Championship Probability Added I
Posted by Neil Paine on December 2, 2009
(Note: This method is in the very early stages, so I'm trying to generate some useful feedback more than anything else. Be warned that the results are very rough.)
If you're a regular visitor here at Basketball-Reference, I think it's pretty safe to say that you're interested in all-time player ranking arguments -- it's okay, I am too. We love to try to make order out of the chaos that is the history of pro basketball, and one of our chief means of doing that is putting together lists and rating players to see where they fit in the larger context of the game. In fact, that was the motivation behind this next series of posts, to try to put together some semblance of an objective player ranking, encompassing the entire Win Shares era (1951-52 to the present).
The main problem I've had with these types of all-time rankings in the past is this: how do you deal with the playoffs, in terms of weighting them against regular-season results? Many fans place a lot of value on the postseason, but without a lot of thought going into why, other than "Player X is clutch," or "I think titles matter the most because as a fan I want my team to win them, so I'll rank Player Y higher than Player Z because he has more rings". What I want to determine, though, is an objective way of valuing the playoffs vs. the regular season, so we can achieve what I'm going to call "Championship Probability Added", or the amount by which a player increased his team's chances of winning a title through his Win Shares.
Here's a thought experiment: on average, how much is each regular-season win worth, in terms of increasing a team's probability of making the playoffs? Well, the average NBA team has a 1/30 16/30 chance of making the playoffs at the start of the season. Win 1 game, and you really don't increase that number very much, but win 20, and then 10 more, then 10 more again, and suddenly you're looking at pretty good odds of playing in the postseason. So not all wins are created equal; the 39th win of the season is worth more than the 29th or the 49th in terms of making the playoffs, because the marginal value is higher in that sweet spot around the 8th seed -- each single win takes on more importance because it can be the difference between playing on or going home. On average, though, not every win comes in that sweet spot; some come early in the year, some even come after a team has locked up a playoff berth. And Win Shares doesn't discriminate between early or late-season wins, so we have to find how much the typical NBA win is worth in terms of playoff probability added. Fortunately, this pretty easy: as we said, at the start of the year, each team has a 1/30 chance of playing in the playoffs, and by the end of the regular season, 16 of those teams have a 100% chance of making the playoffs. League-wide, there were 1230 wins, and 16 teams moved from (1/30) to (1), so the average win increases playoff probability by ((1 - (1 / 30)) * 16) / 1230 = .0126, or 1.26%.
How do we convert this to championship probability? Well, for the average playoff team, 100% playoff probability gives them a 1/16 chance of winning a ring, so replace 1 in the above equation with (1/16), and you get 0.0004, the amount by which each regular-season win (and, consequently, each Win Share) increases your probability of winning a title. (Obviously, this doesn't take into account seedings, etc., but I thought we'd keep things simple at first.) Now, once you're in the playoffs, you start with a 1/16 championship probability and add to it by advancing each round. So one team (the champs) will have increased their CP from (1/16) to 1, another (the runners-up) will have increased it from (1/16) to (1/2), and so on. We could go through all of the different deltas for teams that make each round of the playoffs, but it's easier to realize that in any given season, CPA must equal 1.0 -- if we had 0.0004 * 1230 = 0.492 CPA distributed in the regular season, the playoffs would have to see 0.508 be distributed, or 0.508 / 85 = 0.0063 CPA per playoff win in 2009. That means we can multiply RS Win Shares by .0004 and playoff Win Shares by .0063, and find how much Championship Probability Added each player had.
Note that the ratio of "importance" between the playoffs and the regular season has changed dramatically over time. In 2009, the regular season was essentially as important as the playoffs in terms of your probability of a title, because roughly half the teams in the NBA missed the playoffs -- basically you can consider the RS like a preliminary playoff round. Back in 1952, though, the playoffs were four times as important as the regular season, because out of the league's 10 teams, only 2 missed out on postseason play. This meant that performance in the postseason took on far more importance than it does today, since you were competing with 80% of the league (the equivalent of 24 teams in '09) even after making the playoffs.
