The Two Ways to Win More Games

Last season the Nationals scored 655 runs and allowed 742, for a run differential of -87 and an overall record of 69-93.  Respectively, those outcomes ranked 14th, 12th, 13th, and 14th in the NL.

What's the most efficient way to reduce post-game scenes like this one?

It is obvious that run differential is highly determinative of a team's record.  It is equally obvious that there are two paths to improvement - (1) score more runs; (2) allow fewer runs.  A huge amount of virtual ink has been spilled on nationals.com advocating for improved defense and declaring the team's offense to be adequate.  In reality, though:

• Does it matter whether the team's run differential is improved through RA or RS?
• For a given run differential, how much does the overall level of RA and RS matter?
• Generally, what can be said about RA vs. RS improvements?

The Pythagorean Theorem of Baseball is a good place to start.  PTB relates a team's run differential to an expected (or Pythagorean) winning percentage:

• Generic Pythagorean Winning % = [(RS)^1.83]/[(RS)^1.83 + (RA)^1.83]
• 2010 Nationals Pythagorean Winning % = [(655)^1.83]/[(655)^1.83 + (742)^1.83] = .443

A PWP of .443 equates to 72-90, three games better than the Nats' actual record of 69-93.  There are multiple interpretations for the difference between Pythagorean W/L and actual W/L (clutch play, bullpen quality, residual bad karma from the Jim Bowden days), but it's generally accepted to be driven by statistical noise (more romantically known as 'luck').*

Pythagoras posits the Nats were bad and unlucky in 2010.

Historically, teams will typically finish the season within three games of their Pythagorean win total (in 2010, 12 out of 16 NL teams).  Armed with the PTB, let's return to the questions:

Does it matter whether the Nationals' run differential is improved more through RA than RS?  Not much.  An improvement of 30 runs from last season's levels (our goals are modest here) is worth ~3 projected wins.  This projection is unaffected by the RA/RS combination used to achieve the improvement, until the numbers get quite large.  For example, giving up 60 fewer runs and scoring 30 fewer runs (for a net 30-run improvement) gets you 75 projected wins.  Likewise, scoring 60 additional runs and giving up 30 more (+30 again) gets you to the same place.  As do changes of (RS/RA) 30/0, 0/30, 15/15, etc.  When the numbers get larger, the answer changes a little (as detailed below).

For a given run differential, does the overall level of RA and RS matter?  Slightly.  To establish reasonable parameters, let's utilize the worst of the worst from 2010: the Pirates' 836 RA total and the Astros' 611 RS tally.  In that case, whether the Nats -87 run differential is set at one extreme (611 RS/698 RA) or the other (749 RS/836 RA) matters to the tune of 1.7 projected wins (71.2 wins vs. 72.9 wins):

In short, teams can't significantly improve their fate without improving run differential.  An interesting side note is that if run differential is fixed, then the ideal RA/RS level depends on the quality of the team.  For example, a team expected to be below .500 (like the Nats) would benefit from a move that improves their RS and hurts their RA in equal measure (represented by the positive slope of the blue dots).  The opposite is true for a team above .500; a perfectly .500 team is indifferent.  To reiterate, if the Nats sacrifice pitching/defense for a perfectly offsetting offensive improvement, they will experience a (marginally) higher win total.  Take that, Ladson.

Generally, what can we say about RA vs. RS improvements?

1. Preferences shouldn't matter.  Offense, defense, and pitching are simply means to an end.
2. Prices do matter.   Understanding the per-run price of any move enables a team to maximize their win total, subject to its overall budget constraint.  Constrained optimization is a familiar problem for all of us and the reason why my freezer is full of hamburgers even though I prefer steak.
3. Supply is plentiful.  With the Nationals at 14th in the league in RS and 12th in the league in RA, opportunities abound for improvement.  I see no logical reason why the focus should be on one particular facet of the game. The Nationals are certainly bad enough to accommodate a wide range of improvements.

*This is supported by the fact that - given a large enough sample - actual winning percentage converges to Pythagorean winning percentage, not the other way around.  The 2005 Nationals are a classic example of a team's second-half convergence to its first-half Pythagorean winning percentage.