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Formulas

## Formulas

In this section I will present the formulas underlying the Win Values system.?   There are actually only a few formulas.? Each formula is conceptually simple   but may look complex.?

We seek the formula for a starting pitcher?s Win Value for a game in which   he pitched Z innings,[1] the   score at the conclusion of the Zth inning was RS to RA,[2]   and the game was played in a ballpark with a park factor of PF.? Let me write   this as WinVal(RS,RA,Z,PF).

From the conceptual framework described above, we know that Win Value is   the difference between the team?s expected probability of winning a game given   that the score is RS to RA at the conclusion of the Zth inning, in a PF ballpark,   and the team?s expected probability of winning a game given that the team   has scored RS runs at the conclusion of the Zth inning, with league average   pitching, in a PF ballpark.

Let me write the first win probability as WinProb1(RS,RA,Z,PF) and the second   win probability as WinProb2(RS,Z,PF).? WinProb1 uses the pitcher?s actual   RA information whereas WinProb2 assumes average pitching.? So we have:

[Eq.1]?? WinVal(RS,RA,Z,PF) = WinProb1(RS,RA,Z,PF) ? WinProb2(RS,Z,PF).?

Let?s first turn to how park effects are handled.? It turns out to require   a couple of simplifying assumptions since there are not enough games played   in any league-season with the same park factor to estimate WinProb1 and WinProb2   for different PF?s.

It won?t be immediately clear why we would want to do so, but let?s rewrite   Equation 1 as follows:

[Eq.2]?? WinVal(RS,RA,Z,PF) = WinProb1(RS,RA,Z,PN) ? WinProb2(RS,Z,PN)

??????????????????????????????????????????????? ??? + [WinProb1(RS,RA,Z,PF)   ? WinProb1(RS,RA,Z,PN)]

??????????????????????????????????????????????? ??? + [WinProb2(RS,Z,PN)   ? WinProb2(RS,Z,PF)]

where PN denotes a park neutral setting.? The reason why we write the equation   this way is so that we can consider each of the terms in brackets.? What do   each of these terms represent?? The first term reflects how the win probability   of a team that has scored RS runs and allowed RA runs at the conclusion of   the Zth inning is affected by the park factor.? The second term reflects how   the win probability of a team that has scored RS runs at the conclusion of   the Zth inning with average pitching is affected by the park factor.

Empirically, I have found that the first term is typically small and can   safely be ignored.? Getting slightly ahead of ourselves, I have found that   WinProb1 is reflected in the game?s deficit (RA-RS) rather than having to   consider RS and RA separately.? By analyzing the inning-by-inning runs scored   distributions at various parks, I have found that the probability that a team   overcomes a deficit of a given size does not depend significantly upon the   park factor.[3]?

On the other hand, I have found that the second term is potentially significant,   and merits special treatment.? Dropping the first term in brackets in Equation   2, and defining a new variable, we have:

[Eq.3]?? WinVal(RS,RA,Z,PF) = WinProb1(RS,RA,Z) ? WinProb2(RS,Z)

where, for convenience, I have dropped the PN labels in WinProb1 and WinProb2,   and where ParkAdder(RS,Z,PF) = WinProb2(RS,Z,PN) ? WinProb2(RS,Z,PF).

I will now describe the formulas for each of these three terms.? WinProb1   involves two concepts.? The first is how RS and RA interact in the formula,   and the second is how we ?smear? the run support probabilities.

As described above, WinProb1 will be reflected in the deficit that the team   faces, RA-RS, rather than a different formula for every (RS,RA) pair.? There   are simply not enough games that have the same score in any league-season,   and fortunately, I have found empirically that the win probability of overcoming   a given deficit does not significantly depend upon the actual score.? For   example, the probability that a team trailing 6-3 comes back to win the game   is similar to the probability that a team trailing 8-5, say, will come back   to win the game.[4]

We have previously motivated the notion of ?could have been? runs scored   possibilities for a pitcher?s run support.? The idea is that a pitcher should   be evaluated based not only on how many runs he allowed but also on how many   runs he received that game in run support.? However, we do not want to be   fanatical about fixing his run support.? This would lead to evaluations that   I am not in favor of.[5]? Thus,   we seek a middle ground, and this is where the ?could have been? run support   smeared probabilities come in.

