Picking one game at random, on April 22, 2006, the Brewers played the Cincinnati Reds in Milwaukee, Dave Bush starting for Milwaukee.  The Brewers scored 11 runs in the game, and Bush posted a Game Score of 86.   Question:   Did the Brewers win the game, or lose?

            Well, obviously they won the game; to score 11 runs, get a Game Score of 86 from your starting pitcher—65 is a “gem”—and somehow NOT win the game would be virtually impossible.

            Or, other end of the spectrum. ..on July 15, 2006, the Brewers played the Diamondbacks in Arizona, Chris Capuano starting for Milwaukee.   The Brewers scored 1 run in the game, and Capuano’s Game Score was 15.   Did the Brewers win the game, or lose?

            Well, obviously, they lost; to score one run in a game when your starting pitcher is getting the chapstick beat out of him, and still win the game. .. it’s impossible.   It’s probably never happened in the history of baseball.

            Recently a team in which I have a rooting interesting had a game in which the starting pitcher pitched pretty well (Game Score:  56), while the team scored six runs.   Nonetheless, the team lost the game.     What you may be able to infer from this is that the bullpen did not have a good afternoon.    I have to be careful how I phrase certain things.   Visualize a series of headlines:   James Blames Bullpen for String of Losses.    Reliever Blasts James for Criticizing Pen.    Team Cuts Ties with Statistician.

            We don’t go down certain alleyways, not knowing what may be lurking in the dark.   Anyway, it was a painful loss at the time, and I was thinking about it afterward:  that when the team scores 6 runs and the starting pitcher pitches that well, you probably ought to win the game. . .what, 80% of the time?

            90%, actually; 90.2.

            Based simply on these two data points:

            1)  The runs scored in the game, and

            2)  The Game Score of the starting pitcher,

            one should be able to predict the outcome of the game with a high level of accuracy.  Most real-life combinations of the two have foreseeable outcomes.   Four runs, Game Score of 70, you’re going to win.   Three runs, Game Score of 45, you are going to lose.

            Except for a few “mid-range” combinations. .. .three runs, game score of 57, you might win, you might lose.    One run scored, Game Score of 76, you might win, you might lose.   Six runs, Game Score of 37, you might win, you might lose.  But most of the time, based on the combination of these two facts, you’re either dealing with a win or with a loss.

            What occurred to me next is that one could evaluate a team’s bullpen, then, as the residue or “discrepancy” in this process.    If you win 90.2% of the time when you score six runs and get a Game Score of 56, then who is responsible for the loss when a game like that is lost?   Obviously, the bullpen.   Or, on the other side, if you lose 92.1% of the time when you score four runs and your starting pitcher turns in a “45”, then who gets the credit for that if you are able to win that one?

            It thus occurred to me that one could evaluate a team’s bullpen by combining these other two data points and triangulating the third, but we’ll get to that later. . .not in this article.   At the moment we’re not talking about the bullpen; at the moment I am stuck on the Win Probabilities associated with each combination of Game Score and Runs Scored.    How do we figure out what are the actual Win Probabilities for each combination?

            I have this data base that my son created for me, which I’ve mentioned before.   I took all games played in the major leagues in the last ten years (2000-2009), looked at from the standpoint of each team, and I sorted the data by

            a)  the Runs Scored by the offense, and

            b)  the Game Score of the starting pitcher.  

            There were, you will be fascinated to learn, 112 major league games between 2000 and 2009 in which a team scored exactly six runs, and their starting pitcher posted a Game Score of exactly 56.     The won-lost record of those 112 teams was 100-12, an .893 winning percentage. 

            Why, then, did I say earlier that the win expectation in that situation was .902?

            Well, you probably know the answer to that as well as I do, but this does not relieve me of the responsibility to explain.     To make the data points more reliable, I aggregated the data, combining games in which the combination was 6-56 (6 runs scored, Game Score of 56) with 6-55 and 6-57.    There were 288 games from 2000 to 2009 in which a team scored exactly six runs, and their starting pitcher posted a Game Score between 55 and 57.   The won-lost record of those 288 teams was 259-29, an .899 percentage.    I then interpolated between the data groups, and then applied a “smoothing function” to the data to minimize irregularities. .  .massaging the data.   Common stuff.    Anyway, the ultimate result of this was that 6-56 wound up at .902, rather than .893 or .899.

The “10 Run” column at right is actually ten runs or more.   That probably doesn’t matter for the top half of the chart, but it probably distorts the data a little bit for the bottom part of the 10-run column.

