KEY WORDS: Notational analysis, Australian Rules Football, tactical coaching.

Australian Rules Football (informally known as "AFL" after the Australian Football League) is played with an oval ball on an oval field at high speed, leading to it sometimes being called "What Rules?" by the unschooled observer. Compared to more structured football codes such as American football or rugby league where a "phase of play" always starts in a simply-defined formation, the free-flowing nature of Australian football creates extra dimensions for analysis. This paper describes the qualitative framework for evaluating the phases of AFL and presents empirical interpretation of data from the 2004 and 2005 seasons.

AFL coaches are clamouring for this sort of analysis to inform their strategies and training procedures. They know that being in possession of the ball is important, but this research can show exactly how much it's worth on the scoreboard to take a contested mark, compared with someone from the opposition grabbing the loose ball spilled from the pack. They also know that position is important. They must create opportunities in positions near goal, but their players often have to choose whether to aim at a riskier proposition close to the goalmouth or maintain possession in a worse position. Dynamic programming based on empirically derived parameters can answer this dilemma.

Dynamic programming was first applied to AFL (Clarke and Norman, 1998) to answer the question of whether players should concede a point on the scoreboard in order to gain clean possession afterwards. A new thesis (Forbes, 2006) based on Champion Data's statistics uses a Markov model approach to map out the probabilities of transitions between AFL's phases to predict scoreboard results.

American football, where position is effectively one-dimensional and there are only four phases - the "downs" - has been analysed using dynamic programming in a famous paper (Romer, 2002), and a rating system (Schatz, 2005) called DVOA (Defence-adjusted Value Over Average) evaluates actions with respect to a model of scoreboard value similar to the one created in this paper. The fast-flowing and open sport of ice hockey has recently been modelled using a "semi-Markov" approach (Thomas, 2006).

The modelling undertaken here is largely exploratory - this is a mass of new data which requires further detailed research.

Match equity and field equity
Various authors have employed a plethora of terms to describe the expected value of actions on sporting fields. Studeman, 2004 describes the repeated reïnvention and relabelling of "Win Probability Added" in baseball. Bennett, 2005 has a good simple description of how to value an action that alters the probability of winning the match.

The terminology we use in this paper is derived from the theory of backgammon (Keith, 1996), a game in which the players compete to win points, the first to n points winning the match. We assume teams of equal strength, although much of the reasoning below is still valid for uneven teams. Match Equity is the probability of the team to win the match from this moment, or more specifically:

(1)

The Match Equities of each team in the contest sum to one. A team is always aiming to increase its Match Equity until it reaches one - certain victory. I.e., it is looking for actions which maximise ∆E_M, or at the very least have ∆E_M > 0. As noted in Equation 1, Match Equity is a function of four parameters:

the score margin, m
the time remaining in the match, t
the position on the field, x
the possession state or phase of play,φ

AFL typically has about 50 scores in a match of 80 live minutes. We define s_typ as the typical score of a game (in AFL's case, the goal worth 6 points is dominant), and t_typ as the typical time between scores (approximately 100 seconds in AFL). We can roughly decouple the first two parameters from the others by noting that if we discard any knowledge of x or φ, we can build a satisfactory model of winning probability based only on the time remaining t and changes to the margin m. The phase and location information can be treated as a perturbation of the match-winning probability model E_M.

To model the net potential value on the scoreboard of the current state of play, we introduce

Field Equity:

(2)

where; si is the value of the i^th type of score p_i,q is the probability of the next score being of type i by team q

Changes to Match Equity, Decoupled
	(3)

Table 1 has a simple example of how to estimate the scoreboard value of two well-defined phases: after a goal, and after a behind. This is summarised in Table 2.

The measured residual equity in real matches is non-zero since the stronger team is more likely to be the scorer of both the previous and the next score. It appears irrelevant whether the restart is via the umpire in the centre of the oval, or via a player kick-in at the end of the ground (95% confidence interval for the difference: [-0.02, +0.14]).

Equity Maps
The value of taking a mark and having a set shot at goal directly in front can be seen in this map, with an expected value of more than four points extending all the way out to about 40 metres from goal. A free kick within 25 metres makes the goal a virtual certainty. The tight bunching of contour lines from 40 to 60 metres out along the spine shows the natural limit of an AFL footballer's kick, being about 50-55 metres. To get within one kick of goal, and have the time to execute it, is extremely valuable.

"Directed" (Figure 2) is the second-best phase for a footballer to receive the ball in. Usually he has received a handball in some space and should be able to execute his preferred option. But often he will have to take critical time to swivel as the defence closes in, and it's only within ten metres of goal that the maximum six points can almost be assumed. The gradient we saw at 40-60 metres in Figure 1 is completely missing here, showing the greater difficulty of a snap shot on the run - the attacker wants to be within 30 metres.

An utterly different picture (Figure 3) awaits the player who faces the extreme pressure of a hard ball get. Even within ten metres of goal the expected scoreboard outcome is just 3.5 points. Equity is below zero for the entire defensive zone, but interestingly there is a peak at the top of the forward arc, indicating that perhaps this is one place on the ground where he has two reasonable areas either side of him to shoot out a handball and find a teammate who suddenly has options within range of goal. This circumstance often happens after the centre bounce when a quick kick lands at the congested top of the arc with the opposition still rushing the centre square.