Putting these two ideas together, then, we have:

[Eq.4]?? WinProb1(RS,RA,Z) =? åmSmear(m;RS,Z)   * DWin(RA-m;Z)

where the summation is over m, the runs support possibilities (m goes from   0 to 25, say), Smear(•) is the ?could have been? run support probabilities   and DWin(?) is the probability that a team trailing by a specific number of   runs will come back to win the game.?

The DWin probabilities can be derived empirically for each inning Z (from   1 to 9) using all the inning-by-inning scoring data from all games in the   league-season under study.[6]?   Note that I smooth the DWin probabilities to remove any effect of small samples   or weird games.? That is, I ensure that the probability of winning increases   if the lead increases (holding the inning constant), and I ensure the probability   of overcoming a deficit decreases as the inning increases (holding the deficit   constant).

For the derivation of the Smear probabilities, we will need two new variables.?   Let R(x;Z) denote the probability that a team scores x runs at the conclusion   of the Zth inning.? Let S(w,y;Z,C) denote the probability that a team that   has scored y runs at the conclusion of the Cth inning will score w runs at   the conclusion of the Zth inning.? For our purposes, C will be less than or   equal to Z.? For example, we may be interested in knowing the ?propogation?   probabilities of a team that has scored 4 runs at the conclusion of the 6th   inning.? That is, how likely is such a team to wind up with 7 runs at the   conclusion of the 8th inning, say.

I will not repeat the description of the derivation of these smearing probabilities   via backwards Bayesian bootstrapping.? The interested reader should revisit   the derivation described above in the section on the system?s conceptual framework.?   Suffice it to say here that the backwards Bayesian bootstrapping method requires   me to designate how far back in a game to go for the ?could have been? phenomenon   to kick in.

Remember that Wolverton?s system essentially says to ignore the game?s actual   run support; this is equivalent to pushing the ?could have been? experiment   all the way back to the beginning of the game.? I have described why I do   not want to follow that approach.[7]?   The further back towards the beginning of the game I choose for my ?could   have been? cutoff, the more my method looks like Wolverton?s support-neutral   method.? The closer to the end of the game I choose for my ?could have been?   cutoff, the more my method reflects the pitcher?s W-L record.?

I have experimented with different cutoffs and have reviewed the distribution   of runs scored by inning in great detail.? I have settled upon the following   ?could have been? inning cutoffs: 6th inning for a 9-inning outing, 5th inning   for a 7- or 8-inning outing, 4th inning for a 5- or 6-inning outing, 3rd inning   for a 3- or 4-inning outing, 2nd inning for a 2-inning outing, and 1st inning   for a 1-inning outing.

To save on some notation, let C denote these innings to which we allow the   ?could have been? smearing to begin.? C will depend upon Z, but I will suppress   that in the formula below.

By backwards Bayesian bootstrapping, we have:

[Eq.5] Smear(m;RS,Z) = ån{[R(n;C) * S(RS,n;Z,C)] / åj [R(j;C) * S(RS,j;Z,C)]} * S(m,n;Z,C)

where the two summations, over n and j respectively, go from 0 to 25, say.?   We can derive all the required R and S probability distributions empirically   for each inning Z (1 to 9) using all the inning-by-inning scoring data from   all games in the entire league-season under study.? Therefore, we are able   to derive the required WinProb1 probabilities.

Let?s return to the second term of Equation 3, WinProb2(RS,Z).? Remember   this is the probability that a team that scores RS runs in Z innings will   win the game with average pitching.? There are two elements to consider.?   First, for any run scored we estimate the probability that a team scoring   that many runs at the conclusion of the Zth inning will win the game with   league average pitching.? Second, as above, we smear the runs scored probabilities   based upon RS and the ?could have been? smearing probabilities.? Thus, we   have:

[Eq.6]?? WinProb2(RS,Z) = åm Smear(m;RS,Z) *   AWin(m;Z)

where the summation is over m, the run support possibilities (m goes from   0 to 25, say), and AWin is the probability that a team that scores a specific   number of runs in Z innings will win the game with league average pitching.?   Equation 5 previously gave the formula for Smear(•), so all we need   is AWin(•).

