Does Losing Lead to Winning?

I was reading an issue of Chance News last night and came across the article When Losing Leads to Winning. In short, the authors of this journal article found that, in 6,300 or so most recent NCAA basketball games, teams that trailed by 1 point at half-time went on to win more games than they lost. This they attribute to "the motivational effects of being slightly behind".

Naturally, I wondered if the same effect existed for footy.

This first chart looks across the entire history of the VFL/AFL.

The red line charts the percentage of times that a team leading by a given margin at quarter time went on to win the game. You can see that, even at the leftmost extremity of this line, the proportion of victories is above 50%. So, in short, teams with any lead at quarter time have tended to win more than they've lost, and the larger the lead generally the greater proportion they've won. (Note that I've only shown leads from 1 to 40 points.)

Next, the green line charts the same phenomenon but does so instead for half-time leads. It shows the same overall trend but is consistently above the red line reflecting the fact that a lead at half-time is more likely to result in victory than is a lead of the same magnitude at quarter time. Being ahead is important; being ahead later in the game is more so.

Finally, the purple line charts the data for leads at three-quarter time. Once again we find that a given lead at three-quarter time is generally more likely to lead to victory than a similar lead at half-time, though the percentage point difference between the half-time and three-quarter lines is much less than that between the half-time and first quarter lines.

For me, one of the striking features of this chart is how steeply each line rises. A three-goal lead at quarter time has, historically, been enough to win around 75% of games, as has a two-goal lead at half-time or three-quarter time.

Anyway, there's no evidence of losing leading to winning if we consider the entire history of footy. What then if we look only at the period 1980 to 2008 inclusive?

Now we have some barely significant evidence for a losing leads to winning hypothesis, but only for those teams losing by a point at quarter time (where the red line dips below 50%). Of the 235 teams that have trailed by one point at quarter time, 128 of them or 54.5% have gone on to win. If the true proportion is 50%, the likelihood of obtaining by chance a result of 128 or more wins is about 8.5%, so a statistician would deem that "significant" only if his or her preference was for critical values of 10% rather than the more standard 5%.

There is certainly no evidence for a losing leads to winning effect with respect to half-time or three-quarter time leads.

Before I created this second chart my inkling was that, with the trend to larger scores, larger leads would have been less readily defended, but the chart suggests otherwise. Again we find that a three-goal quarter time lead or a two-goal half-time or three-quarter time lead is good enough to win about 75% of matches.

Not content to abandon my preconception without a fight, I wondered if the period 1980 to 2008 was a little long and that my inkling was specific to more recent seasons. So, I divided up the 112-season history in 8 equal 14-year epochs and created the following table.

The top block summarises the fates of teams with varying lead sizes, grouped into 5-point bands, across the 8 epochs. For example, teams that led by 1 to 5 points in any game played in the 1897 to 1910 period went on to win 55% of these games. Looking across the row you can see that this proportion has varied little across epochs never straying by more than about 3 percentage points from the all-season average of 54%.

There is some evidence in this first block that teams in the most-recent epoch have been better - not, as I thought, worse - at defending quarter time leads of three goals or more, but the evidence is slight.

Looking next at the second block there's some evidence of the converse - that is, that teams in the most-recent epoch have been poorer at defending leads, especially leads of a goal or more if you adjust for the distorting effect on the all-season average of the first two epochs (during which, for example, a four-goal lead at half-time should have been enough to send the fans to the exits).

In the third and final block there's a little more evidence of recent difficulty in defending leads, but this time it only relates to leads less than two goals at the final change.

All in all I'd have to admit that the evidence for a significant decline in the ability of teams to defend leads is not particularly compelling. Which, of course, is why I build models to predict football results rather than rely on my own inklings ...

Pointless v St Kilda

The Swans' 2nd and 3rd quarter performances last Saturday should not go unremarked.

In the 3rd quarter they failed to register a point, which is a phenomenon that's occurred in only 1.2% of all quarters ever played and in just 0.3% of quarters played since and including the 1980 season. Indeed, so rare is it that only one occurrence has been recorded in each of the last two seasons.

Last year, Melbourne racked up the season's duck egg in the 1st quarter of their Round 19 clash against Geelong, leaving them trailing 0.0 to 8.5 at the first change and in so doing setting a new standard for rapidity in disillusioning Heritage Fund Investors. In 2007 the Western Bulldogs were the team who failed to trouble the goal umpire for an entire quarter - the 2nd quarter of their Round 22 game against the Kangaroos.

So, let's firstly salute the rarity that is failing to score for an entire quarter.

But the Swans did more than this. They preceded their scoreless quarter with a quarter in which they kicked just two behinds. Stringing together successive quarters that, combined, yield two points or fewer is a feat that's been achieved only 175 times in the entire history of the game, and 140 of those were recorded in the period from 1897 to 1918.

Across the last 30 seasons only 12 teams have managed such frugality in front of goal. Prior to the Swans, the most recent example was back in Round 14 of 2002 when West Coast went in at half-time against Geelong having scored 4.7 and headed to the sheds a bit over an hour later having scored just two behinds in the 3rd quarter and nothing at all in the 4th. That makes it almost 6-and-a-half seasons since anyone has done what the Swans did on Saturday.

Prior to the Eagles we need to reach back to Round 4 of 1999 when Essendon - playing West Coast as it happens - finished the 1st quarter and the half stuck at 2.2 and then managed just two behinds in the 3rd term. (They went on to record only two more scoring shots in the final term but rather spoiled things by making one of them a major.)

If you saw the Swans games then, you witnessed a little piece of history.

Waiting on Line

Hmmm. (Just how many ms are there in that word?)

