A more sophisticted PLAYOFFS+ metric

by Jason Wojciechowski

As I'm sure you recall, I've proposed that different people have different theories of value. Some of us (myself included) would prefer to strip as much team context as we can out of a player's performance when arguing for individual awards like the MVP and Cy Young. (I called this theory of value WINS+.) Others, though, think that the entire point of the regular season is to help your team get to the playoffs. You see this in arguments that only players on contending teams should be eligible for the MVP.

This, I argued in that post, is an entirely rational position. (I called this theory of value PLAYOFFS+.) The problem is that it's usually applied in an ad hoc fashion, providing excuses for a voter to ignore a better performance because that voter thinks, for whatever unarticulated reason, that the lesser player provided more value by pushing his team near or into the playoffs. I've previously put together a naive but consistent way to implement PLAYOFFS+ -- only players whose teams make the playoffs by a margin slimmer than that individual's WAR total are eligible, and the player, of that group, with the most WAR wins. That approach was rife with problems, some of which I've attempted to address here.

The major difficulty with the naive PLAYOFFS+ implementation is that it zeroes out the contributions of anyone whose team failed to make the playoffs. Thus, a hypothetical 8-WAR player on a 92-win team that is eliminated in a one-game tie-breaker on the last day of the season would not win the award. This is important because every article I've ever seen that espouses the PLAYOFFS+ theory of value speaks of players who help their teams contend, not just players who are on the top four teams.

Further, many PLAYOFFS+ voters will consider exceptional seasons by players on non-contending teams, especially down-ballot. The naive PLAYOFFS+ system doesn't just zero out contenders' contributions, it renders valueless the 10-WAR player on a .500 team whose squad would have been pitiful without him. (Jose Bautista, say.)

Here, then, is a better way. Instead of denying credit to players whose teams aren't either in contention or the playoffs, let's instead give players a bonus for pushing their teams closer to the postseason. Since 1996, for instance, there have been nine teams that have won 82 games. Just one made the playoffs. Fifteen teams won 90 games, eleven of which played in October. What I would do, then, is give an eight-win player on a 90-win team extra credit for the additional probability of making the playoffs that his team had because he was on the team instead of a random Quad-A player.

More specifically, I want to multiiply each player's WAR by (1+p) where p is the difference between the probability that a team would make the playoffs with their actual win total and the probability that a team would make the playoffs if you subtract the player's WAR from the win total. Note that I'm stripping out divisional and league context. I don't care how many wins the AL Central required to win in 1999 when voting for the 1999 MVP award -- I just want to know how many wins it typically takes in baseball as a whole to reach the playoffs, and I build my multiplier from that information.

This doesn't work exactly, however, because, for instance, two out of nine teams with just 84 wins have made the playoffs, while just 1/14 and 1/16 with 85 and 86 wins have. That 2/9 is an aberration, and we certainly wouldn't want to subtract credit from a player who took his team from 84 wins to 85, so I've manually smoothed out the probabilities as follows:

Wins	Actual teams	Actual playoff teams	smoothed playoff probability
0-81	224	0	0
82	9	1	.04
83	19	1	.06
84	9	2	.08
85	14	1	.10
86	16	1	.12
87	8	2	.25
88	21	7	.33
89	12	4	.50
90	15	11	.70
91	10	8	.80
92	11	10	.90
93	9	5	.95
94	6	6	.97
95	15	15	.98
96	9	8	.99
97-162	38	38	1

Rather than giving examples of the metric, I'll jump straight to the MVP ballots for 2011 that the method would produce, which will illustrate much of the range of possibilities. I've used Baseball Prospectus's WARP this time.

First, the American League:

Player	WARP	Team wins	P(playoffs)	Wins w/o player	P(playoffs w/o)	Multiplier	adj WARP
Jacoby Ellsbury	9.0	90	.70	81	0	1.7	15.30
Justin Verlander	6.9	95	.98	88	.33	1.65	11.38
Evan Longoria	6.1	91	.80	85	.10	1.7	10.37
Jose Bautista	10.3	81	0	71	0	1	10.30
Ian Kinsler	6.7	96	.99	89	.50	1.49	9.98

By WARP, the two most dominant players in the league, Ellsbury and Bautista, make the ballot, with Ellsbury's ridiculous season combined with the fact that his team would've been nowhere without him pushing him well to the top. I daresay that this reflects well the way a lot of people think about the MVP award.

