For Context-Dependent Player decisions
, player wins and losses are tied to team wins and losses. In order for this to work, the value of an event has to vary depending on the context in which it occurs – a solo home run in the 8th inning of a game with the batter’s team trailing 8-0 in a game which the team goes on to lose 11-1 is inherently less valuable than a grand-slam home run hit with two outs in the bottom of the ninth inning that gives the batter’s team a 4-3 win.
On the other hand, there is a perfectly reasonable argument to be made that a home run is a home run and that all home runs should, therefore, have the same value. My context-neutral player decisions
are calculated based on this assumption: that a home run is a home run, with the value of a home run dependent only on the run-scoring environment in which it was hit. Calculating Player wins and losses treating all home runs as being of equal value, however, will produce a set of player wins and losses that do not uniquely tie to the team’s won-lost record.
If one insists on valuing plays in such a way as to tie player wins and losses to team wins and losses, I’m not sure that there is a single definitive right way to do it. That is, if somebody were to come up with an alternative set of Player wins and losses that tied to team wins and losses but nevertheless differed from mine, I would be reluctant to call them “wrong”. Nevertheless, I do believe that my numbers here are right. Thinking about how one could value events such that Player decisions tie to Team decisions, it seems to me that there are two extreme positions which one could take.
The first would be a purely prospective evaluation which calculates the value of events based entirely on what is known at the time of the event. This is, essentially, the Win Probability
(or WPA) model of valuation. The problem with such a construction is two-fold. First, a purely prospective evaluation system will produce a different number of player decisions across different games. In particular, events that happen in close games, particularly close, high-scoring games, will have more “value” than events that happen in less close and/or lower-scoring games. Second, and somewhat related, the prospective context in which events take place is determined by what has happened before. A purely prospective evaluation will, in my opinion, therefore tend to under-value the events early in a game which create the context of the later game.
Alternately, one could make a purely retrospective evaluation, calculating the value of events based entirely on what is known at the end of the game. That is, one could argue, for example that every solo home run within a particular game by a team is of equal value regardless of when it was hit or what the score was at the time. I believe that this construction is also flawed. Namely, this misses the fact that the actions of players and managers are not made retrospectively, they are made based only the situation of the game to that point. After a 4-3 win, it is all well and good to point out that the result would have been the same if Mariano Rivera
had pitched a scoreless 1st inning and the rest of the Yankees’ staff allowed 3 runs over the final 8 innings. But, in a real game, if Mariano Rivera actually pitched the first inning of a game, it’s just as likely that the Yankees would have ended up scoring 8 runs over the next 8 innings, winning 8-3 and, to at least some degree, “wasting” Rivera’s use in that game.
My decision is to combine both prospective and retrospective information in determining the ultimate value of the events of a baseball game. I do this by calculating an initial set of Player decisions based on a prospective evaluation using Win Probabilities
. I then adjust these results after the fact
so that the total number of Player decisions is the same across all games. The result is, I believe, a coherent set of Player decisions
which accurately reflect player value. The implications of all of this, in terms of how specific events are valued, may not be all that intuitive. It may, therefore, be instructive to look at some examples of how events are valued in my system.
In 2002, two games at Dodger Stadium ended with a final score of 1 – 0 with the lone run coming on a solo home run. On August 28, 2002
, starting pitcher Odalis Perez
hit a solo home run with two outs in the bottom of the fifth inning off of Arizona’s Rick Helling
. On September 27, 2002
, Paul LoDuca
led off the bottom of the tenth inning with a home run off of San Diego’s Jeremy Fikac
From a context-neutral perspective, Perez’s and LoDuca’s home runs were exactly equal in value: 0.1377 wins, since both home runs took place in the same run-scoring environment – Dodger Stadium in 2002. From a prospective perspective, on the other hand, LoDuca’s home run, which ended the game, was more than twice as valuable, 0.3617 wins, as Perez’s home run, at 0.1691 wins. These values correspond exactly to a WPA valuation system: Baseball-reference.com, for example, reports the WPA of these home runs as being 37% for LoDuca and 17% for Perez.
In retrospect, Perez’s home run was more valuable than it seemed at the time, however, as it turned out to be the difference in the game. The normalization process which I employ reflects this by boosting the final value of this home run, relative to its prospective (WPA) values, to a final value of 0.2531 wins. On the other hand, LoDuca’s home run is reduced in value retrospectively to 0.3363 wins since the high context at the time of LoDuca's home run was created in large part by the events which came before (most notably the fine pitching of LoDuca's teammates Omar Daal
and Eric Gagne
). The final result is that LoDuca’s home run is still more valuable than Perez’s, because it was a game-ending home run, while the Diamondbacks still had four more innings to come back from Perez’s home run. But LoDuca’s home run is not twice as valuable as Perez’s but instead is less than one-third more valuable.
This is all summarized in the table below.
||Two out, bottom of 5th inning
||Leading off bottom of 10th
I think that these results are a reasonable reflection of the relative value of these two home runs.
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