This is the third in a series of blogs (here are Part I and Part II) about modelling the scoring of AFL teams and, with the heavy statistical lifting out of the way, in this blog we can look at the practical uses of what we've discovered so far, which is that:
(1) Team scoring can be modelled by the Score Equation
Score = Number of Scoring Shots x Conversion Rate x 6 + Number of Scoring Shots x (1 - Conversion Rate)
(2) A team's number of scoring shots can be modelled by a lognormal distribution with mean determined by the team's strength relative to its opponent and by whether or not the match is a home game for either team, and with a standard deviation of 5.5 scoring shots.
(3) Teams will convert the scoring shots they produce into goals as if drawing from a binomial distribution. The average team will convert 53.64% of its scoring shots into goals.
What I'll do for the remainder of this blog is consider contests between teams of varying ability, as reflected in the number of scoring shots they're expected to generate and in the rate at which they're expected to convert them. By simulating a huge number of games between the teams, with the scores randomly generated but based on the statistical properties summarised above, I can estimate the probability of either team winning and of a draw.
Let's first consider two evenly-matched, "average" teams playing at a neutral venue. If these teams played a sufficiently large number of games, we'd expect them to average 25.4 scoring shots each (with a standard deviation of 5.5 scoring shots) and to convert 53.64% of them. Given those parameters we'd expect each team to win 49.4% of the time and to draw 1.2% of the time. To dwell on that last number for a moment, what it suggests is that if two equally-matched teams were to play on a neutral venue a large number of games they could be expected to draw only about 1 time in 83 games. That certainly casts an unfavourable light on the $51 price for draws typically offered by bookies. It's a bad enough price for matches involving closely-matched teams, but it's diabolical for games pitting the weak against the strong.
Re-running the simulation, reducing the average number of scoring shots for one of the teams to 22.4 and increasing the average number of scoring shots for the other team to 28.4, thus simulating a matchup between a strong team priced head-to-head at around $1.25 and a weak team priced at around $3.85, then the probability of a draw drops to about 0.96% and a fair price for the draw rises to just over $100. (Almost as though to prove my point, TAB Sportsbet offered $51 on the draw in the Geelong v Port Adelaide game in Round 4, even though Port Adelaide were at $6.25 in the head-to-head market.)
What about inclement weather? Does that make $51 the draw look more attractive? Consider the case where two evenly-matched teams are playing at a neutral venue and where driving and persistent rain reduces the expected number of scoring shots for both teams by one half. In that case, simulation shows that the probability of a draw rises only to about 1.5%, a probability that requires a price of around $67 to be a roughly breakeven proposition.
Okay, but rain will probably also effect conversion rates, so let's drop them to 40% for each team too. Well that does drive the probability of a draw higher still, but only to 1.75%, still not quite enough to make $51 profitable.
So the bookies can still expect to make money on the draw bet even if there's thunder and lightning, and the game degenerates into an above-ground water polo farce played with a waterlogged bar of soap (you know the type of game I mean).
While we're on the topic of rain, let's investigate whether it's 'the great leveller' that it's often claimed to be. This time we'll imagine two teams playing at a neutral venue where, in fine weather, one team is expected to produce 28 scoring shots and the other 22. In this scenario the weaker team will win only 24.4% of games and draw 1% of games and will therefore only earn, on average, 0.99 competition points per game.
Now change the weather - wash the car, plan a picnic, use a toxin on the weeds in the garden that requires 72 rain-free hours to be effective, or do any other thing within your power that reliably precipitates rainfall - and this time we'll reflect the effects of it by reducing the expected number of scoring shots of both teams by 25% and by reducing each team's conversion rate to 45% (so we're assuming steady rather than monsoonal rain now). Using those parameters, the weaker team now wins a startling 29.2% of games and draws 1.2%, meaning that it averages 1.19 competition points per game, about 20% better than it could have hoped for under dry conditions.
So, if you'd backed the weaker team earlier in the week at the $3.85 at which it should have been on offer when fine weather was assumed, you'd find yourself come sodden gameday with a wager sporting a 13% positive expectation. Time to start paying more attention to the weather forecasts, I reckon.
