shields (remember that if the production box is empty it doubles).

Dividing by N we have the rushbuy gold per shield:

N/20 + 2

As a Settler costs 40 shields, and we have 3 shields, there are 37 shields remaining:

Finally we have

37/20 + 2 = 3,85 gold per shield.

[Peaster]

Kramsib, that looks better. You have computed the average cost of a shield when you buy 37 at a time.
But I remind you, it is legal to rush the first row, disband the warrior, and then buy the other 25 shields. That's a better spending strategy, so I don't think I'd pay 3.85 g/s for that group.

Also, in my view, the shields do not all cost the same, but I won't go into that unless someone is interested.

At the moment I am working on food vs shields and hope to offer a fairly complete answer by tomorrow.

[Peaster]

Peaster's Food -vs- Shield Theory

Here's a fairly complete theory of food -vs- shield values for this contest. IMO strategy comes before values. So, finding the correct values is not for placing city workers (which we have to understand first). Instead it is intended for comparing two similar positions in play-testing or in a thought experiment, for example. IMO a 10 percent error is acceptable, though I will try for better.

I am going to focus mainly on values in a new city, or in one that just gave birth, assuming the goal is to produce a settler asap, without regard to increasing city size or keeping the food box full, etc. If there is a forest tile nearby, this means the city will only stay at size 2 for one turn.

EDIT: I wrote this assuming there was only one forest tile nearby. The case of two forest tiles comes up later in other posts. Sorry for not stating this clearly.

I am convinced that every commodity (food, shields, gold) decreases 50 per cent in value every generation, and I will use that idea here. Some of this is debatable, I guess, but it is based on a lot of thinking, posting and testing. As I said, you usually have to decide on some aspects of your strategy before you can compute values.

Case 1: Suppose the city has a forest tile nearby. The fastest way to make a settler [with normal rush buying] is to set the worker on forest for 10 turns, so the food box is full. Assuming we rushed 7 shields on the second turn, we have 37s at that point, and we get a settler on the 11th turn. [Rushing 6 shields would be slightly better, but it isn't possible]. Let's assume this is the best approach - but the following calculation is probably OK for most other strategies, too.

Suppose your city is given +2 food. This gift causes the city to grow a turn earlier than before, which means it gets one extra shield this cycle. And the +2f is still there for the next cycle, but this benefit should be reduced in value by 50 per cent to +1f. So,

Value of +2f gift = 1s + 1f. So, 1f = 1s.


Case 2: Suppose the city has NO forest tile nearby. There is no choice about where to place workers, but the reasoning is similar.

Suppose your city is given +3f, which causes the city to grow a turn earlier than before. We get

Value of +3f gift = 1s + 1.5f. So, 1.5f = 1s in this case.


Adjustment to Case 2: In such a grass-city, food sometimes gets "lost" when the city shrinks. For example, consider a size 2, about to give birth. It doesn't matter whether it has 20 food or 30 food in the box - the results are the same. So, there is a good chance (about 33 per cent IMO) that the +3f gift only has an effect ONCE. In that case, the formula should be 3f = 1s. I prefer to use an "expected value" approach and say

Case 2 revised: 2f = 1s.

Case 3 (maybe not relevant to this contest): Suppose that in case 1, the food box and shield box sizes are changed to 30 each. It is no longer necessary to keep the workers on forests all the time. It will take (30+30)/5 = 12 turns to give birth (assuming a very small rush buy, to compensate for the annoying need to fill the food box before the shield box). With this flexibility in placing workers, 1f=1s.

I mention this case because I once thought we had this flexibility in our contest. It is curious (probably just a coincidence) that the value formula is the same for Case 1 and Case 3. It does offer some hope that value formulas are fairly stable when we must adjust them to fit improved strategies.

Case 4: Berlin in 4000BC. This one is tougher, because rush-buying strategy will have a big effect, and the size of the gift may affect the relative values, too. So, I am less sure about this case, but I think we should assume the player rushes a settler on turn 1. Then a gift of +3f has an effect similar to that in Case 2, but I think the +3f will be lost with the second settler. So, 3f = 1s seems about right.