Figure 4 shows clearly the "hot spot" favoured by AFL coaches. There is volatility of more than six points in contesting a mark or winning a free kick twenty metres out directly in front of goal, rather than letting your opponent have the same.

Also of interest is how cool the wings are - an equity swing of less than 2.5 points for taking a mark over his opponent, as neither player can directly make use of the extra time. A kick from a set shot near the boundary will often travel straight down the boundary to a settled pack, which is very low in volatility.

Calculated as an average over the ground, there is only a boost of 0.3 points to be gained by taking a mark instead of receiving a handball. In modern football uncontested marks across the half-back-line are cheap, with the opponent barely interested in forcing the man to go back and take the set shot. But the advantage is wholly concentrated in the forward-50 arc (Figure 5), with an extra 1.5 points available on the scoreboard for having a set shot rather than a running shot at goal between 25 and 45 metres out. The light areas on Figure 6 show the regions where it is significantly better, at the two-sigma level, for a player to take a mark rather than gather it uncontested.

Average phase equity
The mean net value of each of the phases was calculated by averaging over the 200 zones on the field. This works as a "standard candle" to investigate deviations by teams or in certain situations.

A player's choices: what happens next
Imagine a player who has just taken a mark 70 metres out from goal, on about a 40°-45° angle. It's unlikely he can score himself, and he faces an unenviable choice between bombing it long in hope of improved field position without turning the ball over, or picking out a nearby teammate to do the dirty work for him. This scenario - within six metres - has played out 822 times over the seasons 2004-2005. On average, a team in this position can expect to convert to about two points on the scoreboard (2.06(14)).

It's immediately obvious from Figure 7 below that if the player passes short and keeps it near the boundary, he almost always finds a teammate. Even more encouragingly, the team scores from there virtually every time. On the other hand, directing the ball long into the central corridor seems to be about a 50/50 proposition to hold onto the ball. Is it worth the risk? And should he play on, relinquishing the set shot time to gain some ground by running? Figure 7 shows the results of the 195 marks at the MCG from this position.

The collection of points where a player marked. The plus signs (+) show where he managed to get the ball to a teammate, while the dark squares are immediate turnovers. The grey circles indicate the ball went into the umpire's control. A ring around the marker means that the next score was to the opposition - no ring indicates a score for the marking player's team. The nine diagonal slashes are the rare occasions that the player managed to run to this point and scored for himself.

The results are inconclusive, but they do highlight the dilemma. By choosing to handball, his team keeps the ball 97% of the time. With a short kick (gaining less than 35 metres or not moving closer to goal), the retention rate is 78%, but just 48% with a long kick. And yet the improved position gained from the long kick is worth the risk: a slightly higher equity as more of the scores are goals.

Players who waited rather than played on saw the defence coalesce around their options and could only find a teammate 62% of the time, for a slightly lower equity result. It should be noted that some of these would not have had the choice to play on, due to a close opponent or falling to ground after the mark.

Players and fans understand the scoreboard. Telling them that giving the won ball to the opponent at this point on the field is effectively taking three points off the scoreboard is a strong message, and should foster a new way of thinking about the game. Table 3 has equity estimates for the four phases averaged over the entire field of play.

It has long been noticed that defenders have higher "kicking effectiveness" percentages, a measure of how often they find a teammate as a percentage of total kicks. The pictures in this paper make it obvious why - there is little pressure on them, and a wealth of options to hit. There is an implicit "funnel" in many team sports due to the location of the goals - trying to kick into the neck of the funnel at centre-half-forward is very risky, but as seen by the equity gradient also very rewarding if the team has strong marking forwards in the corridor. It is much easier to advance along the gentle equity gradient in the back half of the ground, the funnel gaping open as teammates have more space to run to. The next step is to identify clubs' equity signatures, and find out where they are breaking down compared to the league standard. Where do they mostly direct the ball? Sydney are known to hug the boundary, but can this tactic be exploited?.

This is just a first look at a huge body of data which is ready for exploitation by AFL researchers. Even these preliminary results are informing AFL coaches about the risks and rewards associated with some patterns of play. Future directions include looking at the effect of speed of play on equity - how much of an advantage is it to be able to advance the ball quickly? Or should the players switch play across the ground to exploit open space? A semi-Markov approach as advocated for ice hockey (Thomas, 2006) could also be useful, to reduce the number of data points needed for conclusive evidence of strategic advantage. Following the lead of baseball, an application to player ratings would be a significant opportunity. Identifying which players consistently increase equity for their team is a major goal.

The author would like to thank the dedicated Champion Data data-capture staff who methodically logged the 350,000 data points used in this analysis.

• Equity analysis provides a way of estimating the net value of actions on the sporting field.
• Combined with spatial data analysis, the relative merits of gaining position or maintaining possession can be judged.
• The advantage of having time and space to use the ball is measured in terms of scoreboard value, and is found to vary with field position.
• Each sport has identifiable areas of the field with high equity gradients, meaning that it is most important to gain territory there.