A few more notes before I show the preliminary leaders in career CPA... First, how much do we discount performance the ABA? In the past I discounted it by roughly 25%, and that's the figure I used here. You'll see, though, that this number could be overstating the ABA's talent relative to the NBA, because a number of ABA rank much higher than the overwhelming majority of basketball fans would place them. The following list also does nothing to balance peak years vs. career performance to combat "compilers". In addition, there's the persistent question of a timeline adjustment, since old-school players (especially from the Auerbach/dynasty Celtics) dominate the list. By recognizing that a higher percentage of the league was still active in the postseason (therefore valuing postseason play more), I'm also indirectly ignoring the fact that even with a higher % of the league active, teams from the 1960s still had to contend with fewer total teams than today's playoff squads. I'll have to develop a way to compensate for all of these problems eventually, but in the meantime, here are the top players in career (total) Championship probability Added, going back through 1951-52, along with their top 3 seasons by CPA:
| Rank | Player | Career | #1 Season | #2 Season | #3 Season |
|---|---|---|---|---|---|
| 1 | Bill Russell | 0.8729 | 0.1012 (1965) | 0.1007 (1960) | 0.0971 (1962) |
| 2 | Wilt Chamberlain | 0.8336 | 0.1158 (1964) | 0.1084 (1967) | 0.093 (1962) |
| 3 | Jerry West | 0.6473 | 0.0915 (1966) | 0.0712 (1969) | 0.0677 (1962) |
| 4 | Kareem Abdul-Jabbar | 0.5507 | 0.0679 (1974) | 0.0553 (1971) | 0.0473 (1970) |
| 5 | Bob Pettit | 0.4853 | 0.0835 (1957) | 0.0713 (1960) | 0.0672 (1963) |
| 6 | Elgin Baylor | 0.4565 | 0.0952 (1961) | 0.0771 (1960) | 0.0563 (1962) |
| 7 | Sam Jones | 0.4519 | 0.0646 (1966) | 0.0634 (1965) | 0.0613 (1964) |
| 8 | Dolph Schayes | 0.4412 | 0.0758 (1955) | 0.0689 (1954) | 0.0521 (1952) |
| 9 | Julius Erving | 0.4381 | 0.0876 (1976) | 0.0568 (1974) | 0.044 (1972) |
| 10 | Cliff Hagan | 0.4125 | 0.1059 (1958) | 0.0651 (1959) | 0.0557 (1960) |
| 11 | Michael Jordan | 0.4111 | 0.0491 (1991) | 0.0487 (1989) | 0.0457 (1996) |
| 12 | Oscar Robertson | 0.4063 | 0.0785 (1964) | 0.0686 (1963) | 0.0365 (1971) |
| 13 | Bob Cousy | 0.3975 | 0.0619 (1957) | 0.0576 (1958) | 0.044 (1956) |
| 14 | John Havlicek | 0.3974 | 0.052 (1968) | 0.0477 (1974) | 0.0472 (1969) |
| 15 | Bill Sharman | 0.3844 | 0.066 (1957) | 0.0605 (1958) | 0.051 (1959) |
| 16 | Magic Johnson | 0.3820 | 0.0417 (1987) | 0.0382 (1988) | 0.037 (1980) |
