A city is size 2, works Fish + Plains-Iron, two very strong tiles. The city has the chance to 1-pop-whip a Courthouse though, now what would you do? As other tiles, you may assume Grasslands, Coast or Plains, so no further resources, and also no nice FPs or even an Oasis. The city has a Granary and a Lighthouse, so the fish is +6 and to regrow from size 1-2, the city needs 22 , from size 2-3 it needs 24 . Plains-Iron is 1 + 5 .
I'll post a picture to illustrate this further if I find the time, basically, all important information is given in the above.
Spoiler:
Answer I came up with was the following: If I whip the Courthouse, I save 2.6 through having it in 1T compared to whipping the city after growth to keep working the Plains-Iron. The Plains-Iron is 1 and 5 , however when I whip, I gain 1 because I lose 1 less and I gain another because the city needs 2 less to grow, but with being smaller, the amount getting stored through the Granary is also slightly smaller, so 5 against 2 and 2.6 already. I then looked how much the city would need to regrow, it'd be 22 or 24, depending on whether whipping now or 1T later, so in this case, 11 are 30 so 1 is roughly 3 .
This brought me to the answer, that the rules of "always work the best tiles" or "don't whip away good tiles" can be very untrue / misleading, because 2 would be about 6 then, resulting in a netto gain of 1 and 2.6 if whipping away the Iron. This surprised even me very much, and I thought I'd post it, because I assume that not really many people would whip away a Plains-Iron because they think it's the best solution. The fact, that 1 can even be (almost) 3 astonished me the most.
The question if building regularly would be best is imo answered by the fact, that the Courthouse gives more benefit than any tile, that the extra population could work, and letting the city grow further would also make the whip less efficient.
Plz post any answers in Spoilers, so that people can really ask themselves this question and find out on their own without accidentally seeing the answer. If I made any errors, I'd be very interested in getting to know them, so TIA eventually.
I answered one turn later. My reasoning was one turn later as then I could still work the 2 good tiles and the 3rd pop really didnt have anything great it could be doing anyway so slow building seems dumb, unless I was banking pop for some other whip
That leaves 1 turn later so I work that plains iron an extra turn or whip now and get -2 gold reduced maintenance for one extra turn. So 5 hammers vs 2 gold seems a better deal. I think I am also minus 1 food by waiting a turn but I am never to sure how to fully calculate all this stuff.
I make it slightly less clear than you calculated:
Spoiler:
2x(30/11) = 5.45, not 6, so the gain from 2 is only 0.55
Also did you factor in 2 from working coast for a turn? I guess after increased city-size maintenance that's probably not huge but might reduce the 2.6 netto a little bit?
It's definitely interesting that at low city size iron itself does not qualify a tile to be unwhippable. I guess I would always look at ice or desert iron as whippable depending on circumstances, but this shows that flat plains iron is too. If the iron was on a hill or especially grassland though, both of those would make the citizen much safer, from me at least!
A third good tile would definitely change things. Also maybe it would be worth slow-building the rest of the CH if MC was coming up soon or if we wanted a library here..
I like to think about this optimization problem a little bit differently. In my opinion, Civ 4 can be confusing because we often forget about the City Centre square.
We like to think that, for example, a wet Corn will earn us +6 Food. But, in fact, it only earns us +4 Food, since the citizen working the Corn eats 2 of that Food.
The City Centre gives us (for most City locations) 2 Food "for free" (i.e. that isn't consumed by a citizen). If you can set aside thoughts of this free Food when calculating an individual square's worth, you can make a much more objective evaluation of the contribution made by the citizen working that extra square.
Thus, in my opinion, the true measure of a square in Civ 4 is how much it gives you after you have subtracted 2 Food.
So, for example, a Grassland Iron Mine at 2 F + 4 H gives us a net of 4 Hammers, since the citizen working that square consumes the 2 Food.
Incidentally, this fact means that a Grassland Iron Mine is worth FOUR TIMES as much production as a Grassland Forest square. This fact is a key point to keep in mind; it's too easy to think that the Grassland Forest square earns us 2 Food and 1 Hammer, especially due to the confusion offered by the City Centre square's free 2 Food, but the opportunity cost of working a square has to factor in the Food required to work that square.
Thus, whenever we are doing comparisons involving whipping, it can be a helpful practice to assume that the whipped citizen would have otherwise worked a 2-Food square, such as a Grassland River Cottage.
