Pythagenpat Using Run Rates

The widespread implementation of seven-inning games during the 2020 season forced a re-examination of some of the standard sabermetric tools. One example is Pythagorean records. It would be foolish to expect that the same run ratio achieved over nine innings would lead to the same expected winning percentage as if it had been achieved over seven innings. Thus, simply taking a team’s composite runs scored and allowed for the season, which consisted of some unique to that team distribution of seven-inning and nine-inning games, and expecting the standard Pythagorean approach to be the best estimate of their overall winning percentage was also foolish.

The approach that one should select to deal with this issue depends on what the desired final estimate is. If one wanted to estimate how many games a team should have won over the course of that season, one reasonable approach would be to develop a proper Pythagorean or Pythagenpat exponent for seven-inning games, and then calculate a team’s estimated winning percentage in seven-inning games using that value, in nine-inning games using the standard approach, and then weighting the results by the percentage of seven-inning and nine-inning games for the team (defining this in terms of the scheduled length of the game and not the actual number of innings that was played in the case of extra-inning seven-inning games).

Tom Tango studied games that were tied entering the third inning to simulate a seven-inning game, and found a Pythagorean exponent of 1.57 was appropriate. Of course that’s fixed rather than Pythagenpat exponent, but you could use the same approach to develop an equivalent Pythagenpat formula, and then apply as described above. 

I decided that I more interested in attempting to estimate what the team’s W% would have been under standard conditions (i.e. nine-inning games as the default, as we normally view a major league season). Thus I was interested in what a team’s W% “should have been” had they played in a normal season. This allowed me to skip the step of dealing with seven-inning games, and instead think about the best way to fit their 2020 data into the standard formulas. Of course, the silly runs scored in extra inning games are a problem, but I chose to ignore them for the sake of expediency (and in hopes that this all would be a temporary problem) and use the team’s runs (and allowed) per nine innings to plug into Pythagenpat.

In thinking about this, I was reminded of a related issue that I have been aware of for a long time, which is the reduced accuracy of Pythagorean estimates (and really all R/RA estimates of W%) as pertains to home and away games. If you look at 2010-2019 major league data and use Pythagenpat with x = RPG^.29, the RMSE of estimate team W% multiplied by 162 is 3.977 (for the sake of convenience I’ll just call this RMSE going forward, but this can be thought of as the standard error per 162 games). If you just look at away games, the RMSE is 6.537, and for home games it is 6.992. 

It should not surprise us that the error is larger, as we have just halved the number of games for each observation, and we should generally expect larger deviations from expectation over small samples. However, it’s not just the magnitude of the error that matters. Over this span, home teams averaged a .535 W% and road teams (of course) the complement of .465. But the Pythagenpat record of home teams was .514, and for road teams .486. One’s first inclination upon seeing this might be to say “Aha! Evidence of home field advantage manifesting itself. Home teams exceed their Pythagenpat record by .021 wins due to [insert explanation...strategic advantage of batting in the bottom of the ninth, crowd support allowing them to will runs when needed, etc.]”

One sabermetrician who encountered this phenomenon and developed a more likely (and indeed, obvious upon reflection) explanation for it was Thomas Tress. His article “Bias Against the Home Team in the Pythagorean Theorem” was published in the May 2004 By The Numbers. Tress provided the obvious explanation that home teams often don’t bat in the bottom of the ninth, which means that they often have fewer opportunities to score runs than they do to allow runs. Tress offers a correction with a scalar multiplier that can be applied to a home team’s runs (and of course also to the road team’s runs allowed) as a corrector.

Tress’ approach is a solid one, but it addresses only the home/road Pythagorean conundrum that we entered on a detour, rather than my primary concern about length of game (this is not a criticism as it was not intended to). The issues are related because the home team not batting in the bottom of the ninth is one way in which game lengths vary from the standard nine innings that are inherently assumed in most metrics (or, more precisely, they assume the average number of innings in the data which was used to calibrate them, which we’ll get to in due course).

I should point out that there is another issue that pertains to home teams that also distorts Pythagorean records, which is truncated bottom of the ninths (or tenths, elevenths, etc.). Foregone bottom of the ninths are more obviously troublesome, but truncated bottom of the ninths (in which a walkoff victory is achieved before three outs are recorded) which leave home teams’ runs totals lower than they would otherwise be, as run expectancy is left on the table when the game ends. I will not be correcting for that here; it is a lesser problem than foregone bottom of the ninths for the sake of Pythagorean records, and there’s no easy fix (one could add to a home team’s runs scored and an away team’s runs allowed the run expectancy that existed at the end of the game, but this is not a correction that can quickly be made with a conventional dataset). You can avoid this problem by using runs created rather than actual runs, as the potential runs are still reflected in the calculation, but that changes the whole nature of the Pythagorean record by opening up a second dimension of luck (“sequencing” of offensive events rather than simply “timing” of runs).