Next let's imagine that we're coaching a currently "average" team. What if our training managed to lift their expected scoring shot production by just half a shot per game to 25.9? Such a team would be expected to win 51.7% of games and to draw about 1.2% when playing against a still "average" team at a neutral venue. Or, put another way, our team could expect to average about 2.09 competition points per game when playing such opponents, 4.5% more than it could have expected to accrue before the training to lift its scoring shot production.
Training instead with a more defensive mindset that succeeded in reducing the expected number of scoring shots of our opponent by half a shot (to 24.9) while leaving our scoring production unchanged (at the original 25.4) would have an almost identical effect on our likelihood to win or to draw. In this scenario too we would expect to reap 2.09 competition points per game.
What if, rather than training to alter our team's or our opponent's scoring shot production we focussed on improving our team's conversion rate. Let's say we practiced kicking technique so much and so effectively that we lifted our team's conversion rate from 53.64% to 55%. Now our team can expect to win 51.5% of games and to draw about 1.2% of them when playing against an "average" team at a neutral venue. That improvement in our team's chances is just a little less than that which we achieved by lifting our expected number of scoring shots by half a shot (or by reducing our opponent's expectation by the same amount).
Further improvements to our team's conversion rate would pay handsome dividends. If we could lift it to 60% then we'd catapult our probability of victory to 58.6% while leaving our probability of drawing at about 1.2%. This represents an expected number of competition points per game of 2.37, a whopping 18.5% higher than what we'd expect if our team's conversion rate merely matched our opponent's.
Kicking straight more often is perhaps the most obvious manner in which the virtue of team consistency can be reflected in our simulations. But there's another, more subtle way. Consider a team that, while it generates the same expected number of scoring shots as its opponent, does so with less variability around that expected value. This team is consistent too in that you can have more confidence that it'll kick something close to the number of scoring shots it should, week after week.
To model this type of consistency let's assume that our nothing-if-not-diverse training methods have produced a team that generates scoring shots with a standard deviation not of 5.5 scoring shots per game, but of 4.5 scoring shots per game. Our eminently pliable and newly trained team can now expect to win 50.2% of games and to draw about 1.3% of them when playing against an "average" team at a neutral venue. As such it will accrue 2.03 competition points per game, which is a mere 1.5% improvement over its pre-trained self.
In fact, even if our team could virtually eliminate the variability in the number of scoring shots it created - say we got the standard deviation down to a ridiculous 0.1 scoring shots per game - it would still only increase its expected competition points per game when playing average teams at neutral venues to 2.06 - a 3% improvement.
The small magnitude of the increases notwithstanding, it's frankly quite startling that any improvement in a team's chances of victory can come solely from a greater consistency in creating scoring shots. Remember, I've not been changing the average number of scoring shots our team or our opponent produces on average, just reducing our team's variability around that average.
A closer analysis of the simulations for this scenario shows that a team with lesser variability in scoring shot production, on average scores exactly the same number of points as its opponent, but tends to win by smaller margins - albeit more often - than its opponent. In the example above where our team's standard deviation was 4.5 scoring shots and our opponent's 5.5, our team's average margin of victory is 24.95 points, but its average losing margin is 25.67 points.
Lastly, while I'm on the topic of victory margins, another thing that the simulation model implies is that the average margin of victory when two equally matched teams play at a neutral venue is 27 points with a standard deviation of 20.8 points. The median margin of victory is 22 points. Those statistics have finally given me some rationale for why I've intuitively felt that an MAPE below 30 points per game and a median APE below 23 points per game are strong margin-predicting performances.
Also, note that only 13.4% of games between evenly-matched teams should be expected to finish with a margin of 5 points or fewer, 47.5% of them should be expected to finish with a margin of 24 points or more, and 28.2% of them should be expected to finish with a margin of 36 points or more.
So, just because your team was thumped by 6 goals on the weekend doesn't mean that they weren't evenly-matched with their opponent. (Be aware though that if it happens for five weeks in a row, it gets a little harder to make the case.)
(Footnote: statistical honesty compels me to report that it might have been possible to make money by wagering on the draw in recent seasons. Over the period 2006 to 2009, which is the only period for which I have reliable head-to-head price date, there have been 6 draws in the 118 games where the teams were both priced at $2.05 or less. That works out to a bit under one draw in 20 games, which would have made for spectacularly lucrative draw wagering if (a) you could have secured $51 the draw, or even a little less, on each occasion, and (b) you recognised this possibility 5 years ago.)