In this way, we can remove the variable difference that is obtained by whipping away squares that provide a different amount of Food than the citizen working that square would have consumed. Doing so allows us to isolate the Food earned due to whipping in a case where each additional citizen earns exactly as much Food as it consumes, so that we can see precisely how much gain we have in terms of Food due to whipping.
Then, when the decision of whether to work that square or to whip needs to be made, we can more directly evaluate the impact of the square's contribution in Food + Hammers + Commerce minus 2 Food against the combined benefits of the extra Food earned due to whipping plus the value earned (or even the extra cost paid for certain costs) by obtaining the whipped item sooner.
So, how much Food is earned by whipping?
Game Speed: Food required to grow when at Size 1, Size 2, Size 3, Size 4, Size 5
Quick: 14, 16, 17, 18, 20, 21 (increments of 2 and 1, in seemingly-weird order)
Normal: 22, 24, 26, 28, 30 (increments of 2)
Epic: 33, 36, 39, 42, 45 (increments of 3)
Marathon: 66, 72, 78, 84, 90 (increments of 6)
Note that I haven't performed the math for other Quick Speed cases. It might be possible that the math could give a value other than 1 Food earned by whipping due to the odd way that the Food requirements for City Size growth will vary. I'd certainly be interested to hear if someone has found a particularly good or bad example for specific City-Size-whipping on Quick Speed.
In Seraiel's example, we compare:
A] Not Whipping (working the Iron Mine)
i. 1F + 5H - 2F (the -2F comes from the citizen who works the Iron Mine)
= -1F + 5H
versus
B] Whipping away the citizen that would have worked the Iron Mine
i. +1F from whipping at Normal Game Speed (see the values above)
ii. Whatever value we obtain from having the item that was whipped a bit sooner than we would have otherwise obtained it
iii. Whatever gains or costs are associated with one turn of being without that population point
Note that iii. could refer to multiple things:
a) Possibly saving 1 Gold on empire-wide City Maintenance if the loss of the population point bring us down below a whole number
b) Possibly paying 1 Gold in Unit Cost if the total population empire-wide gets decreased below the threshold of receiving one additional free unit (a rough heuristic is to think that 4 population points can reduce the Unit Cost value by 1, but this point is only meaningful if you have a sufficiently-large army to actually have a non-zero Unit Cost value)
c) Possibly being the first whipping action, which starts the Whipping Unhappiness counter, which can mean that on a future turn, we may have 1 less Unhappiness in the City at a time when the City would have otherwise had an Unhappy citizen who was not being productive. This point only really matters for your first whipping action in a City, or for a whipping action that occurs when all existing Whipping Unhappiness has expired; once you have started your Whipping Unhappiness counter, then it doesn't matter if you add an extra 10 turns of Unhappiness (different values for different Game Speeds).
For example, say that you have 7 turns of Whipping Unhappiness. If you whip this turn, next turn you'll have 16 turns of Whipping Unhappiness. If you instead whip next turn, next turn you'll equally have 16 turns of Whipping Unhappiness.
But, if you don't have any Whipping Unhappiness, then whipping on the current turn can mean only having 9 Whipping Unhappiness next turn, while whipping next turn would mean having 10 Whipping Unhappiness next turn.
How you evaluate B] ii. can make a large impact on your decision to whip.
For example, we can see an immediate benefit from a Courthouse. Whether that benefit is sufficient to outweigh the cost paid is up to the player to decide. But, will getting a Library one turn sooner help? Maybe, if you're going to stay at 100% Science for the next little while, or want to get that Academy up 1 turn sooner so that you can start on 100% Science 1 turn sooner. But, if you're alternating between 0% Science and 100% Science, does it really matter if you spend 1 more turn at 0% Science and get your Academy 1 turn later? Maybe it will matter, but maybe it won't matter, and thus it could be better to take the extra value earned from working the "good square" for an additional turn.
Whipping a Monument or a Library for Culture sooner might matter. Whipping a Military Unit sooner might matter if it means getting a Military Unit out to stop a Pillaging Barb Unit. Whipping a Barracks one turn sooner is less likely to matter and thus it's probably not worth the price paid.
I also like the point raised that this population point could have gone toward whipping a different build item; there is an opportunity cost to be considered there, especially if you are Happiness-constrained and can only whip a few items before running up against your Happiness cap.
