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Pythagorean Wins
One of Bill James's contributions to
sabermetrics
is
Pythagrean Expectation, which attempts to relate a team's runs scored (RS)
and runs allowed (RA) to its wins (W) and losses (L). James conjectured
Though it is widely accepted, this formulation has two short-comings:
- Estimated wins and losses for all teams based on this equation are
seldom equal. Clearly, since every game has a winner and a loser,
total wins must equal total losses, league-wide. So-called
Pythagorean standings can show, paradoxically, that the entire league is
better (or worse) than itself.
- Using only the ratio of runs scored to runs allowed is problematic.
Consider two situations where two teams (A and B) play two games against
each other:
- Team A scores 2 runs, and Team B scores 1.
- Team A scores 20 runs, and Team B scores 10.
Since ties are prohibited, case (a.) can only occur if A wins a game
by a 2-0 final and loses the other game 0-1. Team A's record must
be 1-1. If we assume runs are equally likely to be scored in
either game, Team A wins at least one game but will win both games about
95% of the time, resulting in 1.95 expected wins. The Pythagorean
expectation predicts A to win 1.6 games in both cases.
Conditional Inference
Both problems of Pythagorean Expectation can be solved by considering wins
and losses conditional on run totals. While the mathematics can be
difficult for real data sets, the premise is straightforward. We'll
illustrate the idea using a round-robin tournament between three teams (A, B and
C) with the following results:
A 3
A 4
B 3
B 2
C 1
C 2
which can be represented in a table.
| |
A |
B |
C |
Runs Scored |
| A (2-0) |
-- |
3 |
4 |
7 |
| B (1-1) |
2 |
-- |
3 |
5 |
| C (0-2) |
1 |
2 |
-- |
3 |
| Runs Allowed |
3 |
5 |
7 |
|
One way to predict win totals based on runs is to consider the permutation
distribution1 of cell entries based on holding row and column sums
constant. For this example, the only other possible tables with the same
run totals are:
| |
A |
B |
C |
Runs Scored |
| A (1-1) |
-- |
2 |
5 |
7 |
| B (1-1) |
3 |
-- |
2 |
5 |
| C (1-1) |
0 |
3 |
-- |
3 |
| Runs Allowed |
3 |
5 |
7 |
|
| |
A |
B |
C |
Runs Scored |
| A (2-0) |
-- |
4 |
3 |
7 |
| B (1-1) |
1 |
-- |
4 |
5 |
| C (0-2) |
2 |
1 |
-- |
3 |
| Runs Allowed |
3 |
5 |
7 |
|
| |
A |
B |
C |
Runs Scored |
| A (1-1) |
-- |
5 |
2 |
7 |
| B (1-1) |
0 |
-- |
5 |
5 |
| C (1-1) |
3 |
0 |
-- |
3 |
| Runs Allowed |
3 |
5 |
7 |
|
However, if we assume that runs are equally likely to be scored in any game,
some of these tables are more likely than others, meaning win-loss records based
on them should be weighted accordingly. For this example, the four run
tables should occur with probabilities2 0.581, 0.116, 0.291 and
0.012, respectively.
This suggests that conditional on run totals, Team A should win 1.872 games3
and lose 0.128 Similarly, the expected win-loss records of B and C are 1-1
and 0.128-1.872, respectively. Note that unlike the Pythagorean
Expectation, these expected records will always satisfy the property that total
wins and total losses each equal total games played.
1. A permutation distribution is the basis for
so-called "exact tests" which require enumeration of all possible data sets
given certain constraints (in this case runs scored and allowed by each team).
2. Table probabilities follow a non-central hypergeometric distribution, and
in general can not be calculated in closed form. However, Markov Chain
Monte Carlo (MCMC) techniques are available for simulating tables with the
correct frequencies.
3. The expectation is calculated by taking a weighted average of wins (which
are directly computed from the table) with weights corresponding to the
probability of the particular arrangement of runs. Here 1.872 = (0.581)(2)
+ (0.116)(1) + (0.291)(2) + (0.012)(1).
Advantages of the Conditional Approach
In addition to preserving league-wide wins and losses, our conditional
procedure provides a complete distribution of wins and losses for each team.
Conversely, the Pythagorean Expectation is merely a single number. The
advantage of having the complete joint distribution of wins is that it allows us
to compute many potentially interesting quantities:
- Win Percentile. Given these run totals, how often would a
particular team win at least as many games as it did?
- Playoff/Division Races. Given these run totals, how often
could a particular team expect to win its division or win the wild-card?
The potential research questions are limitless.
Results
We applied the SportsQuant
conditional win procedure to seven years (1998-2004) of Major League Baseball
data. The following table gives win totals with conditional expectations
in parentheses.