Ignoring the truncated innings issue, there is an obvious approach that should help address both the home field issue and the question of shortened games, which is using a rate of runs scored and allowed that considers outs/innings rather than raw totals or rates (most commonly runs/game) that don’t take into account outs/innings. Since Pythagenpat is built around runs per game determining the exponent, I will take the approach of using runs/9 innings.

Before jumping into the Pythagenpat implications, two points on this definition:

1. It’s easy to know a team’s defensive innings, as it’s just their innings pitched. For offenses, you can use Plate Appearances – Runs – Left on Base (at least for non-Manfred innings), although it’s easier if you can just get opponents’ innings pitched, or opponents’ putouts, since PO/3 = IP by definition.

 2. I am using 9 innings because it is the regulation game length, but it actually corresponds to a slightly longer game than what we actually saw in 2010-2019. For those seasons, the average outs/game was 26.82, which is equivalent to 8.94 innings/game.

I’m using 2010-2019 data for this post not because I think ten years (300 team seasons) is an appropriate sample when conditions of the game have not changed in the last century to an extent that should significantly influence Pythagorean records. The more mundane explanation is that data on actual team outs, home and away, is not easily accessible, and the easiest way I know how to get is through Retrosheet’s Game Logs which are an absolutely fantastic resource. But I didn’t want to spend a significant amount of time parsing them, which is why I limited my sample to ten years. 

My first step was to optimize standard Pythagenpat to work with this dataset, so that any RMSE comparisons we make after building a rate-based Pythagenpat formula are on a level playing field. However, I was quite surprised by what I found - the Pythagenpat exponent that minimizes RMSE for the 2010-2019 majors is .264 (in other words, the Pythagorean exponent x = RPG^.264).

Typically, a value in the range .28 - .29 minimizes RMSE. I was so surprised by .264 that I thought for a moment I might have made an error compiling the data from the game logs, so I checked the Lahman database at the season level to be sure. The data was accurate – this set of 300 teams happen to actually have a lower Pythagenpat exponent than I am conditioned to seeing. 

For the purpose of a proof of concept of using rates, this is not really an issue; however, I certainly question whether the best fit values I’ve found for the rate approach should be broadly applied across all league-seasons. I will leave it up to anyone who ultimately decides to implement these concepts to decide whether a larger sample is required to calibrate the exponents.

With that being said, the differences in RMSE using the lower Pythagenpat exponent are not earth-shattering. Using .264, the RMSE for all games is 3.923, with 7.015 for home games and 6.543 for away games, with the home/road RMSEs actually higher than those for the standard exponent. I provide these values for reference only as the real point of this exercise is to look at what happens for a rate-based Pythagenpat.

First, let’s define some basic terms:

R/9 = Runs/Actual Outs * 27

RA/9 = Runs Allowed/Innings Pitched * 9

RPG9 = R/9 + RA/9

x = RPG9^z (z will be our Pythagenpat exponent and x the resulting Pythagorean exponent for a given RPG9)

W% = (R/9)^x/((R/9)^x + (RA/9)^x)

The value of z that minimized RMSE for this dataset is .244. That RMSE is 3.771, which is a significant improvement over the optimized Pythagenpat that does not use rates. This is encouraging, as if there was no advantage to be had this whole exercise would be a waste of time. I also think it’s intuitive that considering rates rather than just raw run totals would allow us to improve our winning percentage estimate. After all, the only differences between raw runs and rates for a team season will arise due to how the team performs in individual games. 

To with, we can define opportunities to score runs in terms of outs, since outs are the correct denominator for a team-level evaluation of runs scored/allowed on a rate basis. A perfectly average team would expect to have an equal number of opportunities for their offense and defense, but a good team will allow its opponents’ offense more opportunities (since they will forego more bottom of the ninths at home and play more bottom of the ninths on the road), and a bad team will get more opportunities for its own offense. These differences don’t arise randomly, but due to team performance.  So we should expect a slight improvement in accuracy of our winning percentage estimate when we allow these corrections, but it should be slight since foregone bottom of the ninths have a ceiling in practice and a lower ceiling in reality (even very bad teams often forego bottom of the ninths and even very good teams frequently lose at home or at least need a walkoff to win).

