Wednesday, January 11, 2012

Theory of Relativity

"I didn’t have a clue about the winner. I’ve not seen him all year and I didn’t know anything about him or what he was capable of." (Mo Farah, Aug 2011)

Some quotes from Chris Powell in the News Shopper piqued my interest this week.

In particular, I have to take issue with his comment that, “I see the Hillsborough fixture as just another game in our season.”

Now to be fair to Powell, he may be playing some form of mind games, keeping his team relaxed and not lifting energy-sapping nerves unnecessarily.

However if he genuinely means it, then I think he is very much mistaken. The same obviously applies to the upcoming Sheffield United fixture too.

I will term what I will discuss as New York Addick’s, “Theory of Relativity”.

At the start of the season, especially in a relatively hard-to-predict division like League One, the opening game is by definition 100% driven by ‘absolute factors’.

In short, not only does a team not know with great confidence whether it will be competing at say the top, middle or bottom of the table, but it also doesn’t know which of its fellow teams will be competing there either!

For example, whilst plenty of observers might have ‘fancied’ say Huddersfield or Charlton to do well ex ante based on the usual factors, likewise one could have made a similar case for the likes of Preston or Scunthorpe, neither of whom are competing.

Thus on opening day and indeed for the first several games, the target is purely an absolute one. Get as many points as possible, then see where you are.

At the other extreme, when the 46th game of the season kicks off in May, for most of those teams still with something to play for, the target is purely a relative one.

In short during that final game, judging how your team is doing cannot be assessed solely on the basis of the scoreline of its own match.

Defeat may be good enough if other results are going your way, and likewise victory may not be enough.

By definition therefore somewhere between these two extremes of the 1st and 46th fixture of the season, a ‘tipping point’ must be reached where the importance of the absolute number of points achieved becomes superseded by the importance of the number of points relative to rivals.

A good estimate for the fixture at which this tipping point is reached might well be the halfway point of the season, in our case the start of the Brentford game two Saturdays ago, our most recent League game.

However after Charlton had thumped Preston 5-2 on Bonfire Night, the League One table already had a familiar look to it:

Charlton P17 40
Huddersfield P17 35
Sheff Weds P17 33
MK Dons P17 30
Sheff Utd P17 29

Thus with the acknowledged benefit of some hindsight, the 'tipping point' had probably already been reached after just 17 games, given the constituents of the top five.

On an absolute basis since then, most Charlton fans would probably conclude our form has been just fine: 14 points from 7 games.....promotion form surely?

However, whilst after 17 games the combined points gap of the four chasing teams behind us was 33, it is now only 24 (admittedly two have played one game more).

So in an absolute sense we have been doing great, but in a relative sense (and extrapolating), it would be a very close run thing whether we would even keep our noses ahead at this rate especially if we treat the games directly against the quartet as 'just another game'.

In order to explain better my theory, imagine say a 10,000m athletics race in which a pack of five runners has pulled clear of the field.

At the start of the race, a runner (let’s call him Mo) would no doubt have had a time target in mind which if achieved, would likely win him a gold medal (this is the ‘absolute’ target).

Indeed as the first few laps are completed, if a healthy pace has been set then this absolute target is important because the runner will know few will be able to stay in contention at this rate for 25 laps.

However as laggards do indeed begin to drop away and just five runners remain in contention, suddenly Mo’s original absolute target becomes virtually obsolete.

Even if the pace were to drop by a couple of seconds per lap, the remaining runners are too slow and too far back to catch up anyhow.

Now it’s only about the tactics that the other four competitors adopt – the race will now be won or lost purely on ‘relative’ factors.

If one of the runners ‘kicks for home’ then Mo can’t just look at his watch and determine that he remains on course for his original absolute target. He needs to kick too.

Mo may end up winning glory with a time worse than his original ‘absolute target’, or finish 5th with a better time.

Returning to football, the ‘absolute target’ in our case is presumably a points total which usually wins promotion.

Whilst we can make an educated guess about how many points will be required (by extrapolating from current points, or looking at previous seasons), it remains a highly uncertain data point.

