## Sunday, November 22, 2009

### Interesting Killpoints math over on Warseer

Saw this over on Warseer, figured I'd repost it here:

Yes, it's another Kill Point thread. I hope this one shows something new, so bear with me.

We've had a thread on whether KP's are a good mission, and whether they are "balanced." Clearly, on average, the smaller army has an advantage. The pro-KP side will contend that this is the case, but it balances out an equal and opposite advantage for large-unit-count-armies in objective missions, they will have more units to take or contest objectives with, and therefore most armies will gravitate to the same number of units, balancing out the mission.

Theory

This is largely a repost from the previous thread, so feel free to skip this section if you've gone through this (or if you hate algebra!)

Assuming you have two players, X and Y. X has x units, Y has y units and we can assume these are not equal and x>y. X scores b KPs and Y scores a KPs. In order for X to win, b>a. But in order to do so, he must kill the fraction b/y of the enemy army. So assuming that ca=b (where c is a constant), in order to draw a KP mission:

ca/y = a/x

=> c = y/x

Conclusions: player X must kill y/x more of the enemy army than Y.

Now if we look at the objective mission (where things get more fuzzy): Player X has x scoring/contesting units and Player Y has y scoring/contesting units (and x>y again). There are z objectives. Player X loses a units and player Y loses b units. Assuming each unit can only capture/contest 1 objective each (big assumption!), that all units can contest all objectives (another big assumption) and, for simplicity's sake, that one player will win if he has at least one less untaken objective than his opponent, then the number of untaken/uncontested objectives U(x) or U(y) are:

U(x)= (x-a)-z (min 0)
U(y)= (y-b)-z (min 0)

U(x)-U(y)>1 for Y to win
So ((x-a)-z)-((y-b)-z)>1

This can only happen if x-a and y-b>z

Assuming that both suffer equal proportions of losses, (a/x and b/y), then x-a>y-b therefore given the assumptions above, yes, X will be expected to win but only if x-a>z. Therefore the optimum number of units is the minimum possible to leave z scoring/contesting units surviving the game, less if they can (a) take more than one objective or (b) they can move freely about the board and the enemy cannot.

Conclusion: an army with a larger unit count has an advantage in objective missions but only a) if the enemy has less than z units by the end of the game, (and remember that on average z=3and z is reduced by 1 for each enemy unit that is in range of multiple objectives); and b) if the numerically superior army has free reign to move about the board.

Statistics

So now to test it!

So, the predictions are:
• In Kill Point missions, the smaller army is expected to have an advantage.
• In Capture and Control missions, z=2, the advantage to the larger army is negligible, so neither is expected to have an advantage.
• In Sieze Ground missions, the larger army is expected to have an advantage.

In order to test the predictions, I took a random sample of battle reports, namely the first 10 pages of the Warseer Battle Reports Forum (as they were November 19th). Each battle report was scrutinised and the following data recorded:
• Mission Type
• Number of Objectives
• Number of units in each army (taking combat-squads or combined infantry squads into account)
• Outcome of game
• Whether the larger or smaller army won.

Battle reports were screened and only battle reports that met the following criteria were included:
• A detailed list of both armies' units
• Standard 40K missions (no planetstrike/ard boyz/etc)
• An unequal number of units between the armies
• Clear, online written format (I excluded all video reports, for example)
Obviously this screening was done before looking at the results, and needless to say all eligible reports were included. Edit: Also, to prevent bias, only one report per player was used (as much as I could tell). So the results weren't biased by a particular player submitting multiple reports with the same army (where the player skill is not independent from the army size). If a player submitted multiple reports, the first eligible one was used.

Results are attached as an excel file [edit: text file as I'm not allowed to attach excel files, if anyone wants the original .xlsx file pm me]. To summarise:

Sieze Ground
Total Games = 18
Won by smaller army = 7
Won by larger army = 7
Tie games = 4
Chi-square = 0
P = 1

Capture and Control
Total Games = 24
Won by smaller army = 9
Won by larger army = 7
Tie games = 8
Chi-square = 0.57
0.5 > P > 0.1

Annihilation
Total Games = 38
Won by smaller army = 24
Won by larger army = 11
Tie games = 3
Chi-square = 15.36
P < values =" number">Conclusions.

Sieze Ground and Capture & Control missions did not show any significant deviation from random, although C&C did show a slight trend towards the smaller-unit-count player (quite plausible, as small, elite, fearless armies may have an advantage in holding and keeping a small number of objectives). Annihilation missions showed the expected and highly significant advantage to the smaller player, with over double the number of games being won by the smaller side. Two games were noted by the authors as being a VP win for the side that lost by KP.

So: is there an advantage to high-unit-count armies for objective missions that counterbalances the clear advantage for small armies in annihilation missions? Answer: no.

Thanks to Lord Inquisitor for all the stats work, it's been too long since my one stats class.

#### 1 comment:

1. So is there an advantage to high-unit-count armies for objective missions that counterbalances the clear advantage for small armies in annihilation missions? Answer: no.
And the reason for this? Control.

As it currently works, you only need 1 unit to control or contest an objective. Heck, if the objectives are close, one unit can even control multiple. In return, it only takes 1 unit to contest an objective too. Additionally, you only need to control 1 more objective than your opponent to score a "win".

Thus there's no real incentive to take more units for objectives games while KP benefits those armies that take less. What could balance this?
1. Stacking control/contest checks. Example: An objective has 3 units on it, 2 for player A and 1 for B, so control goes to A for having more units total.
2. Make winning an objective mission harder. Rather than only needing 1 to "win", require holding half or more. Thus Capture & Control still on requires holding 1 objective, but winning a 5 objective Seize Ground would mean you need 3. (Or do a tiered win system, but that only tends to matter in tournaments.)
3. Add in more missions/victory conditions. Table quarters, Loot, Assassination... there's lots of things that could open up the scope of missions beyond 1 KP and 2 objectives.
4. Go back to VP. KP was a cute idea to simplify victory calculation, but the reality is that it is a failure. There's just too many ways to exploit the rules advantageously.

Of course, the problem with any of the above is that they require a deviation from the main rulebook missions. This means they're relegated to the world of house-rules, tournament organizer choice, or 6th edition. Still, maybe if enough of us call foul, GW will listen. I'll be over here not holding my breath. :-p

Cheers for a good write-up and detailed research. I'll be back-linking this.