If You Can't Get Better, Get More Unpredictable
/For the weaker team in any contest which rewards only victory (or at least does so disproportionately to the rewards for proximity to victory), variability can be an advantage. Why? Because it makes it more likely that the stronger team's superiority will be swamped by chance.
The notion that altering variability might affect victory probabilities has been addressed before here on MatterOfStats. For example, I explored the potential benefits of improved consistency in one of two evenly-matched teams in relation to Scoring Shot production in a blog back in 2010, and I also unpacked the upside of inconsistency in relation to four-ball best ball golf in a blog on the Probability and Statistics portion of the site in that same year.
It's only been over the last few days though that I've come to think about variability in the way I've expressed it in that first paragraph. A few examples with dice might help to explain the concept, initially, in a relatively simple way.
Example 1
We're going to roll two die, a Blue one and a Red one.
- The Blue die is a standard die and has the numbers 1 through 6 on each of its faces (ie {1,2,3,4,5,6}).
- The Red die has the number 2 on two of its faces, the number 3 on two other faces, and the number 4 on its remaining faces (ie {2,2,3,3,4,4})
Rolled head-to-head against the Blue die repeatedly, we would expect the Red die to win about 12 times in 36, tie about 6 times in 36, and lose about 18 times in 36.
This is not altogether surprising because the Blue die, on average, scores 3.5 per roll and the Red die scores only 3.0.
Now I'm going to give you the opportunity to switch your Red die for another one that also averages 3.0 spots per roll. It's configured a little differently though:
- The new Red die has the number 0 on two of its faces, 3 on two more faces, and the number 6 on its two remaining faces (ie {0,0,3,3,6,6})
Even though the average margin of "defeat" by the Blue die is still half a spot spot per contest (ie 3.5 versus 3.0), this new Red die is expected to win 14 of 36 contests and tie 4 of 36. In AFL terms it increases its expected competition points per contest from 1.7 to 1.8.
If we calculate the variability of the scores produced by the two Red dice we find that the original Red die's score has Standard Deviation 0.89 and the new Red die has Standard Deviation 2.68. On average it still scores half a point fewer than the Blue die, as I've said, but its average loss is larger (3.0 now versus 2.1 with the previous Red die), and its average win is larger too (2.6 now versus 1.7 with the previous Red die). In summary, overall its results are more variable, the standard deviation of its "game margin" rising from 1.9 to 3.0 spots per game.
So, here we've found that the weaker die, the Red one, benefits by becoming more variable despite being no better at all on average. What if the Blue die, galvanised by this revelation, decided to become more variable itself?
EXAMPLE 2
Let's revert to the original Red die, but swap the Blue die for a new one that has the same average score but a higher variance:
- The Blue die is now {1,1,3,4,6,6}
- The Red die is, again, {2,2,2,3,3,3}
This new Blue die has a Standard Deviation of 2.3 spots per roll, up from 1.9 for the original Blue die, while its average score remains at 3.5 spots per roll.
It does not, however, benefit from the additional variability. In fact, the increase in its variability affords the same benefit to the Red die as did the increase in its variability: the original Red die is now expected to win against this new Blue die 14 contests in 36, draw 4 and lose 18.
We find then in this example that the Blue die, whose underlying ability to produce spots remains greater than the Red die's, is harmed by increasing the variability of its spot-production, while in the earlier example we found that the weaker, Red die benefits from a similar ruse.
In these examples, additional variability clearly benefits the weaker opponent.
I don't want to push this point too far in the realm of dice as I'm sure it's possible to find counter-examples where, say, an increase in the variability of a weaker die would have no benefit or might even diminish its chances of defeating a stronger die. These examples here merely hint at the possibility of result-enhancing variability - a possibility that I'll now explore in MoS' more usual domain.
A FOOTBALL EXAMPLE
So those examples are all very well for dice, you might think, but what about for footy teams? Well it probably applies to them too, I'd suggest, as I'll proceed to show.
To do this I'll assume that the Home and Away team scores are distributed as Normal variables about some expected value determined pre-game. This is a slight simplification relative to this post from 2014, which would suggest that there's some mild skew in the scores of individual teams, but it's a reasonable first-order approximation (honest).
Specifically, let's firstly imagine that:
- The Home team's score is distributed as a Normal random variable with mean 110 and standard deviation 25
- The Away team's score is distributed also as a Normal random variable but with a mean of 90 and the same standard deviation of 25
In that circumstance, making the further simplifying assumption that the Home and Away team scores are independent (which is not quite true in practice, but can be safely ignored here), the final game margin from the Home team's perspective has mean 20 and standard deviation of 35.4, and the Away team's probability of victory is just 28.6%.
Now let's imagine that the Away team can devise a method of playing that increases the variability of its scoring without affecting its mean, specifically such that it lifts the standard deviation of its scoring to 30 points.
That change in the Away team's behaviour doesn't alter the mean of the final game margin, which remains at 20 points in favour of the Home team, but it does change the standard deviation of the distribution of that margin, lifting it to 39.1 points. Most importantly, that change in standard deviation lifts the victory probability of the Away team to 30.4%, an increase of almost 2% points.
You can see what's happening diagramatically via the chart above. The increase in the variability of the Away team's score translates directly into an increase in the variability of the final game margin, altering the game's outcome from one being drawn from the Normal distribution shown in cyan to one being drawn from the distribution shown in red. That the Away team fares better in the higher variability scenario is reflected in the fact that the area labelled 2 exceeds the area labelled 1.
So, here too we have an increase in the variability of the performance of the weaker team leading to an increase in its chances of victory. QED.
PRACTICAL IMPLICATIONS
This result only has practical applications if it's possible to adopt strategies in-play that affect the variability of outcomes in a way that has, roughly speaking, equal upside and downside so as not to dramatically affect the average of those outcomes.
The weaker team can afford to take a slight hit to its average performance in order to increase its variability, however, and still improve its overall chances of victory. In the previous example, if we set the Away team's expected score to 88 (rather than 90) when we increase its standard deviation to 30, it's still very marginally more likely to win (28.7%) than when its expected score is 90 and standard deviation 20 (28.6%).
Whether or not a deliberate strategy in football - or in any sport - to be more random but, on average, no worse or only very marginally worse is a viable one in practice, is an assessment I'm not qualified to make.
I'll finish by noting a logical/mathematical consequence of the result we've developed here for football, and that is that increased variability in the performance of the stronger team is not in its interests. If, for example, we take the Home team from earlier and increase the standard deviation of its scoring to 30 points per game, leaving its mean score at 110, and pit it against that original Away team with mean score 90 and standard deviation of 25, we find that the Home team's probability of victory is exactly as it was when we increased the Away team's standard deviation to 30, falling by about 2% points. That's because the standard deviation of the margin is the same in both cases.
Make of all that what you will, but I'd be interested to hear your thoughts of the relevance, if any, of this analysis. Do you think it'd be possible to coach for greater variability of performance without dramatically and adversely affecting average performance? If so, how do you imagine that might be done?