| 17 | Frank Ramsey | 0.3617 | 0.0746 (1959) | 0.0702 (1958) | 0.064 (1957) |
| 18 | Zelmo Beaty | 0.3331 | 0.0565 (1971) | 0.0376 (1966) | 0.0338 (1964) |
| 19 | Tom Heinsohn | 0.3298 | 0.0595 (1957) | 0.0573 (1960) | 0.0423 (1963) |
| 20 | Paul Arizin | 0.3064 | 0.0982 (1956) | 0.0594 (1960) | 0.0485 (1958) |
| 21 | Dan Issel | 0.3046 | 0.0615 (1973) | 0.0551 (1971) | 0.0434 (1976) |
| 22 | Larry Bird | 0.3001 | 0.0483 (1986) | 0.0459 (1984) | 0.0369 (1987) |
| 23 | Shaquille O'Neal | 0.2960 | 0.0428 (2000) | 0.0346 (2002) | 0.0341 (2001) |
| 24 | Artis Gilmore | 0.2894 | 0.0596 (1973) | 0.0532 (1975) | 0.0466 (1976) |
| 25 | Walt Frazier | 0.2818 | 0.0534 (1972) | 0.0515 (1970) | 0.0445 (1973) |
| 26 | Chet Walker | 0.2774 | 0.0646 (1967) | 0.0377 (1965) | 0.0281 (1975) |
| 27 | George Yardley | 0.2751 | 0.0893 (1956) | 0.0521 (1958) | 0.0502 (1955) |
| 28 | Karl Malone | 0.2725 | 0.0311 (1992) | 0.0298 (1998) | 0.0285 (1994) |
| 29 | Vern Mikkelsen | 0.2685 | 0.0672 (1952) | 0.049 (1953) | 0.0439 (1954) |
| 30 | Larry Foust | 0.2605 | 0.0657 (1956) | 0.0442 (1955) | 0.0416 (1953) |
| 31 | Rick Barry | 0.2583 | 0.0473 (1967) | 0.0441 (1975) | 0.0412 (1972) |
| 32 | George Mikan | 0.2566 | 0.0886 (1954) | 0.0845 (1952) | 0.0769 (1953) |
| 33 | Hakeem Olajuwon | 0.2544 | 0.0407 (1986) | 0.0382 (1994) | 0.0265 (1995) |
| 34 | Hal Greer | 0.2540 | 0.0523 (1967) | 0.0436 (1965) | 0.0358 (1968) |
| 35 | John Stockton | 0.2525 | 0.0289 (1997) | 0.0228 (1996) | 0.0219 (1988) |
| 36 | Tim Duncan | 0.2522 | 0.0434 (2003) | 0.0364 (1999) | 0.0271 (2007) |
| 37 | Roger Brown | 0.2411 | 0.0542 (1970) | 0.0493 (1969) | 0.039 (1972) |
| 38 | Scottie Pippen | 0.2402 | 0.0302 (1992) | 0.0293 (1991) | 0.0291 (1996) |
| 39 | Kevin McHale | 0.2398 | 0.0364 (1986) | 0.0325 (1985) | 0.032 (1988) |
| 40 | Moses Malone | 0.2381 | 0.0421 (1983) | 0.041 (1981) | 0.0221 (1977) |
| 41 | Bailey Howell | 0.2379 | 0.0472 (1965) | 0.0433 (1968) | 0.0323 (1969) |
| 42 | Charles Barkley | 0.2309 | 0.0411 (1993) | 0.0235 (1986) | 0.0194 (1997) |
| 43 | Don Nelson | 0.2211 | 0.0352 (1968) | 0.0322 (1969) | 0.0256 (1966) |
| 44 | Reggie Miller | 0.2201 | 0.0316 (2000) | 0.0261 (1994) | 0.0255 (1995) |
| 45 | Wes Unseld | 0.2173 | 0.0357 (1975) | 0.0354 (1971) | 0.0267 (1979) |
| 46 | Horace Grant | 0.2108 | 0.0319 (1992) | 0.0267 (1991) | 0.0232 (1995) |
| 47 | Kobe Bryant | 0.2098 | 0.034 (2009) | 0.0335 (2001) | 0.0251 (2008) |
| 48 | Robert Parish | 0.2091 | 0.0239 (1987) | 0.0217 (1982) | 0.0215 (1985) |
