I've got a confession to make. This article actually has nothing whatsoever to do with Pythagoras, the mathematician – I just liked the title. What we shall really be discussing is something known as "Pythagorean expectation", an equation devised by Bill James in the early 1980s that relates a team's number of wins to its total runs scored and runs allowed. James named his equation "Pythagorean" because of what he saw as its resemblance to the Pythagorean theorem in mathematics. The original James formula was:

Win Ratio = runs scored^{2}/(runs scored^{2 }+ runs allowed^{2})

Thus to use a simple example if a team scores 750 runs and allows 750 runs, its win ratio is 750^{2}/(750^{2}+750^{2}) = 0.5 (or 81 wins in a 162-game season), which obviously seems to make sense. But what happens if a teams manages to find an extra 10 runs from somewhere? Now we have 760^{2}/(760^{2}+750^{2}) = 0.5066 which when multiplied by the 162 games gives us 82.07 wins rather than 81. It is from this kind of exercise that the 'rule of thumb' 10 runs = 1 win was derived.

Naturally, in a low-scoring run environment an extra 10 runs counts for slightly more than it does in a high-scoring run environment, which explains why Fangraphs gives a runs/win of just 9.117 for 2014, so in fact last year 10 runs were actually "worth" 1.1 wins.

In his 1982 *Baseball Abstract* Bill James decided that an exponent of 1.83 rather than 2 was slightly more accurate, making the equation:

Win Ratio = runs scored^{1.83}/(runs scored^{1.83 }+ runs allowed^{1.83})

but fundamentally the optimal exponent depends on the league run environment for the period you wish to analyze. For the period 1996 onwards (the "wild card" era, excluding the strike-shortened 1994 and 1995 seasons), you can run a linear model and determine that an exponent of 1.89 produces the most accurate results, so:

Win Ratio = runs scored^{1.89}/(runs scored^{1.89 }+ runs allowed^{1.89})

Now let's run our version and see what comes out for the period from 1996 onwards. But first, let's take a quick visual look at the relationship between runs and wins:

Every dot represents one season for one team. The red dots represent the Indians and the blue dots everyone else. (There is no particular significance in the fact that red dots are larger than the blue ones – I just wanted them to stand out more.)

The first thing to notice is that this is a pretty strong linear relationship - larger positive run differentials are strongly correlated with higher winning percentages and negative run differentials strongly correlated with lower winning percentages. If we build a simple linear model with winning percentage explained by run differential, the resulting coefficient of determination (or R^{2}) is 88.3. In other words, 88.3% of the variation in wins is "explained" by differences in run differentials.

We also note that the standard deviation of the expected number of wins is roughly 3.9 wins. Thus we can state that around two-thirds of teams will fall in a range within plus or minus 3.9 wins of their expectation – a discrepancy of 10+ wins will be quite rare indeed.

Over the 19-year period under discussion, the Tribe won 1581 games, whereas the formula predicts they should have won nearly 1583, that's a difference of a mere two games overall. In 2011 they outperformed their win expectancy by 4.92 games, whereas in 2006 they were extremely unlucky, recording just 78 wins (despite a run differential of +88) when the formula would have predicted 89 wins. You can see just how much of an outlier these two seasons were in the context of this graph, with the 2006 Tribe season representing one of the biggest overall outliers:

But the question arises, can we perhaps identify one or more issues that might be causing these discrepancies between Pythagorean expected wins and actual wins? Are there forces at play that can cause a team to over-perform or under-perform its expectation?

#### Is every run worth the same?

One of the most obvious potential flaws of the Pythagorean approach is that it only considers runs on a seasonal aggregate basis. In practice, a few extra runs scored or conceded in a blowout will have no bearing on a team's final win total, but those runs count for exactly the same in the model as those from close games. It is certainly possible that the distribution of how those runs were scored should be taken into consideration.

If we dig deeper into that 2006 Indians season, we can see clear evidence of this – the Tribe's average margin of victory that year was 4.6 runs, whereas the average margin of defeat was just 3.2 runs. The Tribe was often winning by blowout and then losing the close ones. Specifically, in games decided by a single run, the Tribe only went 18-26, whereas in two-run games they went 11-14, while in three-run games they went 10-15. In games decided by more than three runs they went 39-29.

Back in August 2005 Ray Ciccolella examined the effect of a team's distribution of runs on their Pythagorean expectation. His conclusion was startling: "Teams with the highest runs scored ratio [runs scored per game/standard deviation of runs scored per game] and lowest runs allowed ratio [runs allowed per game/standard deviation of runs allowed per game] exceeded their predicted wins by an average of 3.6 wins per team. Teams on the other extreme missed their predicted number of wins by an average of 2.3 wins per team." In other words, teams with a lower volatility of runs scored and a higher volatility of runs allowed performed best.

Over the course of a season, a team that scores four runs every single day will outperform one which scores 10 runs one day and two runs the next. In fact, if you take a (roughly) league average offense scoring four runs every single day and take them up against the score of every losing team during the 2014 season, they would score a remarkable 120.9 wins. "Consistent" offense is desirable whereas consistent defense/pitching isn't something a team needs to strive for (by and large if you concede 10 runs you might just as well have conceded 20). So that's more ammunition the next time that we all bemoan the Tribe's "inconsistent" offense.

#### Offense or defense?

In *Baseball between the Numbers* Nate Silver and Dayn Perry make the interesting argument that "...good pitchers have a structural advantage against good-hitting *teams*. That advantage comes to forefront in the playoffs, when all the teams can hit pretty well." The idea is that the pitchers can render hitters less than the sum of their parts by distributing hits and walks in an even fashion, preventing the "expected" number of runs from being scored. "Great pitching, average offense" beats "great offense, average pitching" 2-3 percent more often than one would "expect".

Furthermore, in the book *Extra Innings* Dan Turkenkopf demonstrates that *for a winning team*, reducing 20 runs allowed is preferable to increasing 20 runs scored (but *vice-versa for losing teams*). Even if the Tribe's offensive production and pitching were to remain the same, the potential benefits of saving runs this year through improved fielding are evident – 40 runs saved would be worth much more than four wins.

But it certainly isn't advantageous for a team to be too imbalanced towards either offense or defense, however good that offense or defense may be. In a separate article in *Baseball between the Numbers*, Steven Goldman shows that the best hitting team of all time (he chooses the 1931 Yankees) would only win 86 games when paired with the worst-ever pitching team (the 1996 Detroit Tigers were his pick), and that the best-ever pitching team paired with the worst-ever pitching team fares even worse. Thus building a balanced team (one that is better than average on both offense and defense) is the optimal way to generate wins.

OK, that's it for Part I (don't worry, all the essential, technical stuff was in that part!). In the second half of this article we shall move on to a discussion about the influence of managers and the role of the bullpen, review each teams's 2014 Pythagorean expectations and then put forth a run differential "plan" for the Tribe in 2015.

*Incidentally, if anyone is interested in exploring win expectancy in more detail, I would strongly recommend* Analyzing Baseball Data with R* by Max Marchi and Jim Albert (CRC Press). I based part of my analysis in this article on chapter 4 of their book. Marchi was hired last Spring to join the analytics team at the Cleveland Indians, so he is obviously a good egg. *

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