I play tested the position with a food van replacing the warrior. The results in 3000BC were approx as before, which tends to confirm the theory (15f = 5s). But if I buy a van in this contest, I might use it in a different city, more in need of food than Berlin.

Fine print: IMO these formulas are useful in an average value sense, but there will be exceptional situations where they don't apply. For example, the size 2 city with 20 food [discussed in the adjustment to Case 2] has no need at all for +10f, so the marginal value of food is zero in that case. Likewise, a new city in Case 2 can use +2f, +3f, or +4f equally well [full box in 6 turns].

[rjmatsleepers]

I've just realised why we are coming to different conclusions about some aspects of this. I was looking at the position at the start of the game (when we are in despotism and some tiles give a reduced value for food). I will need to rethink my approach for forms of government where we get 3 food from the irrigated grassland. (But I'm not sure when )

Presumably, no-one doubts that the city should be founded on a grassland tile?

RJM at Sleeper's

[Grigor]

Peaster -

in your 4000BC analysis (fundy, 100g), I would suggest that the point of comparison is that at which all 3 cities have a complete first row. That reflects the time cost of using all your money on th first turn.

I haven't done that analysis yet.

[Grigor]

Peaster - regarding your elaborate case analysis: I have lookd at one of th crucial points of the test game - the end - and found an interesting point in a common situation - a city with access to two forests:

After rushing the first row, worker placement does not change the result

That is not precisely true, but there are three strategies. I begin at 1550BC with 3 food in the box and 10 shields. In the next 8 turns I can do the following:

Strategy 1 - Build at maximum speed
This means 1 turn at 3 food and 2 shields (avoiding food waste) and 7 turns at 2 food and 3 shields. I end up with a full food box and 33 shields.

Strategy 2 - grow at maximum speed
This means 5 turns at 3 food and 2 shields plus 1 turn at 2 food and 3 shields filling the food box, plus 2 turns at 1 food and 5 shields, working both forests. I end up with the same 33 shields! but with 2 extra food in the box.

Strategy 3 - mix and match
This would be 3 turns at 3 food and 2 shields plus 4 turns at 2 food and 3 shields filling the food box, plus 1 turn at 1 food and 5 shields working both forests. I end up with 1 extra food in the box but the same 33 shields!

The implications for the earlier portion of the game are unclear except that in 2-forest cities, 10-turn settler building is possible for 35g. (19g for the first row and 16 for the last row)

[Peaster]

@RJM - Despotism? The posted save starts you off in monarchy, but I suggest you change that to fundy. Yes, I build all cities on grass until the endgame.

@Grigor 1: Not sure what you meant... Is this about playtesting from 4000BC? [I have been playing to 3000BC, when there are usually 6 cities]

@Grigor 2: First, 2 minor questions:

Did you really mean 1550BC ? If the city takes 10 turns to make a settler, there is not enough time left for the settler to make a city. I guess this is not relevant to your main point - just curious.

In Strategy 2, you state that after filling the box, you can get two turns at 1 food and 5 shields (which I call 2<1,5> ). But in the growth year, the AI always places one of my workers on grass, so I get a turn at <2,4>. Do you know a way to prevent that?

I was mainly interested in the 3500-1800BC period, and didn't really consider cities with 2 forests, or the possibility of having 35g per city to spend. These cities are different because they can still produce 5 useful stock at size 2 [with one forest you can get <2,4>, but the food is not useful for the current cycle]. So, Strategy 2, to reach size 2 quickly, is probably the best option (especially if you are right about the 2<1,5> )

Assuming this is best, my food-shield analysis for such cities is

+3f = +3s + 0.5f, so that 0.84f = 1s.

[But if you apply this after about 1800BC, it should be 1f=1s]

Finally, I am curious about your 4-cycle strategy. If the game ended in 1200BC or 800BC instead, could you adapt the strategy to that?