Many of the excellent analysts on the blog have argued in recent days that Phillies have less to gain than other teams at the deadline due to their status as a "playoff lock".   I tend to believe that the opposite is true, and I have put together a quick model that I'd like to share.

Let's consider 4 teams:
Team A:  In 50/50 battle to make playoffs, and then 50/50 in each playoff series.
Team B: Lock to make playoffs, then 50/50 in each playoff series.
Team C:  Lock to make World Series, 50/50 in World Series (highly hypothetical)
Team D: 98% to make playoffs, 69% in first round, 63% in LCS, 54% in World Series (estimate of Phillies situation, see also Footnote 1)

Each team is considering acquiring a rent-a-player who would provide, let's say, 2.4 WAR (improvement) per 162 games. This translates into 2.4/162 = 0.9 wins for the regular season stretch drive, and it gives a team that was in a 50/50 two-team playoff race a 56.7% edge if you run a simple binomial distribution comparison (Footnote 2).  Let's be generous and boost this up to 58%, for reasons described in footnote 3. 

The acquired player also provides a single game winning percentage improvement of 2.4/162 = .0148. I.e.,  a .500 ballclub is now a .5148 ballclub.   This single game winning percentage equates to a 5 game series win percentage of 52.77% and a 7 game win series win percentage of 53.23% (if the series was 50/50 to start with).

Here are the results of the acquisition:
Team A:   Was .5*.5*.5*.5 = 6.25% to win World Series.
Team A:  Now .58*.5277*.5323*.5323 = 8.68% to win World Series.  Gain = 2.43%
Team B:  Was .5*.5*.5 = 12.5%
Team B:  Now .5277*.5323*.5323 = 14.95%.  Gain = 2.45%
Team C:  Was 50%
Team C:  Now 53.27% Gain = 3.27%
Team D:  Was .98*.69*.63*.54 = 23.00%
Team D:  Now .985*.715*.6606*.5787 = 26.92% Gain = 3.92% (see footnote 4)

Conclusion: The non-"playoff lock" teams do get a bigger boost in terms of their chances in a relative sense, since 8.68/6.25 is 39% better, verses 26.92/23.00 = 17% better.   But, teams that are World Series favorites have so much to gain or lose by even a small percentage change in their chances, that overall there are more gains to reap at the deadline.  It is still a crapshoot, but the stakes are so high that a small odds tilt makes a big difference.

(Footnotes)

(1)  At first blush, these Phillies playoff probabilities may seem high, particularly the short series ones.   But, 69% in five game series translates to 60.43% single game probability, and I actually think that is pretty fair average for the Phillies rotation matching up against NL Central teams or a Posey-less Giants team. 

(2)  The 56.67% comes from comparing the binomial distribution of (60 trials, p =30/60) verses (60 trials, p= 30.9/60) to see how often the distribution with the additional 0.9 WAR is better.  Assuming the teams are both really good makes little difference in the exercise:  (60 trials, p = 36/60) verses (60 trials, p = 36.9/60) is a 56.77% situation.

(3)  (a) Playoff chases involve head to head matchups, and the acquired player will subtract wins from rivals in those games.   (b) In a multi-way race for multiple playoff spots, the marginal value of one win is likely to be higher than in a two-way race (I believe) due to the fact that the "bar" for playoff eligibility is likely more fixed.  In the 2 way race, the single opponent can get hot and render the marginal WAR less valuable.

In any event, this tinkering actually argues against my overall conclusion.  

(4)  It does seem a little weird that the marginal WAR benefits the already-favored playoff team a little more than a team that was 50/50;   notice the .5787 verses .5400 compared to .5323 verses .5000.      This is accurate, though, in a probabilistic sense.  The thing that might be slightly off is that an already successful winning ballclub does not get 2.4 WAR from the player, they perhaps only get 2.1 WAR as they find additional wins marginally difficult to achieve.   I don't believe this difference would alter the overall conclusion though.