Should we root against the Sox or the Yankees?

Hello! I apologize in advance if any of the following is poorly written or difficult to follow; my statistics terminology is spotty at best this far removed from my last math class. An additional disclaimer: I know there are more and better models out there, I just couldn't find one that told me exactly what I wanted.

Work today was breathtakingly slow, and my boss was out of the office, so instead of working I was on the open thread and following the Yanks/Sox game all day. This, as was discussed on the thread, presented a dilemma: who do I want to see lose (alternatively, who am I least upset to see win?).  Despite my youthful, one-time, misguided, and even then only partial affection for the Sox, I now hate them almost as much as the Yankees. That being the case, the emotional/intangible benefit to seeing them beat the Yankees is so small as to be insignificant.

With no sentimental investment in the game (other than perhaps hoping for injury...no, we don't do that, right?), all that's left is the impact on the O's. But how do you quantify that? Is the marginal value of an additional game lead worth more over the Yankees or the Sox? In this particular case, we are (were) looking at the difference between a .5 game lead over the Yankees combined with a 5.5 game lead over the Sox vs the alternative of a 1.5 game lead over the Yankees and a 4.5 game lead over the Sox. As a note, when I say "game lead" I'm referring to the standard "games behind" calculation which is (while we're in first place anyway) ((O's wins - O's losses) - (Badguy's wins - Badguy's losses))/2

Instinctively it seems as though we should always root for the team with fewer wins in this situation. This is because we would expect that the marginal value of each additional game lead (or deficit) is less as that lead gets larger. In other words, the difference between a 1 game and 2 game lead over a divisional opponent is greater than the difference between a 31 game and a 32 game lead. But how do you quantify that? Considering the end goal of the regular season is a playoff spot, and we can to some extent calculate how likely each team is to make the playoffs, playoff-likelihood seems like a good measure.

For simplicity's sake, I disregarded the wild-card in this analysis, so when I say playoff-likelihood (PL) I really just mean the chances of winning the division. Using an assumed "true" winning percentage (TW%) and current records, I came up with an expected win total and the associated variance for each team. By then assuming a normal distribution around the expected wins, I could calculate the odds that any one team finishes with more wins than any other. For instance, over a 162 games season, a hypothetical team with a TW% of .500 that has a 5-1 record would have a 64% chance of finishing with more wins than an equivalent team with an 0-6 record, and a 55% chance to finish ahead of an equivalent 4-2 team.

To get from that to PL, I tried to come up with each team's probability of finishing ahead of every other team in the division, then multiplying those percentages. However, I was too lazy (or not clever enough to find an easier way) to lay out the odds of every permutation to make this work, so instead I took the easier route of just simulating 100,000 seasons and taking that result. For simplicity's sake (again), I assumed all games are independent events even though they're not (as the teams involved play one another).

Onto some results:

For a base case, let's assume that every team has a TW% of .500. So every team has an expected win total of its current wins plus half of the remaining games. In this case, every team will have a similar variance (around 40.5, or a standard deviation of 6.3 wins) because the TW% is equal for every team and there's 96% of the games left to be played so it dominates the actual record.

 Results:

(The way to read this table is that the O's before the game had a 31.03% chance of winning the division, and would have a 30.51% chance after a Yankees win and a 31.28% chance after a Sox win.)

Here the results are exactly as we'd expect. The marginal value of the 1.5 game lead vs. the .5 game lead is far greater to our hypothetical O's team than the marginal value of the 5.5 game lead vs. the 4.5 game lead; and our hypothetical Red Sox win is worth an additional .76% chance of winning the division to the O's (and and 2.35% bump for the Sox). Also note that the delta is the difference between the playoff odds for each team in the case of a Sox or Yankees win, not the difference from before the game to after it.

However, this is not the full story! Because as I'm sure you all know and have been wanting to yell at me the entire time you've been reading this (and I know a bunch of you will have tl;dr'd by now) These teams are not all equal, and we shouldn't expect or assume that they each have a 50% chance to win or lose every game for the rest of the season. So, what impact does some weighting have on our beloved O's PL?

Using PBR me ASAP!'s generously donated estimates of each team's "True Winning %" (Sox - .587, Yanks - .516, O's - .500, Rays - .481, Jays - .449) I reran and recalculated these odds. And feel free to quibble with those TW% numbers, if there's any interest I will re-run this analysis over the course of the season and change these estimates to match our changing expectations (I'll probably be doing this anyway, as I've already wasted a couple hours building this tool).

 Here are the new results:

Now the glove is truly on a different foot! If the Sox actually are a .587 team that's just had a run of bad luck to start the season, and the Yankees are merely a .516 team that has started hot; each Red Sox loss is far more valuable to the O's playoff chances than each Yankee's loss, so even despite the fact that the Sox are further behind in the standings now, the O's playoff odds are 1.17 percentage points better in the case of a Yankees win than a Sox win.

So, what is the end result of this study? If we as O's fans believe that the Sox are a dominant force this season, and we want to always have our rooting interests aligned with the O's best chances of winning the division, then at this juncture of the season, we should be cheering more at each Pedroia error than at each Doucheria strikeout.

And yes, I know the game has ended and the Sox won, but this took me some time to put together, OK?!

Thanks for reading!