Better yet, the reductions in RMSE for home games (5.779) and road (5.215) are larger, which we might have expected as the impact of foregone bottom of the ninths will not be as smooth across teams when considering home and road separately. When using this rate approach, the expected W% for all home teams in the dataset is .536, compared to the actual home W% of .535. So there is no evidence of any home field advantage in converting runs/runs allowed to wins that does not get wiped away by taking opportunities to score/allow runs into account, contrary to what one might conclude from a naïve Pythagenpat analysis.

A further note is that if you calculate a team’s total expected wins as a weighted average of their home and road rate Pythagenpats, the RMSE is a little better (3.754) than just looking at the combined rate. This also should not surprise, as we have sneaked in more data about how a team actually distributed its runs scored and allowed across games by slicing the data into two pieces instead of one. If we calculated a Pythagenpat record for every game and then aggregated, we should expect to maximize accuracy, but at that point we are kind of losing the point of a Pythagorean approach (we can make the RMSE zero if in that case we replace Pythagenpat with a rule that if R > RA, we should assign a value of 1 expected win and if R < RA we should assign a value of 0 expected wins).

Again, I would consider this a demonstration of concept rather than a suggestion that this be implemented with a rate Pythagenpat exponent of .244. My hunch is that the best value to use over a broad range of team-seasons is higher than .244. Also, I think that for just looking at total season records, a standard approach is sufficient. If you ever are working with a situation in which you can expect to see significant discrepancies between the number of foregone bottom of the ninths for a team and its opponents (as is definitely the case when considering home and away games separately, and may be the case to a much lesser extent for extremely good or extremely bad teams), then you may want to consider calculating Pythagenpat using run rates rather than raw totals.

More Mundane Comments on the Playoff Structure

In the previous post I briefly mentioned my dislike of the five-game series format currently used in the Division Series and formerly used in the LCS. But what is the real difference between a five and seven game series? If we make some simple assumptions about team quality, how often will the better team win a series of X length? Common sense tells us that the longer the series, the more likely the better team will win, but let's attempt to quantify that. (Actually, it's not attempting, since given the assumptions that I will make, the answers are simple probability, and it's also tough to classify it as an "attempt" since many people have done it before).

First, let's start with the assumptions:

* each game result is independent of the other games in the series (this assumption is likely weaker for the post-season than for the regular season, as the series status has a great influence on how the manager approaches the game, particularly with regards to pitcher usage).

* there is no home field advantage

* the probability of a win for the teams is the same from game-to-game--we are not making any allowances for the aforementioned home field advantage, the identity of the starting pitcher, etc.

With these assumptions in place, we can use the binomial and geometric distributions and the principles behind them to crudely model series of X length. Throughout the rest of the piece, I will refer to "better" or "correct" outcomes. Please understand that I am using these terms in conjunction with the stated assumptions--we know the precise probability of each team winning, and therefore we absolutely know which team is better and ideally will win the series. Obviously, in real life situations we do not know with certainty which team is better. Which is the point--if a playoff format does a poor job of rewarding the better team when we are certain about its identity, it will be even less efficient at that task when we don't know which team is better.

First, let's look at the probability of a team winning the series, given that it is of X length. This can be done with the binomial distribution. For example, for a seven-game series, we simply add up the probability that a given team will win all seven games (even though they will not all be played), six out of seven, five out of seven, and four out of seven. This is the probability that they will win the series.

I will present the probabilities for each interval of .01 in W% between .51 and .65. I have limited the range because realistically in playoff series we will rarely see matchups in which one team is a heavy favorite over the other. The most unbalanced realistic playoff matchup would pit a .700 team against a .500 team, with an expected W% of .700. And that is assuming that the team's sample W%s are their true talent W%s, which would be unlikely for a .700 team. Again, these W%s are the expectations for a single game between the two teams.

I figured the probabilities for series ranging in length from one to fifteen games. I went up to fifteen games because fifteen games was the actual length of the World's Series in 1887, even if the series was not treated with the full championship reverence of today's World Series:



I bolded the 53% line because I'm going to use it as the "average" playoff series--I realize this table is tough to read with fifteen different scenarios. The explanation for why I chose that particular W% is explained below--it's not profound by any stretch (*).

