Measurements of speed and land use in milked games

[SirPleb]

Aeson's comment "It will be interesting to see what percent of max turn score you can squeeze out" got me thinking. The game score as a percent of max turn score seems a very interesting metric for HOF games. I wondered what other metrics might be used along with it.

I don't know if the following leads anywhere. But I have come up with just two metrics which seem to, between them, describe a fair bit about a milked game. The percentage Aeson mentioned is the most important one. The other is a measurement of land use efficiency. Combining these with two factors which are constant for any given map results in an interesting way to fully break down the final score.

The four factors (two variable, two constant for a given map) are:

DominationLimit Constant for map. The number of tiles which must be claimed to trigger domination.

Difficulty Constant for map. Value from 1 to 6 corresponding to Chieftain through Deity.

LandUse (FinalPerTurnScore / Difficulty / DominationLimit) The higher this number is, the more score the player has squeezed out of the claimed tiles.

GrowthSpeed (FinalGameScore / FinalPerTurnScore) The higher this ratio is, the faster the player's score increased throughout the game.

LandUse is affected by milking efficiency and by the quality of land available. I think that on most maps played with the same settings (often pangaea, wet, warm, 5Byears for HOF games) the average quality of land available is pretty much the same. Given that, i.e. the same map settings in two different games, the LandUse value should be a reasonably good measurement of milking efficiency. Note that the LandUse number can be thought of as the average number of points obtained per tile on the map. For a single irrigated grassland tile, LandUse would be 4 - one point for the territory, 2 for the happy citizen working it, and 1 for the entertainer resulting from the two excess food. For an irrigated plains tile LandUse would be 3.5 - one for territory, two for happy citizen, 1/2 an entertainer from the one surplus food. And so on. City tiles get LandUse of 2 points - one for territory, one for the two excess food from the city tile.

GrowthSpeed is affected by a number of things: speed in reaching the domination limit, speed in reaching maximum perturn score (all tiles improved, all cities fully grown), and happiness and population levels during the expansion and conquest phases. Of these factors I think that speed in reaching the domination limit is usually the most significant by a fair margin. The ratio of NumberOfCivs to DominationLimit is probably also a major factor because it affects the limits of initial expansion, but for HOF purposes that's usually fairly constant, with huge maps and 8 rivals.

The following formula is an interesting way to look at final score as a compound of the four factors:

GameScore = GrowthSpeed * LandUse * DominationLimit * Difficulty

Here are the values for some of my milked games:

                       growth     land
  game                 speed      use
  GOTM  7 Iroquois      0.53      3.11
  GOTM 18 Celts         0.63      3.45
  HOF 47K game          0.59      3.61
  HOF Palace exploit    0.65      3.64

It seems I'm still improving a bit at milking. (But my land use in GOTM 7 probably wasn't as bad as the number suggests - its geography was much less friendly than my HOF games.)

GOTM 7 Iroquois was a deity map with a rough start position so I think the 0.53 speed wasn't bad.

GOTM 18 was a Monarch level game which is probably why I could reach the high speed of 0.63 there. I think that the highest acheivable GrowthSpeed value will go down as the difficulty level increases.

I think that the improvement in speed from 0.59 to 0.65 between my two HOF games mostly comes from using the Palace rank exploit

[Mazarin]

These are some very interesting points, SirPleb

I'm also looking forward to some data from Moonsinger's deity game that should have a higher Max turn score, but I wouldn't be surprised if her GrowthSpeed was significantly lower due to the high number of jungle near to her starting position.

[Aeson]

The game score as a percent of max turn score seems a very interesting metric for HOF games.

I have been thinking about adding an entry for % maxturnscore for milked entries. It gives a pretty good idea of how wellplayed a milked game is, without the distraction of the domination limit numbers.

It would be interesting to see how land use stacks up against the average use available per tile on the maps. As you know I use the average to predict a max turn score for each map. If there is a relatively constant relationship between attainable use per tile and the average it would help refine the prediction quite a bit.

