Cherishing Inconsistency Where It's Welcome

In an earlier blog we demonstrated the benefits of consistency in football - moderate benefits if the consistency came in the form of generating scoring shots with less variability than teams of otherwise similar ability, and significant benefits if it came in the form of converting more of those opportunities into goals.

Indeed, consistency's a characteristic that sports commentators reserve for their warmest - and often longest - soliloquys, and players and teams, once they've reached an acceptable level of performance, announce as though scripted that they're now "striving for consistency". So, surely, consistency is always a good thing, isn't it?

Well no, not always. There's a sport - or a version of it - that rewards a modicum of inconsistency. It's four ball, better ball golf, which I'll abbreviate to 4BBB for the remainder of this blog. For those of you unfamiliar with this golfing variant all you need to know is that the contest is played between two teams each comprising two golfers. All four golfers play every hole and a team's score for a hole is the lower of the two scores of its team members. So, if the two golfers on Team A make a 4 and a 5, while those on Team B make a 3 and a 7, Team B wins the hole because their lower score is 3, which beats the lower score of 4 for Team A.

When the low score for the two teams is the same, the hole is said to be 'halved' and, effectively, nobody wins it. The team that wins the contest is the team that wins the greater number of holes of those played. For our purposes we'll assume the teams play 18 holes each and that all participants play off what's called "scratch", which means that their final score for a hole is the number of strokes they took to complete the hole, which is not adjusted in any way to account for the differing abilities of the participants. 

(I know, by the way, that this form of golf is usually called "four ball, best ball" but the choice on each team is between two scores so it should really be "better" rather than "best". Pedantry, like mould, once it takes hold pervades everything.)

Okay. Now imagine the following scenario. You're scheduled to play a 4BBB against two equally-talented golfers. Both of them can be expected to double-bogey 5% of the time, bogey 20%, par 50%, birdie 20% and eagle on 5% of holes. You're of the same calibre as these two opponents and can be expected to shoot double-bogeys, bogeys and so on in the same proportions. 

You have a choice of two partners for the match: 

• Partner C (for 'Consistent') who is of the same calibre as you and your opponents and,
• Partner E (for 'Erratic') who is less consistent and a little less talented. He double-bogeys 5% of the time, bogeys 33%, pars 30%, birdies 30% and eagles only 2% of the time, which means that he shoots, on average, a smidgeon over-par. His inconsistency manifests in an excess of bogeys and a scarcity of pars and eagles relative to you and to your opponents, but, importantly, it also manifests as a relative excess of birdies.

All players have the characteristic that their score on any given hole is unaffected by their own scores on other holes and by the scores of the other golfers on this same hole or on previous holes. In statistical terms, this means that each golfer's score on a hole is 'independent' of their own and other's scores on this and on any previous holes in just about every way you can think of.

Which partner - C or E - offers you the better chance of victory?

As you've probably already guessed by now, it's Partner E. With him you can expect to win 32.5% of holes, halve 36.3% of holes, and lose 31.2%. Paired with your consistency and pitted against your opponents', it's his inconsistency that makes him valuable to you. He shoots over par more often than you or your opponents do (he'll average about 1.6 over par on average over the 18 holes), but your consistency often saves him when he does that. Vitally, he also breaks par more often than anyone else and, since it's the low score that wins the hole, that makes him, in the current scenario, an asset often enough to be valuable. 

Greater inconsistency, however, is not always beneficial. To pick an extreme example, if Partner E produced birdies 26% of the time - so he still breaks par more often than do you or your opponents - but double-bogeys the other 74% of he time, then the pair of you could expect to win only 29.1% of holes, halve 33.3%, but lose 37.6%. In aggregate, then, you'd lose about 8.5% more than you'd win and you'd end up drinking in the 19th far more often to commiserate rather than to celebrate.

Taking the consistent you out of the picture for a moment, it's true that mutual, but again independent, inconsistency can be beneficial too. If, for example, two players like partner E paired up against the consistent duo, they'd win 33.6% of holes, halve 34.1%, and lose 32.3%. In total then, they'd win 1.3% more than they'd lose, the same nett result as a consistent you and a partner of type E would achieve. 

The best partner of all, though, is one whose scores are negatively correlated with yours. So, for example, imagine a partner who, when you shoot bogey, tends to shoot par or better and, conversely, when he or she shoots bogey or worse, you tend to shoot par or better. In this case we're now breaking the previously stated assumption that the scores of each golfer are independent. 

So let's return to the situation where you and your partner face that same consistent duo and let's assume that, overall, both you and your partner generate eagles, birdies and so on at the same rate as they do. Now, as an example of negatively correlated scoring, you and your partner's scores have the following characteristics: when you shoot eagle, your partner shoots bogey half the time and double-bogey the other half; when you shoot birdie, your partner shoots par 20% of the time, bogey 75% of the time and double-bogey the other 5%; when you shoot par, your partner shoots eagle 4% of the time, birdie 26%, par 62%, bogey 5%, and double-bogey 3%; when you shoot bogey, your partner shoots eagle 5% of the time, birdie 20% and par 75% of the time; and when you shoot double-bogey, your partner shoots eagle 40% of the time, and birdie the other 60%.

Those percentages mean that your score and your partner's are highly negatively correlated, and that makes you a formidable pair. Against the consistent duo you can now expect to win 34.7% of holes, halve 39.6% and lose only 25.8%. On balance, you'll win about 9% more holes than you'll lose. Remember : you and your partner both, on average, tend to produce eagles, birdies and so on at exactly the same rate as your both of your opponents. It's just the emergent property that is your negatively correlated scores that makes you so devastating to encounter.

In summary, inconsistency can be good in 4BBB, but only if it's moderate inconsistency. Negative correlation is even better (as indeed it is, and for similar reasons, if you're looking for a valuable asset to add to your portfolio).