However the ‘relative target’ is simply actual automatic promotion, which can be defined in very simple and stark terms:

To finish the season behind no more than one other club

Alternatively, given the form of the top five and the gap that exists behind them, I think realistically this target can simplified even further:

To finish the season ahead of three or more of the current top five

Whilst this ignores the prospect of say a Carlisle or Stevenage making a run for promotion, I think my athletics analogy above is useful here - under almost all realistic scenarios, they are already too far behind.

When assessing the importance of the upcoming two games, allow me for simplification and for the timebeing (see below) to assume all other games involving the chasing quartet are postponed.

If Charlton win both games, the table would look like this on 21st Jan:

Charlton P26 60
Sheff Utd P26 50
Sheff Weds P25 49
Huddersfield P25 47
MK Dons P24 46

At that point, we are not just looking good....we are virtually home and dry, even with three of the other sides having games in hand.

The 63% probability of automatic promotion discussed in my last post has probably become 85%.

Conversely, let's consider if we lose the next two games.

At this point I should also mention that the 'postponed games' alluded to above will actually (weather permitting!) be as follows:

Sheff Utd: Bury (A)
Sheff Weds: Hartlepool (H)
Huddersfield: Oldham (H), Brentford (H)
MK Dons: Carlisle (H), Notts C (A)

Imagine a nightmare scenario (but hardly an unlikely one) then in which all of the above games are won by a top five club, and we lose the two Sheffield games above.

Then the League table looks like this on Jan 21st:

Sheff Utd P27 56
Sheff Weds P26 55
Charlton P26 54
Huddersfield P27 53
MK Dons P26 52

At this point after factoring in the devastating impact on confidence, I could make a serious case that we would be fifth favourites for promotion in just nine days' time!

My Theory of Relativity is not a pessimistic one, far from it.

I am not for one moment suggesting we are not more than capable of taking apart both of the Sheffield sides.

But crucially it is vital to understand that it is only in these games that we can directly impact the form of our rivals; they really are 'six pointers'.

Have two in a row, and you effectively have twelve points at stake in just three hours of football.

It would be akin to our athlete Mo having the ability to trip up two of his five rivals on the back straight!

My Theory of Relativity thus simply states that when form has become more relative than absolute (ie. when the tipping point has been passed), believing that fixtures against your direct rivals are 'just another game' is a gross mistake.

Which given Chris Powell is an intelligent man, is why I suspect he doesn't believe it at all.

Sunday, January 08, 2012

The 27 Club

"There are known knowns, there are things we know we know. We also know there are known unknowns, that is to say there are some things we know we do not know. But there are also unknown unknowns - there are things we do not know we don't know."(Donald Rumsfeld, Feb 2002)

The relief that we didn't actually win at Fulham was palpable (enough indeed to encourage me to write a rare blog post), and now the priority can rightly return to League One matters.

Readers of my We Are Leeds? blog will recall my concern that our season was resembling theirs rather too closely from 2009/10.

Their remarkable start to the season (they had accumulated a whopping 57 points from 24 games, compared to our own impressive 54) only went downhill rapidly after their FA Cup exploits misallocated vital focus and energy.

Thus I can only commend Chris Powell's team for skilfully balancing the need to make the 7,000 travelling Addicks proud, whilst at the same time ensuring they went out of the competition.

More evidence surely of a truly great side.

Looking forward, there are now four possible outcomes for this season:

Outcome A: automatic promotion
Outcome B: promotion via play-offs
Outcome C: play-off defeat
Outcome D: finish outside top six

As a brief aside, if I asked you the following questions, what would your answer be?:

Question 1: which of the following events is more likely? Charlton suffer Outcome C (play-off defeat), or you randomly select a diamond from a full pack of cards?

Question 2: which of the following events is more likely? Charlton suffer Outcome C (play-off defeat), or a family of four has two sons?

I instinctively sense that most people would answer both questions by stating that the latter of the two options is more likely.

If so, then I fear that you are wrong for reasons I will now explain.