HOF Palace exploit 0.65 3.64

The 0.65 is the same as I was able to squeeze out with my 63k game. Your use is a couple tenths higher than mine was. Here's a metric for you... Point value gained from each claimable tile (FinalScore/DominationLimit):

52842/3717=14.216303
63535/4752=13.37016

[SirPleb]

Originally posted by Aeson
I have been thinking about adding an entry for % maxturnscore for milked entries. It gives a pretty good idea of how wellplayed a milked game is, without the distraction of the domination limit numbers.

I think that would be interesting information indeed

Originally posted by Aeson
It would be interesting to see how land use stacks up against the average use available per tile on the maps. As you know I use the average to predict a max turn score for each map. If there is a relatively constant relationship between attainable use per tile and the average it would help refine the prediction quite a bit.

I'm not sure but my guess is that there's a useable but imperfect relationship. I think the average would well reflect the differences resulting from map wetness, warmth, and age settings - the impact of those settings on average land quality is probably in step with the impact on attainable high quality tiles. But the map shape is also a factor, affecting the number of coastal and sea tiles. That will be reflected in the average since the coastal tiles will count but I'm not sure if the affect on the overall average is the same as the effect on attainable average. I've been thinking that continents might actually yield the highest potential score. I've been getting better at claiming a high proportion of sea to coast tiles recently. I figure that claiming more than one sea per two coastal results in better score than claiming grassland instead of the coastal. A continents map should maximize that potential. But maybe the total effect from this is too small to matter anyway.

Originally posted by Aeson
Here's a metric for you... Point value gained from each claimable tile (FinalScore/DominationLimit)

That's another interesting one. I'd suggest further dividing it by difficulty level to simplify comparisons between the difficulty settings. This metric is different from %maxturnscore in that it doesn't assume the final milked position to be optimum. I think %maxturnscore is perhaps only really useful if presented along with the landuse number. Without landuse, %maxturnscore can be misleading - could be a high number just because maxturnscore is lower than it might have been on the map.

Hmmm, on thinking this through I just realized something.

Let's call this one:
PerformanceFactor = (FinalScore / DominationLimit / Difficulty)

And I'll call %maxturnscore "GrowthSpeed" (for anyone trying to make sense of all this so that it matches my earlier post.)

It happens to be true that:
PerformanceFactor = GrowthSpeed * LandUse
(In case anyone is interested in why I've put an explanation a bit further down.)

So the PerformanceFactor metric is a combination of the other two aspects of performance, presenting their net result in a single number.

Now that I've realized that, my preference (if you do add entries to publish any of these numbers for milked games) would be to see both GrowthSpeed and LandUse. By having the two numbers there's a more detailed picture of how the score was acheived. But this discussion of PerformanceFactor highlights that GrowthSpeed on its own is incomplete. So having both numbers seems nice to me

For those who want it, here's a description of how these numbers interact and why PerformanceFactor = GrowthSpeed * LandUse.

Definitions:
DominationLimit = number of tiles which trigger domination, constant for map
Difficulty = value from 1 to 6 corresponding to Chieftain through Deity, constant for map
FinalGameScore = game score at 2050AD
FinalPerTurnScore = game's internal "per-turn" score for 2049AD or 2050AD

Formulas:
(1) LandUse = FinalPerTurnScore / Difficulty / DominationLimit
(2) GrowthSpeed = FinalGameScore / FinalPerTurnScore
(3) PerformanceFactor = FinalGameScore / DominationLimit / Difficulty

Multiplying (2) by (1) we get:
GrowthSpeed * LandUse
= (FinalGameScore / FinalPerTurnScore) * (FinalPerTurnScore / Difficulty / DominationLimit)
= FinalPerTurnScore / Difficulty / DominationLimit
and that's the same as the formula for PerformanceFactor.

Furthermore, since we now have:
FinalGameScore / DominationLimit / Difficulty = GrowthSpeed * LandUse
we also know that it is true that:
FinalGameScore = GrowthSpeed * LandUse * DominationLimit * Difficulty

[superslug]

I figure that claiming more than one sea per two coastal results in better score than claiming grassland instead of the coastal. A continents map should maximize that potential. But maybe the total effect from this is too small to matter anyway.

That's an interesting idea, SirPleb. However, I would be greatly shocked to find out that the total effect of this would be significant enough to offset the fact that you would have one continent that doesn't receive the benefits of wonders such as the Pyramids, which have a blanket effect upon all cities on a pangaea map.