AWin is estimated empirically for each inning Z (1 to 9) using all the inning-by-inning   scoring data from all the games played in the league-season under study.[8]?   Note that I smooth the AWin probabilities to remove any effects of small samples   or weird games.? That is, I ensure that the probability of winning increases   if more runs are scored (holding the inning constant).? I also ensure that   the win probability when scoring any number of runs at the conclusion of Z   innings decreases with Z (holding the number of runs constant).

The remaining term of Equation 3 is the Park Adder.? There is insufficient   data to estimate a separate park adder for every possible park factor and   every possible run scored.? I therefore pool the park data of each league-season   into three groups: hitters parks, neutral parks, and pitchers parks.? This   allows me to estimate the park adder as a percentage of the park factor of   the home park for each of the possible run scored figures (say from 0 to 25).?   I then smooth these park adders to remove any effect of small samples or weird   games.

An example will help here.? Suppose that I find that the home park affects   the probability of winning a game when scoring exactly 5 runs by .009 per   percentage point of the home park?s park factor.? For concreteness, let?s   use the Oakland Coliseum in 2000 which had a Total Baseball Park Factor of   97.? This implies that runs were 6% less prevalent in games at the Oakland   Coliseum compared to a league neutral park.? Multiplying the 6 by the .009   yields an estimate that a team scoring 5 runs at Oakland Coliseum had a .054   higher chance of winning the game than a team that scored 5 runs in a league   neutral park.

The last step to deriving the Park Adder is to pro-rate the change in win   probability by the number of innings the pitcher pitched in the game.? Algebraically,   then, we have:

where PAddPct(RS) is the per percentage change in the probability of winning   with RS runs (in the above example this was the .009).? The middle term reflects   the effect of the ballpark on runs scored and the fact that only half a team?s   games are played at home (in the example above PF was 97, so that the middle   term is 6).? (Z/9)+ is Z/9 capped at 1 to properly handle partial   games and pitchers who pitch more than 9 innings.

Finally, we have completed our formulaic journey.? Equations 3-7 provide   the formulas underlying the Win Values system.? Since formulas can be rather   dry and imposing, I next turn to giving numerical examples of all of the terms   appearing in these formulas.

[1]?   Since my Win Values system relies upon the inning-by-inning runs scored distributions,   the system treats partial innings pitched, including facing one or more batters   in an inning without recording an out, as a complete inning.? In addition,   no distinction is made between earned runs and unearned runs.

[2]?   RS is his team?s runs scored and RA is how many runs he allowed.? The evaluation   is based solely upon the score at the conclusion of the last inning the pitcher   appeared in.? Accordingly, the performance of the pitchers who relieve the   starter is not considered germane to the evaluation of the starter?s performance.

[3]?   Mathematically, this would be considered a second-order term.

[4]?   Let me give a representative example.? Using the AL 2000 season, when leading   after 7 innings by one run, the probabilities of winning the game based upon   the score are as follows.? 73% if 1-0; 82% if 2-1; 80% if 3-2; 84% if 4-3;   71% if 5-4; and 72% for combined 6-5, 7-6, 8-7, etc., where this last grouping   is necessary to achieve a sufficient number of games to make the comparison   valid.

[5]?   Remember, if we ?fix? run support at RS, then a pitcher who wins a game 6-2   would be deemed to have the same win probability as a pitcher who wins a game   6-5.

[6]?   Interleague games are included where appropriate as follows.? For distributions   of runs scored, runs scored by a team in the league under study are included.?   For distributions of runs allowed, runs scored by the opponent of a team in   the league under study are included.

[7]?   See footnote 4.

[8]?   See footnote 12

Rob Wood Posted: August 19, 2002 at 06:00 AM | 0 comment(s) Login to Bookmark
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