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WARNING

I believe that the data below has errors in the defense strength, so is therefore not solvable. I will update it when I have more information. Thank you.

I play a game (Empire: Four Kingdoms) in which soldiers attack castles where other soldiers defend, and I'm trying to solve for the equation that they use in order to calculate losses.

I have some data points but am having a hard time determining the actual formula.

In general terms, if the attack strength stays the same, adding more defenders will lower the defenders losses (as may be imagined), but it is not linear (i.e. doubling the defending strength will not halve the losses).

Here are some data points that I have so far (in all scenarios below, all attackers were lost, but at this point I am only concerned with the defenders losses):

Qty Attackers, Qty Defenders, Attack Strength, Defense Strength, Qty Defenders Lost

52, 78, 11,954.8, 13,459.68, 69
52, 182, 11,954.8, 31,658.59, 44
52, 138, 13,041.60, 33,506.83, 35
52, 103, 11,954.80, 23,549.26, 39
52, 64, 13,041.60, 32,487.89, 17
136, 190, 17,912.83, 56,612.55, 7
42, 94, 4,869.23, 23,403.63, 7
42, 86, 4,869.23, 25,792.09, 4
42, 87, 4,956.00, 24,399.58, 5
42, 82, 4,956.00, 16,448.96, 13
46, 70, 4,724.38, 28,339.08, 5
46, 65, 9,537.00, 11,953.70, 45
46, 65, 9,537.00, 12,773.50, 41
52, 247, 8,816.08, 47,454.43, 15
52, 232, 10,032.36, 98,829.79, 8
52, 224, 8,816.08, 61,920.54, 12

Community
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lnafziger
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  • This seems more like a stats.SE question. – Mario Carneiro Feb 21 '15 at 04:51
  • The simplest possible model would be $$ Aw + Bx + Cy + Dz = L, $$ where $A,B,C,D$ are constants, not necessarily positive, and $w,x,y,z$ are your first four variables, Since you keep the first fixed at 52, you have no way of finding $A.$ Finding the numbers is called linear algebra. Finally, it is someone else's program, we cannot be sure it is that simple. – Will Jagy Feb 21 '15 at 04:52
  • @WillJagy To add to this, realize that there are only five data points and five constants, so you are very likely to overfit with only this data. I would recommend at least 10-15 data points before you can make any reasonable guesses. – Mario Carneiro Feb 21 '15 at 04:54
  • @MarioCarneiro, agreed. – Will Jagy Feb 21 '15 at 04:58
  • @MarioCarneiro Well, I can get more data points, but I feel like it should be a pretty simple formula (maybe I just need to square something or take a square root) and have been trying to solve it intuitively. I'm hoping that someone else's intuition is better than mine though since it hasn't been working, lol. – lnafziger Feb 21 '15 at 05:05
  • @lnafziger You are probably right about that, but there just isn't enough data with this many variables in play. Try coming back with more data. – Mario Carneiro Feb 21 '15 at 05:07
  • Oh, and I'm not sure that the number of attackers or defenders actually plays a part in this (in fact, I'm pretty sure that we simply lose a certain percentage of defenders, based on the ratio between attacking and defending strength). I included them just in case. – lnafziger Feb 21 '15 at 05:07
  • @MarioCarneiro I have added more data points per your request. – lnafziger Feb 21 '15 at 05:46
  • Is the WARNING recent? – Cesareo Sep 06 '23 at 08:02

2 Answers2

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I tried some fits based on your guesses about the formula, but I did not find enough improvement to justify them over the basic guess $D=d\dfrac{s_a}{s_d}$ where $d$ is the number of defenders, $s_a,s_d$ are the attacker and defender strengths, and $D$ is the number of defender deaths. However, this is not a very good fit either. Although it follows the rough contours, it is often off by $10$ or more, with the worst outlier being the datapoint

$$a=136,\ d=190,\ s_a=17\,912.83,\ s_d=56\,612.55,\quad D=7,\quad \overset{\sim}D=57.99$$

making me suspect that there is a large random component to the data (which needless to say makes fitting it much more difficult). (Of course the best proof that there is a random component is getting different outputs for the same input, but barring that you can make certain claims about the complexity of the function if you get enough closely spaced input data with widely different output values.)

Mario Carneiro
  • 27,438
  • I know from experience that there is no random component because the same attacks have the same results over and over. Thank you for looking though! – lnafziger Feb 21 '15 at 13:12
  • @lnafziger Are you sure that the four variables you listed are the only relevant ones? – Mario Carneiro Feb 22 '15 at 01:29
  • Well, I've been looking at it more and may need to revisit how I calculated the defense strength. There are a lot more variables, but I thought that I had them figured out and unfortunately it's looking as if I don't have them quite right yet. I will keep working on it and update the question when I have better information. Thanks! – lnafziger Feb 22 '15 at 02:10
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You get the same results most of the time not all of the time. There is a random variable used in calculating the result. That is from 30, 000 attacks experience.

user251106
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  • There is some type of exponential function involved that applies to the ratios. There is also a prioritization of losses towards particular defender types and a strength threshold below which none of the defender types are lost. This threshold is firstly determined by total number of defenders ie there has to be sufficient to win, and then it is determined by defender to melee ratio – user251106 Jun 28 '15 at 13:45
  • That exponential function is what I'm trying to solve for. :) – lnafziger Jun 29 '15 at 14:58