The Gains Formula by Makka

Abstract
The aim of this research was to discover the formulae, equations and limits placed upon the gains of the Traditional March attack within utopia, and share it with the public. The development of this research has spanned several rounds and used a variety of data collection methods, as well as a different number of analytical methods to process the results. Ultimately, the smallest data set with the simplest analytical approach proved to be the most successful.

Introduction
For many rounds, since the gains formula changed significantly from the old system to the new, there has never been a release of the formula to the public. There are a couple of people that claim to have the formula already. Several tell the truth, but none share. I chose to make my own formula public, in order to assist every utopia player and kingdom for planning and strategy and I hope it assists everyone in some way or another.

Acknowledgements
A small number of people assisted in my research efforts on this project and I wish to give them all credit here. Some assisted with data, some with relevant conversations, and others just helped me.

Thank you for data collection: MorbidAngel, Clampy, Jobo, Midgardsorm, TheHawk, MokMok, Mr_V, and Darksel.
Thank you for the endless discussions on topic and off: Realest, Catwalk, Bytes, LightNing, my kingdom, and #strategy.

To those I have missed my thanks and my apologies.

Method
As I have already mentioned, the simplest method proved to be most successful in my research. In the interests of assisting anyone doing their own research in the future, I will discuss the process at some length in this section. If you have no interest in the methodology, or my successes and failures, feel free to skip ahead to the results section and good luck to you.

I started out by gaining access to the CSV (comma separated value) data available from Utopia Temple some time ago, which was shared in the hopes of someone discovering the formula. Using that data, I started by focusing on a single round as changes could affect gains, then I accounted for all of the modifiers in each attack (with a series of IF statements in excel) by dividing the reported gains value by each of their effects to get a RAW gains value.

• The need for using raw gains was apparent from the beginning since it puts all the data on an equal level, whether they had science or were an Orc, it no longer mattered, those variables were eliminated from further consideration.
• The only unknown variables I had left to deal with after this stage were the presence of Guard Stations (due to survey inaccuracy) and kingdom networth, so I simply deleted all data containing guard stations and with a tip from a friend ignored kingdom networth for the data sets that were in war.

The following images are at various stages of development.

The first image is prior to ignoring OOW kingdom data, which as you can see still gives a few indicators, but absolutely nothing substantial for finding any accurate formula.

1.

This second image is after I focused on war data.

2.

Subsequently, by cleaning up based on discrepancies I found in many of the attack submissions with regards to stance, relations and ridiculous inputs, I arrived at the third image.

3.

Seeing these images, you might have come to the same conclusion I did at first; that gains appeared to be linear until it reached a cap.

However; testing this in game proved that a formula following anything remotely like the values indicated in those charts were completely inaccurate. The only accurate thing about their indications were the apparent cap on raw gains around 12% relative acres, but I did not know this at the time.

It was at this point when I gave up for quite some time and worked on other projects such as optimising builds and province setups, then I started working on a wave and war planner, in which excel macros would extract data from pimp from both the self kingdom and an enemy kingdom, then assign targets for each province in the self kingdom based on max gains and whether or not they could be broken. There were many more limits and functions, but that’s the basics of it, in any case, it got to the point where I really needed a more accurate gains formula, so work was renewed.

Age 41 saw me collecting gains data myself in both WoL and Genesis, harassing everyone I could find in any channel to help me collect data also. Very few assisted, so I am thankful to those who did.

I followed the same plan with my first hand data, accounting for all the variables I could, and not recording attacks that contained variables I could not account for with 100% accuracy. I ended up with the following page of data for in war attacks in WoL.

4.

As you can see, the data followed a nice smooth line. It took very little effort (aided by the trendline function in excel) to extrapolate a best fit equation for all those values. I eliminated those attacks that looked to be clearly on the bottom feed limit, and then manually adjusted the formula to more rounded and obvious decimals, ending with the displayed equation of -0.5x^2 + x – 0.38 that matched every data point I had to within 1 acre… Good enough for me.

A little research on bottom feeding and I just slapped an estimated limit on there, not being too fussed about that. Someone may find that it is not the right cap but it's a minor issue for me compared to the main 60% to 140% range equation in war.

Taking that equation, I then started gathering out of war data on the Genesis server. Granted, the formulas might differ between the servers, but sending zero thief infiltrate ops to 50 provinces between both kingdoms out of war as opposed to 10 in genesis, plus the added problem of having the data change by someone logging in and doing something while I’m collecting so much information made it worth the switch.

So by updating every province in the kingdom, rechecking the kingdom page, recording the defending provinces networth and the updated kingdom networth for both kingdoms, along with confirming the defender had absolutely no guard stations using two surveys, I collected new data for a new problem.

