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Wednesday, June 1, 2011

To Balance Your List or Not To Balance Your List?

This is a continuation of my previous musings from last week. The thought? Overall, will a balanced list perform better at tournaments than an unbalanced one? First, what does balanced mean?

In the context of this thought experiment, the term "balanced" means "equal chance to beat all types of lists". Thus, the perfectly balanced list means that it has a 50/50 shot of winning every game it plays, versus 100% of all other lists, assuming all other things being equal.

Thus, an unbalanced list must be a list which has an uneven chance of beating some lists and losing to others. For example, you might tailor a list to win a high percentage of the time versus most other kinds of lists you expect to face but expose weaknesses to a lesser percentage of lists. For example, an unbalanced list might be a list which wins 70/30 of the time versus 70% of other lists and only 30/70 of the time versus the remaining 30% of lists.

Here's a more concrete and less math blah example! Let's say you come up with a killer Tyranid list that rocks most Marine lists provided they don't have Land Raiders in it. If we could somehow calculate the number of Marine lists without Land Raiders and came up to 80%, and your Tyranid list wins 80% of the time versus Marine lists without Land Raiders, we can say that your list wins 80/20 versus 80%. If your Tyranid list only wins 10% of the time if the Marine list contains Land Raiders, your list has a 10/90 versus the remaining 10%.

Obviously, an unbalanced list will perform better with a higher win percentage versus a higher percentage of armies and a lower losing potential versus the lowest percentage of armies.

There's so much to investigate on this topic because the notion is exciting. Keep in mind though that all of these numbers, at the end of the day, are mostly bogus. I mean, seriously. How can we calculate your "win %" with "all things being equal" versus some arbitrary "percentage" of lists out there. We can't, but you can make a serious go of it. You could look at lists brought to tournaments in the last year, play against them with someone about equal skill as you, several times each, and get a rudimentary percentage. It won't be perfect, but it's better than nothing. In any case, I just wanted to come right out and say, at the end of the day, any results we come to will be shaky at best!

So, in the previous article I showed that a list with a high percentage chance to win versus a high percentage of armies, and a lower percentage chance to win versus a low percentage of armies actually performs better in a 4 round tournament than a "balanced" list most of the time.

The next thing I wanted to know was if common wisdom is right... shouldn't the more rounds you play in a tournament mean the balanced list does better? Turns out, common wisdom is wrong. Not just wrong, but completely wrong. Indeed, the numbers show us that the more rounds you play, the more likely the unbalanced list is to outperform the balanced list.

Here's some numbers for you to chew on.

List A)

Let's assume that List A has a 70% chance of beating 70% of all lists. It has a 30% chance of beating 30% of the remaining lists. (Also seen as 70/30 V 70% and 30/70 V 30%)

List B)

Let's assume that List B is a "balanced" list and has a 50% chance of beating 100% of all other lists. (Also seen as 50/50 V 100%)

Assuming Win/Loss Tournaments. Also, assuming that a Positive (+) matchup for A is one of the 70% of the lists A does well against. A Negative (-) matchup for A is one of the 30% of the lists A does not do well against most of the time.

2 Round Tournament:

A) 49% (2+) : 49% to win.
42% (1+/1-) : 21% to win.
9% (2-) : 9% to win

B) 25% to win (0.5 * 0.5)

List A outperforms List B 49% of the time. Thus, List B is a better choice for a 2 Round Tournament.

4 Round Tournament:

A) 24% (4+) : 24% to win
41% (3+/1-) : 10% to win.
26% (2+/2-) : 4% to win
8% (1+/3-) : 2% to win
1% (4-) : 1% to win

B) 6% to win

List A outperforms List B 65% of the time. Thus, List A is a better choice for a 4 Round Tournament.

6 Round Tournament:

A) 12% (6+) : 12% to win.
30% (5+/1-) : 5% to win.
32% (4+/2-) : 2% to win
18% (3+/3-) : 1% to win
6% (2+/4-) : 0% to win
1% (1+/5-) : 0% to win
0% (6-) : 0% to win.

B) 1.5% to win.

List A outperforms List B 74% of the time. Thus, List A is a better choice for a 6 Round Tournament.

8 Round Tournament:

A) 6% (8+) : 6% to win.
20% (7+/1-) : 2.5% to win
30% (6+/2-) : 1.5% to win.
25% (5+/3-) : 0.5% to win.
14% (4+/4-) : 0.2% to win.
5% (3+/5-) : 0% to win.
1% (2+/6-): 0% to win.
0% (1+/7-) : 0% to win
0% (8-) : 0% to win.

B) 0.4% to win

List A outperforms list B 81% of the time. Thus, List A is better for an 8 Round Tournament.

Seeing a pattern? You should be noting that not only is A more likely to outperform B the higher the rounds go, the % by which A is more likely to outperform B goes up.

So, I did a little study after realizing that. What about lists that are closer to balanced or further away? Well, it ended up like this:

The Gray Line represents the balanced list. As the rounds go up, it begins to under-perform versus unbalanced lists.

The Red Line represents an unbalanced list with a very high win % versus a very high % of lists. and a low loss % versus a low % of lists.

The Green and Blue lines represent unbalanced lists with progressively lower win % versus lower % of lists OR higher loss % versus a higher % of lists, respectively.

The way it works is just like you would expect. The better your unbalanced list is versus most other lists, the better it will do. The way that is measured in this experiment is by the "SPEED" in which it will start to outperform the balanced list where the "SPEED" is the number of rounds in the tournament.

So, a better performing unbalanced list will start to outperform the balanced list in less rounds. A lesser performing unbalanced list takes more rounds to outperform the balanced list.

Eventually, though, all unbalanced lists will outperform the balanced and approach 100% of the time as tournament rounds go up.


  1. I tried to add this on your last post but it got lost in the warp! Here goes again...

    This example is not useful because you are comparing two lists with different win percentage expected values. Of course the 70/30 list wins more - its expected value to win is:
    .7 x .7 + .3 x .3 = 0.58

    Whereas the balanced list has an expected value of:
    .5 x 1 = 0.50

    So as you play more games, the 70/30 list is going to soar ahead. Instead you need to redo the above with a list that has an expected value of 0.5 but not evenly distributed. A list that had a 60% chance of winning against 60% of lists and a 35% chance of winning against 40% of lists would have such an expected value:
    .6 x .6 + .35 x .4 = 0.5

    What happens then?

  2. Don't be the guy who shouts out the ending of the movie. ;) We'll get to that!


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