| 49 | Ed Macauley | 0.2082 | 0.0429 (1957) | 0.0362 (1953) | 0.0343 (1955) |
| 50 | Elvin Hayes | 0.2079 | 0.0369 (1978) | 0.0329 (1975) | 0.0265 (1974) |
| 51 | Bobby Jones | 0.2071 | 0.0356 (1976) | 0.026 (1982) | 0.0252 (1980) |
| 52 | David Robinson | 0.2058 | 0.0306 (1999) | 0.025 (1995) | 0.0197 (1996) |
| 53 | Clyde Lovellette | 0.2054 | 0.0406 (1955) | 0.034 (1954) | 0.0338 (1959) |
| 54 | Jimmy Jones | 0.2045 | 0.0542 (1974) | 0.0415 (1969) | 0.0394 (1968) |
| 55 | Tom Sanders | 0.2026 | 0.0404 (1966) | 0.0327 (1965) | 0.0325 (1962) |
| 56 | Walt Bellamy | 0.1912 | 0.0438 (1965) | 0.0194 (1970) | 0.0184 (1967) |
| 57 | Mel Daniels | 0.1895 | 0.04 (1973) | 0.0331 (1972) | 0.0298 (1971) |
| 58 | George Gervin | 0.1852 | 0.0358 (1976) | 0.0295 (1979) | 0.0197 (1975) |
| 59 | Slater Martin | 0.1817 | 0.0519 (1953) | 0.0329 (1954) | 0.0276 (1952) |
| 60 | Dick Barnett | 0.1803 | 0.0383 (1963) | 0.0285 (1970) | 0.0204 (1961) |
| 61 | Lenny Wilkens | 0.1797 | 0.0371 (1967) | 0.0276 (1964) | 0.019 (1966) |
| 62 | Bob Dandridge | 0.1795 | 0.0297 (1979) | 0.0285 (1972) | 0.0284 (1974) |
| 63 | Maurice Cheeks | 0.1792 | 0.0255 (1982) | 0.0237 (1983) | 0.0208 (1980) |
| 64 | Willis Reed | 0.1779 | 0.0475 (1970) | 0.0427 (1969) | 0.0275 (1967) |
| 65 | Clyde Drexler | 0.1771 | 0.0324 (1992) | 0.0283 (1995) | 0.0228 (1991) |
| 66 | Neil Johnston | 0.1751 | 0.0719 (1956) | 0.0362 (1958) | 0.0314 (1954) |
| 67 | James Worthy | 0.1703 | 0.0297 (1987) | 0.029 (1985) | 0.0265 (1988) |
| 68 | Chauncey Billups | 0.1692 | 0.0337 (2005) | 0.0293 (2004) | 0.024 (2009) |
| 69 | George McGinnis | 0.1674 | 0.0473 (1975) | 0.0415 (1973) | 0.0337 (1974) |
| 70 | Dennis Johnson | 0.1673 | 0.0256 (1979) | 0.0213 (1987) | 0.0188 (1978) |
| 71 | Larry Costello | 0.1663 | 0.0393 (1959) | 0.0349 (1961) | 0.0232 (1962) |
| 72 | Louie Dampier | 0.1661 | 0.036 (1975) | 0.0275 (1976) | 0.0226 (1973) |
| 73 | Dirk Nowitzki | 0.1647 | 0.039 (2006) | 0.0223 (2003) | 0.0188 (2001) |
| 74 | Bob Lanier | 0.1633 | 0.0278 (1974) | 0.0238 (1976) | 0.0158 (1980) |
| 75 | Nate Thurmond | 0.1621 | 0.0493 (1967) | 0.032 (1964) | 0.0131 (1973) |
| 76 | Dave Cowens | 0.1617 | 0.0365 (1976) | 0.0309 (1974) | 0.0244 (1973) |
| 77 | Robert Horry | 0.1611 | 0.0217 (1995) | 0.021 (1994) | 0.02 (2002) |
| 78 | Patrick Ewing | 0.1607 | 0.0296 (1994) | 0.0191 (1993) | 0.0184 (1992) |
| 79 | Freddie Lewis | 0.1604 | 0.0336 (1969) | 0.0255 (1972) | 0.0242 (1975) |
| 80 | Jeff Hornacek | 0.1594 | 0.0292 (1996) | 0.0199 (1997) | 0.019 (1990) |