II.  Fairly Useless Little Heuristic

            Multiply the Runs Scored by the Team times the Game Score of the Starting Pitcher.   If the total is 562 or higher, you win the game.    There were 2,758 major league games from 2000 to 2009 in which the product of those two was 562 or higher, and the won-lost record of those teams was 2,758-0.

            If the product of these two is zero, obviously, your chances of winning are zero, since virtually all of the games in which the product of these two is zero are shutouts, and you can’t win a game in which you are shut out.    But if the product is more than zero but less than 50, your chances of winning are still zero.  There were 3,209 games during the last decade in which a team had a product greater than zero but less than 50, and the won-lost record of those teams was 0-3,209.

            The point at which you probably have a “winning product” of these two numbers is about 193.    You can’t actually get to 193; 193 is a prime number, so there’s no combination that produces 193 when multiplied together.   Above 193, you’ve probably got a winner.   Below 193, you’ve probably got a loser.

            You can estimate the chance of getting a win, then, by this little formula:

            Game Score times Runs Scored

            Minus 50

            Divided by 286

            = Chance of getting a Win

            But obviously, that’s just a quick-and-dirty approximation, not really useful.   Six times 40 is 240, 5 times 48 is 240, 4 times 60 is 240, 3 times 80 is 240.    But if you get six runs and a Game Score of 40 (6-40), your win expectation is .607; 5-48 is .671; 4-60 is .780; and 3-80 is .958.    We’re not dealing with perfectly inverse relationships between the two elements.   

            Another way in which this little study was very satisfying is that, while I did the research based entirely on the games from 2000 to 2009, I realized after having done so that the results would probably apply almost perfectly to any other decade, even a decade in which the normal run levels were very different.    I initially assumed that it wouldn’t apply “out of range”, but then thought about it and realized that it probably would.

            Why?   First, Game Score norms are fairly stable over time—not perfectly stable, but not radically unstable.   And second, to the extent that they are unstable, the two elements are operating on parallel scales, both based on runs scored and runs allowed.    It is true that your winning percentage with a “60” Game Score would be higher in the 2000-2009 era than in the 1960s, for example—but that’s not the issue.    The issue is, what would be your winning percentage with a “60” Game score and X runs scored by the offense?   THAT number, I would predict, would be almost precisely the same in every decade, in that there is not very much that can cause it to vary.

III. Applying This to the Evaluation of Bullpens

            Next it occurred to me, as I was drawing up the big chart above, that the “residue” or “discrepancies” in the outcomes of games had to be, in most cases, almost entirely attributable to the bullpens.    I thus thought that I could use this chart to focus on the assignments and success levels of each bullpen, thus assigning a “bullpen success rate” to each major league team.

            I still think this could be done, but. ..it doesn’t work at this level, for reasons that are probably intuitively obvious to most of you, but which I was in denial about.    Josh Beckett and Jimmy Anderson.    On July 27, 2002, Jimmy Anderson of the Pirates pitched 7 innings, giving up 12 hits, 3 runs, 3 earned runs, 1 walk, 1 strikeout.   That’s a Game Score of 41.   On April 23, 2002, Josh Beckett pitched 7 innings, giving up 3 hits, 3 runs, no earned runs, 11 strikeouts, 3 walks.   That’s a Game Score of 73.

            I don’t have any mixed feelings about the fact that we gave Josh Beckett a much higher Game Score than Jimmy Anderson.    He’s a much better pitcher, and he pitched a much more impressive game—11 strikeouts, 3 hits, versus 1 strikeout, 12 hits.   Still, the fact is that both Beckett and Anderson pitched 7 innings, and allowed 3 runs.     Thus, given a certain number of runs to work with, their bullpen’s chance of winning is the same in one case as the other.   If the team scored five runs and had Beckett’s game, we would assume that the bullpen should win the game 96% of the time.   If they had Anderson’s game, we would assume that the bullpen should win the game 43% of the time.   That’s an intolerable discrepancy, so we just have to abandon that approach to evaluating bullpens.

            We abandon it, even though it works pretty well in practice.   I evaluated every team’s bullpen from the last ten years based on the assumption that one can triangulate the performance of the bullpen from the two factors on the chart above, and it actually gets very reasonable results.    The study says that the worst bullpen of the last ten years was the Cubs’ bullpen in 2002.   Look them up.   They were ghastly.   If it’s not the worst bullpen of the last ten years, it’s got to be close.  

            But still.  . .I can’t go with it.   I’ll get back to that issue later.    I just thought the chart above was kind of fun, and I thought I would share it with you.