Where Ellsbury takes the Red Sox from no chance to strong contention, Verlander pushed the Tigers from contention to near-lock and is rewarded with a big multiplier. Longoria is another Ellsbury, but lesser. Jose Bautista shows how a dominant non-contender is treated by the system, letting his WARP speak for itself.

Now, the National League:

Player	WARP	Team wins	P(playoffs)	Wins w/o player	P(playoffs w/o)	Multiplier	adj WARP
Ryan Braun	6.9	96	.99	89	0.5	1.49	10.28
Albert Pujols	5.6	90	.70	84	.08	1.62	9.07
Matt Kemp	8.1	82	.04	74	0	1.04	8.42
Lance Berkman	4.6	90	.70	85	.10	1.60	7.36
Justin Upton	5.0	94	.97	89	.50	1.47	7.35

Braun had the third-highest WARP in the NL this year behind only Matt Kemp and Clayton Kershaw. Kershaw doesn't appear on the ballot, but he finished with an adjusted WARP of 7.28, indistinguishable from Berkman and Upton, basically tied for fourth.

The one type of player not illustrated above is that on a truly dominant team, one that is well into the "1" portion of the probability table above. Cliff Lee, the best player by WARP on the Phillies, is a good example.

Player	WARP	Team wins	P(playoffs)	Wins w/o player	P(playoffs w/o)	Multiplier	adj WARP
Cliff Lee	6.6	102	1	95	.98	1.02	6.73

That score ranks eighth in the NL, behind the five listed above, the aforementioned Kershaw, and Ian Kennedy (7.06). A full spreadsheet for 2011 is here.

Note that the biggest possible multiplier is 2, and would result from a player on a 97-win team having 16 WAR. Unlike the previous PLAYOFFS+ implementation, then, the worst possible disconnect from pure individual WAR to adjusted WAR would be a player half as valuable as another winning the award. (In the naive implementation, you could end up with no MVP at all, a significantly worse outcome.) And note the extremity such a situation would require: a 16.1-WAR player would beat a 32-WAR player if the 32-WAR player was on an 82-win team and the 16.1-WAR player was on a 97-win team.

This is obviously absurd, so we can set some realistic parameters on the "worst" possible outcome that using this system would produce. The greatest WAR season in history by Fangraphs was Babe Ruth's 15.4 in 1923. Lou Gehrig is the only player besides Ruth to ever top 13 fWAR. Baseball Prospectus's WARP does not go back that far (it appears to start around 1954?), so their top scoring season is Barry Bonds with 12.2 in 2001. Let's be generous and say the highest possible WAR since Babe Ruth is dead is 13.

If that 13-win player were on such a bad team (or such a good one, like the Mariners' 116-win squad) that he got no multiplier, what kind of season could beat his for MVP? Unless I've done something wrong, the worst possible season that could beat this 13-win year is a 7.2-WAR performance for a 93-win team. This configuration results in a multipler of 1.83 and thus a 13.18 adjusted WAR, just nudging our hypothetical Babe Ruth. I haven't run the numbers, but I'd be surprised if anything near that bad has happened since 1996.

If you've made it this far, I do want to say that this isn't my theory of value. I don't like the fact that Cliff Lee's value for an award is dependent on how well Roy Halladay pitches. Forget about defensive support, luck, or anything else that we might try to adjust for when Lee is pitching -- using PLAYOFFS+ causes us to adjust our picture of Cliff Lee based on stuff that happens when he's in the dugout cheering his team on.

My purpose here is to show what type of analysis I would hope PLAYOFFS+ people might engage in -- honest, neutral weighting of individual performances so that their votes aren't simply exercises in ad hoc argument for whoever catches their fancy in a given year.