Calculating the true probability of Charlton's four season outcomes is virtually impossible even with massive computing power, because of the sheer number of relevant factors and influences, plus the so-called 'unknown unknowns' as Donald Rumsfeld referred to above.

As a proxy there are however numerous bookmakers willing to lay bets on some combination of Outcomes A-D.

Of the major bookmakers, seven are willing to lay odds on Charlton being promoted (whether automatically or via play-offs), and these odds range from a not-very-generous 2/7 (Stan James) to a more reasonable 4/11 (Sky Bet).

The probability equivalent of these odds for those (like me) that prefer to think in those terms are 78% and 73%.

I usually prefer to use Betfair's odds as the best proxy for actual probabilities as they better resemble a true market-driven view, and theirs are 1.39 or 72% in the above vernacular.

Given Betfair's odds are (as usual) similar but slightly better to those available on the High Street, I am comfortable taking 72% as being an excellent estimate of Charlton's actual probability of promotion.

However this 72% encompasses both Outcome A (automatic promotion) and Outcome B (promotion via play-offs).

In order to attach probabilities to all four outcomes, I need to use this 72% as the basis to strip out the probabilities of each separately, via some reasonable assumptions.

I will begin with Outcome D because it's the easiest.

Charlton are currently 16 points clear of Stevenage in 7th place.

With 22 games left, for Charlton to finish behind Stevenage we would have to experience form at least 0.72 points per game worse than the Hertfordshire side (and more for any of the other chasing clubs behind them).

This is equivalent to over 33 points on a full '46-game equivalent' basis.

Obviously such differentials in form are more likely as the number of games remaining falls, but still this appears a near-impossibility.

However as a Charlton fan used to disappointment, and in order to be intellectually 'honest', I will assign a 1% probability to Outcome D.

Now let's consider the dreaded play-offs.

If Charlton have to participate in the play-offs, then it is bound to feel like a meaningful disappointment given how long we would have spent in the top two positions.

The negative impact on dressing room sentiment and fan morale, allied with the tautological conclusion that our form must have been relatively poor in the closing weeks, are likely to offset the mere fact that our squad will likely be the strongest of the four clubs involved.

Conversely, given the way the League table is transpiring (with a large gap below 5th), the team which finishes 6th will feel as though it has a 'bonus free option' on promotion, and will feel under no pressure whatsoever (and probably play better as a result).

Putting these two views together, I am comfortable with the conclusion that each of the four play-off clubs will have an equal 25% chance of promotion.

What this means for the probability of Outcomes A-C is as follows, where PA below is the 'Probability of Outcome A' or automatic promotion:

PA + (0.25 x (0.99 - PA)) = 0.72

0.75PA + 0.2475 = 0.72

PA = 0.63 or 63%

The above equations strip out what aspect of the 72% probability of promotion is attributable to automatic promotion or Outcome A (63%), and what probability is attributable to promotion via the play-offs (9%, ie. 72% - 63%).

They do so simply by stating that the probability of promotion is equal to the probability of automatic promotion, plus the probability of promotion via the play-offs (which requires that we actually compete in the play-offs, ie. weren't promoted automatically or positioned outside the top 6!).

Thus the probability of Outcome A is 63%, and of Outcome B is 9%.

The slightly uncomfortable remaining conclusion therefore is that the probability of Charlton losing in the play-offs (Outcome C) at the end of the season is 27%, the balancing figure:

Probability of Outcome A: 63%
Probability of Outcome B: 9%
Probability of Outcome C: 27%
Probability of Outcome D: 1%

If you agree with my analysis to a decent degree, then I'd be keen to know whether these probabilities are similar to your own 'gut feeling' about the remainder of this season.

I wrote up this analysis because I instinctively feel that a 27% likelihood of more play-off heartache seems too high, yet I feel as though I've proved that it isn't.

And now every time I walk past parents of two boys (or look at my own two for that matter), I shut my eyes and see Nicky Bailey's play-off penalty disappearing into the night sky, and say, "Oh no Lord, please not again!"