It's Tuesday evening around 7pm and there's still no Line market up on TAB Sportsbet. In the normal course this market would go up at noon on Monday, and that's when the first match is on Friday night. So, this week the first game is 24 hours earlier than normal and the Line market looks as though it'll be delayed by 48 hours, perhaps more.

Curiouser still is the fact that the Head-to-Head market has been up since early March (at least) and there's an historical and strong mathematical relationship between Head-to-Head prices and the Line market, as the following chart shows.

The dark line overlaid on the chart fits the empirical data very well. As you can see, the R-squared is 0.944, which is an R-squared I'd be proud to present to any client.

Using the fitted equation gives the following table of Favourite's Price and Predicted Points Start:

Anyway, back to waiting for the TAB to set the terms of our engagement for the weekend ...

Marginally Interesting

Here are a handful of facts on AFL margins:

• The largest ever victory margin was 190 points (Fitzroy over Melbourne in 1979)
• Every margin between 0 and 150 points has been achieved at least once except margins of 136, 144, 145, 148 and 149 points.
• Last season, no game finished with a victory margin of 25 points
• No game finished with a margin of 47 points in the previous 2 seasons
• No game finished with a margin of 67 points in the previous 5 seasons
• No game finished with a margin of 90, 94 or 98 points in the previous 8 seasons
• No game finished with a margin of 109 points in the previous 12 seasons
• No game finished with a margin of 120 points in the previous 17 seasons
• No game finished with a margin of 128 points in the previous 39 seasons
• No game finished with a margin of 161 points in the previous 109 seasons
• At least one game has finished with a margin of 6 points in each of the previous 48 seasons
• At least one game has finished with a margin of 26 points in each of the previous 42 seasons

Draw Doesn't Always Mean Equal

The curse of the unbalanced draw remains in the AFL this year and teams will once again finish in ladder positions that they don't deserve. As long-time MAFL readers will know, this is a topic I've returned to on a number of occasions but, in the past, I've not attempted to quantify its effects.

This week, however, a MAFL Investor sent me a copy of a paper that's been prepared by Liam Lenten of the School of Economics and Finance at La Trobe University for a Research Seminar Series to be held later this month and in which he provides a simple methodology for projecting how each team would have fared had they played the full 30-game schedule, facing every other team twice.

For once I'll spare you the details of the calculation and just provide an overview. Put simply, Lenten's method adjusts each team's actual win ratio (the proportion of games that it won across the entire season counting draws as one-half a win) based on the average win ratios of all the teams it met only once. If the teams it met only once were generally weaker teams - that is, teams with low win ratios - then its win ratio will be adjusted upwards to reflect the fact that, had these weaker teams been played a second time, the team whose ratio we're considering might reasonably have expected to win a proportion of them greater than their actual win ratio.

As ever, an example might help. So, here's the detail for last year.

Consider the row for Geelong. In the actual home and away season they won 21 from 22 games, which gives them a win ratio of 95.5%. The teams they played only once - Adelaide, Brisbane Lions, Carlton, Collingwood, Essendon, Hawthorn, St Kilda and the Western Bulldogs - had an average win ratio of 56.0%. Surprisingly, this is the highest average win ratio amongst teams played only once for any of the teams, which means that, in some sense, Geelong had the easiest draw of all the teams. (Although I do again point out that it benefited heavily from not facing itself at all during the season, a circumstance not enjoyed by any other team.)

The relatively high average win ratio of the teams that Geelong met only once serves to depress their adjusted win ratio, moving it to 92.2%, still comfortably the best in the league.

Once the calculations have been completed for all teams we can use the adjusted win ratios to rank them. Comparing this ranking with that of the end of season ladder we find that the ladder's 4th-placed St Kilda swap with the 7th-placed Roos and that the Lions and Carlton are now tied rather than being split by percentages as they were on the actual end of season ladder. So, the only significant difference is that the Saints lose the double chance and the Roos gain it.

If we look instead at the 2007 season, we find that the Lenten method produces much greater change.

In this case, eight teams' positions change - nine if we count Fremantle's tie with the Lions under the Lenten method. Within the top eight, Port Adelaide and West Coast swap 2nd and 3rd, and Collingwood and Adelaide swap 6th and 8th. In the bottom half of the ladder, Essendon and the Bulldogs swap 12th and 13th, and, perhaps most important of all, the Tigers lose the Spoon and the priority draft pick to the Blues.

In Lenten's paper he looks at the previous 12 seasons and finds that, on average, five to six teams change positions each season. Furthermore, he finds that the temporal biases in the draw have led to particular teams being regularly favoured and others being regularly handicapped. The teams that have, on average, suffered at the hands of the draw have been (in order of most affected to least) Adelaide, West Coast, Richmond, Fremantle, Western Bulldogs, Port Adelaide, Brisbane Lions, Kangaroos, Carlton. The size of these injustices range from an average 1.11% adjustment required to turn Adelaide's actual win ratio into an adjusted win ratio, to just 0.03% for Carlton.

On the other hand, teams that have benefited, on average, from the draw have been (in order of most benefited to least) Hawthorn, St Kilda, Essendon, Geelong, Collingwood, Sydney and Melbourne. Here the average benefits range from 0.94% for Hawthorn to 0.18% for Melbourne.

I don't think that the Lenten work is the last word on the topic of "unbalance", but it does provide a simple and reasonably equitable way of quantitatively dealing with its effects. It does not, however, account for any inter-seasonal variability in team strengths nor, more importantly, for the existence any home ground advantage.

Still, if it adds one more finger to the scales on the side of promoting two full home and away rounds, it can't be a bad thing can it?