One takeaway from this chart is how silly it is when folks talk about locks to win a playoff series. Even in a situation in which one team has a 65% chance to win each game (which is a big mismatch in the playoffs--a .500 team against a 105 win team or a 90 win team against a 112 win team using Log5), that team only has an 80% shot at winning a seven-game series. Even if you more than double the series length to fifteen games, there's still an 11.3% chance of an upset.

When sabermetrically-inclined people say that the playoffs are a crapshoot, this is the kind of thing they're generally talking about. It's not that you have no way of knowing which team is better or estimating the degree to which they are, it's just that even in a case where you have clear superiority, the short length of the series makes an upset quite feasible.

It was quite amusing during the Roy Halladay sweepstakes to hear commentators talk about how the Phillies were a lock to win the pennant if they got Halladay. Just like it was amusing to read about how the Cubs were going to march right through the weak NL to the pennant last year, or how the Tigers were going to trounce the Cardinals in the World Series. I wish I knew one-twentieth as much about baseball as those folks think they know.

Let's express that table in a more useful form by showing the marginal probabilities for each extension of series length. For example, the team that wins 51% of their games will win a one-game playoff 51% of the time. Expanding to a three game playoff will lead to them winning 51.5% of the time, an increase of .5%. If we expand to a five game playoff, they will win 51.9% of the time, an additional increase of .4%. This will enable us to see the benefit to lengthening series in terms of ensuring the better team wins:




As you can see, the added benefit starts diminishing quickly and for the normal range, essentially levels out after you make the move to seven games. Of course, these are the marginal outcomes, so longer series are still "better"...but less so with every additional pair of games.

Since the marginal benefit levels off after lengthening to seven games (for the nearly even matchups at least--the more lopsided matchups continue to show significant increases), it seems like as good of a point as any at which to stop.

Of course, I have approached this solely from the perspective of encouraging correct outcomes. This is not the goal of a league--if it was, there would be no need for any kind of playoffs at all. The league is going to act in a way so as to maximize its profits. Which is well and good, but I am examining this from the personal perspective of what I'd like to see and/or what will produce the best outcomes.

There is one thing that overlaps between my perspective and the economic interests of the owners, and that is the desire for a competitive series. Close series encourage higher ratings, and longer series means more ticket revenue. For a fan, there's nothing more exciting that a decisive game for the world championship after a hard fought series. While I have a strong preference for better outcomes, I can't completely suppress the desire for a winner-take-all finale.

So, given the underlying assumptions of this post, let's look at the probabilities of a decisive game, given a series of X length. This is done with the geometric distribution, and I have included the formula (**) because I think many fewer people are familiar with it than the binomial distribution--just speaking for myself, I know the binomial function by heart but have to look up the geometric function just to be safe:



A five-game series with a fairly normal matchup will produce a game five about 37% of the time; a seven-game series about 30% of the time. So for a roughly 1% increase in the likelihood of the better team winning, you give up decisive games in 7% of your series.

The next step, moving from a seven-game series to a nine-game series, would result in roughly the same increase in the likelihood of the better team winning while sacrificing another 4% of series without a grand finale.

All told, it shouldn't be too surprising that the probabilities here can be read to suggest that MLB has correctly identified the series lengths that provide the best combination of practicality, uncertainty of outcome, and producing desired outcomes. The extra benefit in terms of desired outcomes from expanding to longer series is relatively small, and is offset by a larger percentage drop in the expected proportion of series with decisive games.

Finally, let's take a look at the potential value of home field advantage in a five-game series. I previously looked at World Series HFA (i.e. seven-game series), and the same principles will apply here. I have not looked at the empirical data in this case and will only be discussing theoretical results.

First, we can use the geometric distribution to calculate the percentage of series that are expected to go X games, assuming that each game is a 50/50 proposition (in other words, not considering HFA):



Just as is the case for a seven-game series, the probability of a full-length series and one short of it are equal. This makes logical sense, of course; in order to create this situation the first three games must have produced a 2-1 series. There is a 50% chance that the team that has already won two wins, ending the series and a 50% chance that the team behind forces a decisive game.

Unlike a seven-game series, it is impossible for the team with on-paper home field advantage to play more road games than home games, as the format is 2-2-1 (Obviously, I'm talking about the current format; I'm aware that it was sometimes different in the past). Theoretically, on-paper home field advantage results in a true home field advantage 62.5% time, and the other 37.5% of the time there is no HFA for either team.