| |
1998 |
1999 |
2000 |
2001 |
2002 |
2003 |
2004 |
| Angels |
85
(81.7) |
70
(66.5) |
82
(80.4) |
75
(75.8) |
99
(106.7) |
77
(79.8) |
92
(94.0) |
| Astros |
102
(112.6) |
97
(99.7) |
72
(80.3) |
93
(90.6) |
84
(87.9) |
87
(97.3) |
92
(94.3) |
| Athletics |
74
(73.7) |
87
(86.8) |
91
(96.4) |
102
(110.4) |
103
(99.5) |
96
(97.2) |
91
(87.6) |
| Blue Jays |
89
(87.7) |
84
(83.7) |
83
(75.5) |
80
(82.8) |
78
(79.1) |
86
(89.0) |
67
(67.5) |
| Braves |
106
(112.4) |
103
(103.6) |
95
(93.3) |
88
(92.7) |
101
(100.2) |
101
(101.1) |
96
(98.4) |
| Brewers |
74
(67.7) |
74
(71.9) |
74
(70.7) |
68
(72.6) |
56
(55.9) |
68
(61.2) |
67
(64.2) |
| Cardinals |
84
(84.9) |
75
(76.8) |
95
(95.2) |
93
(97.7) |
97
(99.0) |
85
(90.6) |
105
(105.6) |
| Cubs |
90
(86.2) |
67
(60.1) |
65
(64.0) |
88
(90.8) |
67
(74.0) |
89
(87.1) |
89
(97.0) |
| Devil Rays |
63
(63.6) |
69
(64.1) |
69
(66.9) |
62
(54.6) |
55
(51.2) |
63
(63.9) |
70
(64.5) |
| Diamondbacks |
65
(62.1) |
100
(109.2) |
85
(85.9) |
92
(99.1) |
98
(99.2) |
84
(85.1) |
51
(45.6) |
| Dodgers |
83
(79.7) |
77
(81.6) |
86
(89.9) |
86
(82.7) |
92
(90.2) |
85
(83.5) |
93
(90.9) |
| Expos |
65
(62.9) |
68
(64.0) |
67
(61.0) |
68
(62.7) |
83
(93.1) |
83
(80.3) |
67
(63.3) |
| Giants |
89
(94.6) |
86
(85.8) |
97
(102.5) |
90
(87.5) |
96
(102.7) |
100
(95.8) |
91
(90.6) |
| Indians |
89
(89.9) |
97
(98.1) |
90
(96.9) |
91
(90.1) |
74
(69.0) |
68
(70.7) |
80
(81.1) |
| Mariners |
76
(81.2) |
79
(75.7) |
91
(96.5) |
116
(117.1) |
93
(95.2) |
93
(101.4) |
63
(65.1) |
| Marlins |
54
(50.3) |
64
(60.6) |
79
(72.1) |
76
(80.7) |
79
(72.4) |
91
(88.7) |
83
(83.4) |
| Mets |
88
(89.1) |
97
(99.0) |
94
(89.8) |
82
(71.3) |
75
(78.6) |
66
(65.6) |
71
(74.8) |
| Orioles |
79
(85.1) |
78
(85.5) |
74
(66.8) |
64
(63.2) |
67
(67.2) |
71
(71.7) |
78
(82.6) |
| Padres |
98
(96.1) |
74
(71.7) |
76
(73.1) |
79
(78.1) |
66
(61.2) |
64
(61.4) |
87
(89.1) |
| Phillies |
75
(79.7) |
77
(80.2) |
65
(65.5) |
86
(84.5) |
80
(78.5) |
86
(93.1) |
86
(88.3) |
| Pirates |
69
(72.3) |
78
(79.5) |
69
(69.5) |
62
(55.5) |
72
(68.5) |
75
(75.1) |
72
(72.0) |
| Rangers |
88
(89.2) |
95
(91.2) |
71
(66.4) |
73
(72.3) |
72
(76.2) |
71
(64.3) |
89
(89.2) |
| Red Sox |
92
(99.2) |
94
(96.0) |
85
(87.1) |
82
(84.0) |
93
(104.8) |
95
(98.6) |
98
(102.7) |
| Reds |
77
(79.7) |
96
(100.5) |
82
(89.2) |
66
(66.7) |
78
(72.5) |
69
(57.2) |
76
(62.1) |
| Rockies |
77
(77.5) |
72
(67.3) |
82
(89.2) |
73
(82.9) |
73
(66.4) |
74
(76.2) |
68
(70.4) |
| Royals |
72
(58.2) |
64
(73.0) |
77
(75.1) |
65
(65.1) |
62
(62.6) |
83
(77.1) |
58
(58.3) |
| Tigers |
65
(63.7) |
69
(64.0) |
79
(80.5) |
66
(62.3) |
55
(44.3) |
43
(40.0) |
72
(78.9) |
| Twins |
70
(70.6) |
64
(60.5) |
69
(68.7) |
85
(81.7) |
94
(87.6) |
90
(86.2) |
92
(89.5) |
| White Sox |
80
(73.5) |
75
(69.7) |
95
(93.0) |
83
(81.5) |
81
(87.9) |
86
(90.5) |
83
(85.1) |
| Yankees |
114
(117.7) |
98
(101.7) |
87
(87.5) |
95
(92.1) |
103
(104.4) |
102
(101.4) |
101
(91.9) |
Results
Over the seven seasons studied, we ranked each team according to wins in
excess of those expected. Teams with actual win totals greater than their
conditional expectations overachieved while teams whose win totals lagged their
conditional expectations performed poorly. A few interesting observations
are worth making.
- The 2001 Seattle Mariners set the record for most wins in a season
(116), however our results indicate that they were even better than
their gaudy record suggests.
- Although the Expos, Devil Rays, Brewers and Tigers were perpetual
laughing-stocks, they ranked 1st, 3rd, 4th and 6th, respectively in
terms of win differential. This underscores the lack of talent on
those teams. Their dismal performance still exceeded expectations.
- The Red Sox were 30th (dead last) of the teams over the course of
this study. Without exception, the team won fewer games than would
be expected for a team with their offensive and defensive abilities.
Their regular-season performance in 2004 (the season in which they
eventually won the World Series) was typical -- namely 5 wins fewer than
expected.
- There is growing sentiment that Joe Torre is a Hall-of-Fame caliber
manager. However, the Yankees rank 14th (of 30 teams) in win
differential, indicating that they win about as many games as they
should given their perennially talent-laden roster.
Reference
Annis, D. H. (2005) Approximate
Conditional Inference for Evaluating Managerial Performance. (unpublished
manuscript)
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