The following image is a collection of out of war data, there are quite a number of discrepancies here and I cannot claim to have this portion of the equation 100%, but it seems to be quite accurate for the most part, and is at least a far better estimate than current available formulae.

5.

Something interesting I found was that I was consistently getting an estimate of +1 on the actual gains in game. I attempted to find the problem by rounding the modifiers to various decimal points, and a roundup or round down did not solve the issue. So a simple -1 acre for gains estimates was attached for all hits outside of war. This issue along with the others will likely be resolved now a lot more people will focus on refining my equations.

I haven’t described how I came to every conclusion or every step in discovering this formula, but I am sure what I have discussed is enough for anyone wishing to review my work or work on other game mechanics problems themselves. I will make my actual excel documents available on a case by case basis to those who would like them.

Results
My preliminary conclusions are based solely on the data I have made available and discussed in the method section above, and as such, with the obviously limited range of data and inevitable inaccuracies, I make no hard and fast conclusions.

Hopefully my work will produce accurate estimates, more so than current equations available, and provide a basis to launch a more focused project into refining the formula completely. Here are my preliminary equations:

Where:

x = (Defending province NW) / (Attacking province NW)

w = (Defending KD NW) / (Attacking KD NW)

Raw Province Networth Effect Equations:
0 < x < 0.6:

Raw Province Relative Gains = 0.035

0.6 < x < 1.4:

Raw Province Relative Gains = -0.5x^2 + x - 0.38

1.4 < x < ∞:

Raw Province Relative Gains = (No Data! No Estimate)

*Notes* I believe the top feed cap is the same as the bottom feed cap; However I have no data to confirm that assumption so I will not make it part of my results at this time.

Raw Kingdom Networth Effect Equations:
Well established in war:
Raw Relative Gains = Raw Province Relative Gains

Outside of war:
0 < w < 0.425:

Raw Relative Gains = 0.6 * Raw Province Relative Gains

0.425 < w < 0.825:

Raw Relative Gains = (w + 0.175) * Raw Province Relative Gains

0.825 < w < ?:

Raw Relative Gains = Raw Province Relative Gains

? < w < ∞:

Raw Relative Gains = (Insufficient Data! No Estimate)

*Notes* I personally do not believe there is a top feed BONUS as such, I think hits upwards of 0.825 KD ratio will have no deficit but no advantage either. However; I have no data to confirm that assumption so I will not make it part of my results at this time..[/i]

Modifiers:

• Race Effect = Orcs: (0.3 + 1)
• Time Effect = (Added Hours * 0.03) + 1
• Relations Effect = Hostile = (0.1 + 1); Kingdom Networth Effect is present
• Established War = (0.1 + 1); No Kingdom Networth Effect
• Stance Effect = Peaceful = (1 – 0.15); Aggressive = (0.1 + 1); Else = 1; War overrides stance: 1
• Gains Science = (1.4 * √(Gains Books / Acres) / 100) + 1: Libraries effect must be included
• Guard Stations = 1 - (2 * BE * ((Guard Stations / Acres) * (1 - (Guard Stations / Acres))))
• Dragon Effect = Emerald = (1 – 0.1); Else = 1
• Recently Hit:
  
  • GB OOW:None = 1; Little = 0.9; Moderate = 0.75; Heavy = 0.4; Extreme = 0.2
  • GB War: None = 1; Little = 1; Moderate = 0.95; Heavy = 0.9; Extreme = 0.85

Total Modifiers = Race * Time * Relations * Stance * Gains Sci * GS * GB * Dragon

Total Land Gains:

Acre Gains = Raw Relative Gains * Total Modifiers * Defenders Acres

If Acre Gains > Defenders Acres * 0.2 then Acre Gains = Defenders Acres * 0.2
All Acre Gains are rounded after modifiers to the nearest acre
If relations are not War; Acre Gains = Acre Gains – 1

Future
With this project nearing completion, I will now be making myself available to assist with ongoing research projects with the rest of the uTimes utolab crew. I have an interest in working on plunder gains as well as a breakdown on massacre damages. I would also like to investigate some of the less known aspects of thievery and magic, but if there are any projects that people feel necessary for immediate attention, please feel free to post in the uTimes forum research lab thread.

Comments
Obviously there are some gaps in my work, but I felt it was the time to release it to everyone, I am losing interest in the project and it has progressed far enough to be quite a good use for the general community.

For those wishing to use my formula in their own work be that irc scripts, bots, programs or online tools, all I ask is that you provide some recognition of myself (Makka) or at the very least do not claim my work as your own. I put a great deal of time and effort in to this research for everyone in the communities benefit, this is my only request.

If you have any questions or comments regarding gains and my equations, or need assistance implementing them, please feel free to contact me on irc.utonet.org in the uTimes research channel #utolab, or email me directly at makka32@gmail.com

Thank you.