| 81 | Rudy LaRusso | 0.1578 | 0.0301 (1966) | 0.0237 (1963) | 0.0209 (1960) |
| 82 | Jerry Lucas | 0.1526 | 0.0396 (1972) | 0.0211 (1965) | 0.021 (1966) |
| 83 | Paul Silas | 0.1524 | 0.0303 (1976) | 0.0239 (1974) | 0.0151 (1973) |
| 84 | K.C. Jones | 0.1512 | 0.0393 (1965) | 0.0317 (1962) | 0.0308 (1966) |
| 85 | Harry Gallatin | 0.1507 | 0.05 (1953) | 0.0355 (1952) | 0.0259 (1954) |
| 86 | Tom Gola | 0.1478 | 0.0462 (1956) | 0.04 (1960) | 0.031 (1958) |
| 87 | Red Kerr | 0.1457 | 0.0356 (1957) | 0.0288 (1955) | 0.0207 (1956) |
| 88 | Gail Goodrich | 0.1448 | 0.0346 (1972) | 0.0234 (1973) | 0.0215 (1971) |
| 89 | Isiah Thomas | 0.1445 | 0.0286 (1988) | 0.0277 (1990) | 0.0231 (1987) |
| 90 | Billy Paultz | 0.1444 | 0.0361 (1974) | 0.0248 (1976) | 0.0233 (1972) |
| 91 | Willie Wise | 0.1430 | 0.0398 (1971) | 0.0341 (1974) | 0.0291 (1973) |
| 92 | Dick McGuire | 0.1422 | 0.0425 (1953) | 0.0403 (1952) | 0.0226 (1959) |
| 93 | Adrian Dantley | 0.1407 | 0.0292 (1988) | 0.0219 (1984) | 0.021 (1987) |
| 94 | Bill Bridges | 0.1407 | 0.0319 (1966) | 0.0243 (1969) | 0.0174 (1973) |
| 95 | Richie Guerin | 0.1404 | 0.0359 (1964) | 0.0326 (1966) | 0.0147 (1967) |
| 96 | Gary Payton | 0.1402 | 0.0292 (1996) | 0.0167 (1997) | 0.0165 (1998) |
| 97 | Gus Williams | 0.1393 | 0.0321 (1979) | 0.029 (1980) | 0.0281 (1978) |
| 98 | Terry Porter | 0.1380 | 0.0336 (1992) | 0.0286 (1990) | 0.0239 (1991) |
| 99 | Rasheed Wallace | 0.1379 | 0.0201 (2000) | 0.0176 (2004) | 0.0175 (2005) |
| 100 | Bobby Wanzer | 0.1375 | 0.0497 (1952) | 0.0292 (1955) | 0.0287 (1954) |
December 2nd, 2009 at 3:32 pm
Great stuff, Neil, and this is actually an idea I had had before so I'm pleased to see someone trying to put it into action.
I'm a bit confused as to why Michael Jordan ranks (relatively) so low - and really, all the modern players, for that matter. I get that the postseason counts more for the older players, but since Jordan was so dominant in both the regular and postseason, shouldn't he rank closer to Russell/Wilt regardless of how you weight it?
December 2nd, 2009 at 3:41 pm
I think it's because Jordan's (and any modern player's) Win Shares represent a smaller slice of the entire NBA's win total than somebody who played in the 1950s/60s. When Wilt Chamberlain had 25 WS in 1964, the entire NBA had just 360 wins -- in effect, Wilt represented a full 6.9% of the entire league's wins! Meanwhile, when Jordan had 21 WS in 1988, the league had 943 wins, so MJ "won" just 2.2% of the NBA's games at his peak.
