Actually, if you plug the above numbers into the Matchup Formula, you get an a priori out probability on a line drive single to left field in the 2005 National League of
14.02%. If you do the same for the a priori probabilities of a single, double, and triple, however, the sum of these probabilities is 101.96%. The probabilities are then normalized to sum to 100%, which yields the final probability of an out of 13.75%.
3.  Use of Matchup Formula in Allocating Credit for Player Game Points
For those components where multiple players share credit for Player Game Points, such as pitchers and catchers with respect to stolen bases
, the relative credit is divided between the relevant players through a process described
The major drawback to Player Won-Lost records that are tied to team records as developed here is that, for a particular play, the pitcher and catcher are assumed to bear equal responsibility – not in terms of equivalent Player Game Points, but in terms of the fact that wins are credited to both pitchers and catchers for plays in which the defensive team earns wins and losses are debited to both pitchers and catchers for plays in which the defensive team earns losses. In reality, it is perfectly reasonable to envision a scenario whereby, for example, a pitcher does a terrible job of holding a baserunner on and is only saved by a perfect throw from the catcher to catch the runner stealing. In such a case, it may be more reasonable to credit the pitcher with a loss for his role in preventing stolen bases while crediting the catcher with more wins than he currently receives. Another example of this would be a catcher
who, while normally excellent at preventing wild pitches and avoiding passed balls, has the misfortune of regularly catching a knuckleball pitcher.
In terms of Context-Dependent Wins and Losses (pWins/pLosses), where the object is to ensure that Player Wins and Losses relate perfectly to team wins and losses, such a situation is largely unavoidable. If one wants to neutralize individual player records in order to move beyond team records, however, then, at a seasonal level, one could use the Matchup Formula to adjust for the performance of the other players with whom a particular player shared credits.
Suppose, for example, that a pitcher compiled a
(basestealing) winning percentage of 0.515 but that the catchers with whom he shared that
credit compiled an average winning percentage (weighted by the number of
points which they shared with this particular pitcher) of 0.535.
In such a case, the Matchup Formula can be used to adjust the pitcher’s
winning percentage. Here, the pitcher’s winning percentage (0.515) would correspond to W1
in the Matchup Formula above. The average winning percentage of his catchers (0.535) would correspond to O1
, the context in which the pitcher performed. Plugging these values into the Matchup Formula would produce an adjusted
winning percentage for this pitcher of 0.480.
In order to properly adjust both pitchers’ and catchers’ winning percentages in this way, one needs to use an iterative process. That is, one first adjusts pitchers’ winning percentages given the winning percentages of their catchers. One would then need to adjust catchers’ winning percentages given the adjusted winning percentages of their pitchers. Having adjusted the catchers’ winning percentages, however, one would want to re-estimate adjusted winning percentages for pitchers using these new adjusted catcher winning percentages. This process would continue until neither pitcher nor catcher winning percentages change between iterations. This process is repeated four times here. These results are used in constructing Context-Neutral, Teammate-Adjusted Player Won-Lost records (eWins and eLosses) as well as to determine the appropriate allocation of Player Game Points across players.
4.  Adjusting Player Game Points for the Level of Competition
The final way in which the Matchup Formula could be useful in adjusting Player Game Points would be to adjust Player Game Points based on differences in the average level of competition faced by different players. That is, if two batters compiled identical offensive winning percentages (say 0.510), but one faced pitchers with an average winning percentage above 0.500 (say 0.505) and the other faced pitchers with an average winning percentage below 0.500 (say 0.495), the former batter would actually be a better hitter. The Matchup Formula, in fact, would say that the first batter (0.510 versus 0.505 pitchers) actually accumulated an adjusted winning percentage of 0.515 while the second batter, with a 0.510 winning percentage against 0.495 pitchers, accumulated an adjusted winning percentage of 0.505.
As with the shared Player Game Points
, adjustments of this type would have to be made through an iterative process. As of now, I have not yet made any such adjustments.
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