In regards to B] iii., there are a lot of subtle factors that may come into play. In Seraiel's case, he could theoretically earn 4 Gold: 1 Gold from whipping immediately if doing so brought his empire-wide Maintenance to a lower whole numbered value, 0 Gold lost from having a small army, and 3 Gold earned if the empire-wide Maintenance was lower than a whole numbered value due to the 2.6 amount. Alternatively, he might have only earned 2 Gold if the fractions didn't work in his favour, and possibly only earned 1 Gold if he also had a large army and some bad luck in terms of his empire-wide total population in regards to the Unit Cost formula.
Another consideration that makes it worthwhile to separate out the Food earned from whipping is when you whip at a time other than just before growth or just after growth.
For example, let's say that you were to whip when your City would grow in 2 turns' time, and the reduced Food requirement to grow to a smaller City Size still didn't allow you to grow in 1 turn's time due to whipping. In this case, we actually only earn the 1 Food from Whipping for one of those turns. Thus, we can no longer say that it's 2 Food earned on each turn.
Instead, we'd say that we earned 1 Food from whipping, and we spent 2 turns earning 1 Food per turn from not paying the 1 Food to work the Iron square, so we only earned 3 Food, 2.6 Gold per turn (which may be anywhere from a net of 1 Gold to 4 Gold per turn) for 2 turns, and lost 5 Hammers per turn for 2 turns.
So, what's my answer to the question?
Spoiler:
The answer is that whipping is situational. If you're playing a competitive Succession Game of the "Month" game, where you have a team of players who can number-crunch or run test games, it may be worth your while to see if whipping can bring you ahead.
For example, if by whipping, you can grow to Size 3 one turn sooner, you might earn back one turn of working a square, and if it's equally-useful as the Iron Mine (say, it's a G H Copper Mine that you don't currently have the Worker turns to improve but you will have the Worker turns to improve it by the time that you can grow to Size 3 after having whipped), you can earn back the value of working that Iron Mine and still get your build item sooner, plus possibly save on Maintenance.
But, in general, I still believe that is a good heuristic (which is a fallible, but generally applicable approach that you can use) to say that it is often preferable to work a "good square" instead of whipping away the citizen on it. I then make exceptions to that rule whenever I can justify needing that build item 1 turn sooner.
For example, what if getting a Missionary for Organized Religion in another City allows a player to complete a key Wonder one turn sooner? In that case, it's a justifiable cost to pay to lose 1 turn of working that "good square."
If I really cared about the extreme micro of a particular game, I could try to figure out all of the related math, including Maintenance paid or saved, and when the City will grow to various City Sizes on future turns, but to be honest, I don't really play this way, and I don't think that most players would play this way, either, so I'll stick with an approach of not whipping away "good squares," while keeping an open mind about possible cases of when it may actually be of benefit to whip away such a square.
Excellent post Dhoomstriker. Especially the point of a city maybe growing 1T earlier and thereby regain 1T on a certain tile is very important, if reaching the advanced levels of micro. I also didn't consider the possibly lower maintenance by troops. Saving 1 is something, that gains importance, especially on the lower difficulty levels, as there's less gold available.
The point about the playability I find important but also interesting. I've begun to make most of the calculations you talk of in my games and reached completely new levels of insight by that. What helps me the most, is to stay away from formulas like 1 = 2 = 3 , though those can still be valuable in certain situations, like when i. e. trying to evaluate which tile is better in overall. Calculating how many are truely created through the whip, by simply seeing how much food half of the food bar is, and knowing that that amount generates x (value of 1 population, so 20/30/44/90) , brought me the insights, that 1 can near 3 , but also that even a Plains-Mine might be a better choice than a Green-one or even a coastal tile, like i. e. in a Buro-capital with an Academy when the city is a) large and / or the Plains-Mine is b) next to a river while the grassland-one isn't and especially c) if an extra is generated by whichever multipliers.
Some players might not want to make such efforts for such little gains, but like you said with the example of earlier city-growth, gains can cause gains again, and now imagine, that the 5 you fought for actually allow you to finish Oxford 1T earlier, then those 5 become 60-100 ! There are many more examples I could mention. Being able to reach a GP 1T earlier and thereby i. e. being able to start a GA 1T earlier and thereby save a whole turn is one that can very easily be imagined imo..
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and to regrow from size 1-2, the city needs 22 
.
through having it in 1T compared to whipping the city after growth to keep working the Plains-Iron. The Plains-Iron is 1 
.
from working coast for a turn? I guess after increased city-size maintenance that's probably not huge but might reduce the 2.6
! There are many more examples I could mention. Being able to reach a GP 1T earlier and thereby i. e. being able to start a GA 1T earlier and thereby save a whole turn is one that can very easily be imagined imo..