December 2nd, 2009 at 4:48 pm
Neil,
I think the most fair way to reconcile the issue of different eras and league sizes, without making any judgments about the relative strength of those eras, is to simple award an amount of championship credit in each league proportional to the number of teams, i.e. award 1 championship now, but award only 33% (10/30) of a championship in seasons where there were only 10 teams. This is not the most elegant way to do it, but it will account for the changing number of teams.
December 2nd, 2009 at 6:00 pm
That's a pretty solid idea, Dave, I'll give it a shot in Part II tomorrow.
December 2nd, 2009 at 6:01 pm
@ David Lewin:
I think it would be best to have both numbers available, so we can look at it either way.
December 2nd, 2009 at 6:29 pm
This is interesting. I'm sort of ignoring the pre-merger numbers because they don't fit with my understanding of the NBA today - I just didn't witness that game - but what I'm seeing is otherwise pretty solid. Michael #1, Magic #2, Bird #3 (higher peak years than Magic but not as good overall career), Shaquille #4... of course those efficient but non-explosive producers on great teams like Grant and Horry wind up ranked really high which is pretty typical of this sort of thing.
Looking at peak years works a little better actually. Jordan - Bird - Duncan - Shaq - Moses - Magic - Charles - Hakeem - Dirk - Gumby - Kobe ... that pretty much fits with my memory of playoffs past (though of course the point of a metric is to NOT need to rely on memories of playoffs past). I'd like to see peak year averages - say top 5 or 6 - just to see if that gives a better range of players and weeds out some of the consistent but not exceptional players who are the list now.
I look forward to round 2.
December 2nd, 2009 at 9:27 pm
I have an incredibly hard time believing that Hakeem didn't peak far higher, particularly given those who surrounded him at the time of the first ring in '94.
I understand metrics remove preference and bias by weighing undeniable facts... But is there anybody here, amongst us, who doesn't see something wrong with this? Of course, almost every metric turns the odd superstar into an ugly duckling.
December 2nd, 2009 at 9:29 pm
You have more of a problem with Hakeem's placement than Cliff Hagan over MJ and Dan Issel over Bird & Shaq?
Like I said, he results are very rough -- I'm just trying to get some feedback about where the process can be improved at this point.
December 2nd, 2009 at 11:12 pm
No.
I specifically didn't mention that as you've already addressed the issues of historical context, as have others. I was merely pointing out what the metric was propagating in regards to the modern era - and more specifically - an era that I witnessed.
December 3rd, 2009 at 8:18 am
Seems to me at the beginning of the season, each team has a 16/30 chance of getting to the playoffs, not 1/30 (in a 30-team league and 16-team playoffs). This should be adjusted to league size and playoff size changes over the years.
December 3rd, 2009 at 8:45 am
Actually, you're right. So when you make the playoffs, it moves your playoff prob from 16/30 to 1. I'll update the numbers to fix that.
December 3rd, 2009 at 9:15 am
On second thought, I don't think that mistake impacts the championship probability calculation, though -- because after you make the leap from playoff prob. to championship prob., the 1/30 number is actually correct as the initial expectation for the avg. team.
December 3rd, 2009 at 10:48 am
Two sides of the same coin, there... the numbers cancel out and you end up with the same thing.
December 3rd, 2009 at 11:28 am
Is there any way to account for the relative strength of each conference? I understand the basic premise of having to count the 19th win the same as the 39th, but is there some way to take into account the relative increase in difficulty of making the West playoffs this last decade versus the east?
December 3rd, 2009 at 12:32 pm
You could, but then you'd have to go through and work it out on a team-by-team basis. It's something that's probably not worth doing at this stage, but if I develop this method more, it could be a valuable addition.