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Monday, June 6, 2011

Balanced versus Unbalanced Part 3

So, in the last article, I did an analysis of an unbalanced list versus a balanced list and their respective, projected performance across rounds of a win/loss tournament. By looking at the analysis, you would think that there never is a reason to take a balanced list to a tournament. However, as you will note, I also noted that as the percentages changed, so too did the results. The reason is that our study was completely correct, but these changes in the effectiveness of each list, especially those that seemingly approached a balanced list slowing down in performance showed a clear problem with the entire study.

The real reason why our unbalanced lists keep seeming to outperform our balanced lists in these examinations is not because of some hidden truth. It’s actually a good bit simpler than that. The real reason is that, statistically, the unbalanced lists we have been examining are just flat BETTER than our balanced list.

How so? Are you saying that somehow our unbalanced list is better than a balanced list? No. What I’m saying is that our assumptions about how an unbalanced list and a balanced list, using our previous examples, is fundamentally flawed because we are comparing a jet plane to a bi-plane.

The reason why we could see this happening was that as we change the win percentages on our unbalanced list, the rate at which they would outperform the balanced list also changed. This was clearly indicative that not all unbalanced lists or equal, and indeed, it was also clearly shown that the unbalanced list was not equal to the balanced list in terms of quality as they began to approach each other.

As was pointed out in the comments of the previous post, statisticians use a concept called “expected return” to describe what the expected result of a scenario will be in the long term. For example, we all know that a coin has a 50/50 chance to land on heads or tails. Thus, our expected return of Heads when flipping a coin is 50%.

Likewise, our balanced list has an expected return of 50%, being that is has a 50/50 chance of winning any game. However, our list that would win 70% of the time versus 70% of opponents and 30% of the time versus 30% of opponents “appear” to be a good logical comparison, but it really isn’t. The truth is, the expected return of “WINS” for that list is actually (0.7 * 0.7) + (0.3 * 0.3). Thus, the expected return from this list is actually 58% wins and 42% losses. The reason why it is outperforming our balanced list is simply because it’s a better list! It has nothing to do with the balanced or unbalanced nature of the lists. If you compare two lists with an equal expected return, you will find out that the results vary as the round go up as to which list is better but actually tends to favor the balanced list. The truth is, in the long run, they are probably fairly equal.

So, what you are saying is that, this is a sham? No, what I’m saying is that the problem is that we are so focused on balanced versus unbalanced in this discussion when what we should be focused on is the concept of expected return. (Congrats to the commenter from last post for figuring this out early!) The real truth is, balanced or not, it’s the expected performance of the list which is what is going to carry the day across many rounds of tournaments.

So, how is this useful? In some ways it really isn’t which is in and of itself is a useful result. As we said last time, at the end of the day, any result we would come up with would be shaky at best, and what we have discovered is that the assumption about making an arbitrary comparison between what seems like an unbalanced list versus a balanced one is not what we need to do doing. If you want to know if a more balanced list performs better or worse than a more unbalanced list, the way to do that really is as simply as playing a lot of games with it and seeing which one wins more.

The reason why this is true is the only way you can even start to gather data about the overall effectiveness of the list is to play it. Then, the better list will be the one with the highest expected return against across a variety of games.

The reality is, at least in terms of this discussion, a better list is simply a better list.

The next thing to discover will be if two lists which have equivalent expected win rates but different unbalances will perform the same, better, or worse. Evidence so far supports that as the rounds go on, the lists which approach more even percentages tend to perform better.

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