An individual player’s value can be measured by how many goals they contributed. This is not as simple as considering only goals and assists. Any throw will change a team’s likelihood of scoring. A foul will also affect the chances of a goal. This is because a team’s chance of scoring on an individual possession depends on the position of the disc on the field. Ultiworld produced a graphic showing probabilities of scoring at different points on the field for the 2012 NexGen tour, this graphic can be found at http://cache.ultiworld.com/wordpress/wp-content/uploads/2013/02/Screen-Shot-2013-02-06-at-10.29.48-AM.png
The Ultiworld graphic does not show the value of disc positions in the pro leagues, however. The pro field is wider and longer than the field NexGen played on. Therefore, a new chart of values will be made for the pro leagues. A factor to consider in creating this chart is the ease of getting the goals contributed statistics calculated. The statisticians at a pro game have to work at a relatively fast pace to record the flow of the game (who throws to whom). They would not be able to record which zone the disc is in if the chart was divided into very small zones like Ultiworld’s is. So, due to the limited technology available to the game day statistical staff, the values chart will be simpler than that of Ultiworld.
We will write the value of a position on the field as V(x,y). In this value function, x can have three values, R, C, and L. These represent the right, center, and left thirds of the field, from the perspective of the offense. The value of y has a range of the integers from -19 to 80, inclusive. This represents the yard line that the disc is at, with 0 being a team’s own goal line, and 80 being a team’s opponent’s goal line. The zones extend backward from the line, so a disc that is .5 yards in front of that team’s goal line is considered to be at y-value of 1. V(x,y) is equal to the probability of the offense scoring from position (x,y). The value, in goals, for any position (x,y), for the offense is V(x,y); for the defense, it is 1-V(x,y).
When a player has the disc, there are 6 different outcomes that may result. They are a completion, a dropped pass, a blocked pass, a throwaway, a stall-out, and a turnover travel. Being stalled out and turning it over via travel have the same effect, so there are 5 outcomes that must be considered. For this explanation, assume that the Boston Whitecaps are on offense against the Philadelphia Spinners. Jeff Graham has the disc at point (R, 60). If he is stalled out or has a travel turnover called on him, the Spinners take possession at their point (L, 20). Graham’s goals contributed for this play would be equal to (1-V(L,20)) – V(R, 60). If Graham throws the disc out the back of the end zone in the center of the field, the Spinners would take over at their point (C, 0); Graham’s goals contributed for this play would be (1-V(C, 0)) – V(R, 60). These situations place the change in value entirely on the thrower. The other three situations have value changes allocated to the thrower and another player.
Let’s say in the situation above, Graham throws to Josh Markette, who is defended by Mike Baer. The throw is made to arrive at Boston’s point (R, 65). If Markette catches it, Bostom will have the disc at (R, 65). But if Markette drops it or Baer gets a D, the Spinners will have possession at (L, 15). We will say that in a situation where a catch, drop, or D is the outcome of the throw, that a play was made on the throw. In the 2012 season, plays were made on 18,048 throws. Of these plays, 16,820 were catches, 919 were D’s, and 309 were drops. The probabilities of the outcomes are as follows: P(catch) = .932, P(D) = .051, P(drop) = .017. Given the probabilities, we can calculate the expected value of the throw described above for the thrower. This value is .932*V(R, 65) + .068*(1-V(L,15)). We will generalize this function as the value of a play, P(x, y). The function P(x, y) is defined as .932*V(x, y) + .068*(1-V(x’, y’)), with (x, y) being the point where the play is made from the offensive team’s perspective, and (x’, y’) being the point where the play is made from the defensive perspective. So, in the situation described above, Graham earns P(R, 65) – V(R, 60) goals contributed. There are three possibilities for the play.
- Markette catches the throw. Markette would earn V(R, 65) – P(R, 65) goals contributed.
- Markette drops the throw. Markette would earn (1-V(L,15)) – P(R, 65) goals contributed.
- Baer D’s the throw. Baer would earn V(L, 15) – (1-(P(R, 65)) goals contributed.
Throws are not the only thing that will affect the position of the disc. A foul will also affect a player’s goals contributed. A 10 yard foul on the offense with the disc at point (x,y) will cause the player to earn V(x, y-10) – V(x, y) goals contributed.