In order to add HFA into the mix, we need to identify all the possible series sequences, which I will not reproduce here. Suffice it to say that from the perspective of the winning team, there is one series sequence that produces a three-game series (WWW), three that produce a four-game series (WWLW, WLWW, and LWWW), and six that produce a five-game series.

I will assume a home field W% of .573, which is the empirical World Series statistic. I believe that the "true" parameter is likely lower, for reasons discussed in the earlier post, but I'll use the sample statistic for the sake of discussion. Retaining the assumptions of evenly matched teams and independent game outcomes, the probability of the team with on-paper HFA winning a five-game series is 52.66%, compared to a 52.31% chance in a seven-game series. So HFA is theoretically more important in a shorter series (no surprise, but we've estimated the degree).

It should also be noted that we would expect the empirical home field advantage in a five-game series to be even stronger because in those series, the on-paper advantage usually goes to the team with the better record. The same applies to LCS games, but not to the World Series as on-paper home field advantage is chosen without regard to the specific teams competing.

That's it, except for the asterisked digressions.

(*) There is no particularly compelling reason to use 53% as a default W%; I just wanted a line that you could focus on that was reasonably telling, because the whole table is a bit much.

Anyway, I chose 53% because in the World Series (for 1923-2008 with a few years excluded), empirically the mean W% of the team with the better record has been .635, and the team with the lesser record has a mean of .594. Regressing 30% to .500, this results in .595 and .566. Log5 tells us that a .595 team should beat a .566 team 53% of the time. And there you are.

(**) The geometric distribution gives the percentage of time a certain number of failures (x) occur before a certain number of successes (r) occur for a binomial process. In the case of a baseball series, r is the number of wins for the victor in a series (3 for a five-game series, 4 for a seven-game series, etc.), x is the number of wins for the series loser (in a five-game series with a decisive games, x = 2; for a seven-game series with a decisive game, x = 3). We also need to know the probability of a success (P), and calculate the number of combinations using the combination function C(x + r - 1, x).

To find the probability of a decisive game for the series as a whole (with either team winning), we need to do a calculation for each team, which is why the results are summed below--one for the winner and one for the loser. Let G be the number of games in a full length series, W the number of wins for the winning team in such a series, L the number of losses for the losing team in such a series, and P the probability of one of the team winning an individual game. Then the probability of a decisive game is:

C(G - 1, L)*P^W*(1 - P)^L + C(G - 1, L)*P^L*(1 - P)^W

For example, the probability of a seventh game in a series (G = 7) in which one team has a 55% chance of winning each game (P = .55) is (W is 4 and L is 3, of course):

C(6, 3)*(1-.55)^3*.55^4 + C(6,3)*.55^3*(1-.55)^3 = 30%

One other thing to note is the expected number of games in a series. This is found by taking summing G*P(G) for all possible series outcomes. So in a five game series, the expected number of games is:

3*P(3 games) + 4*P(4 games) + 5*P(5 games)

Reverting to the assumption that each game is a 50/50 proposition, the expected number of games in a five-game series is 4.125. The expected number of games in a seven-game series is 5.8125. You can see that there are diminishing returns going on; despite lengthening the possible length of the series by two games, our expectation is that the actual number of games will only increase by 1.6875 games.

The probability of a decisive game hints at this as well, but this is another way you could attempt to quantify the real observed benefit of lengthening a series.

On the World Series Home Field Advantage

A week ago, the American League once again defeated the Neanderthal League (*) in the All-Star Game, securing home field advantage for the World Series. The "This time it counts" mantra about the game is premised on the notion that home field advantage is a significant thing to have (or at least the hope that TV viewers will believe that it is). So it is only natural to look back through history and see how home teams have fared in the World Series.

Let's start off with some theoretical calculations based on a few assumptions. Assume that the two teams are evenly matched, that there is no home field advantage, and that the outcome of each game is independent of any other. Therefore, each team has a 50% chance to win each game, and we can calculate the expected frequency of a 4, 5, 6, or 7 game series using the geometric distribution (I apologize for this digression as many of you know this better than I do):

P(x+r game series for one team) = C(x + r - 1, x)*(1 - p)^x*p^r

Where x = number of failures before r successes, p = probability of success, and C is the combination function

In this case, our successes are victories by the eventual series winner (always r = 4), x is losses by the eventual series loser (0-3), and p = .5.

C(x + r -1, x) is the number of different of distinct sets of wins and losses that can occur in the series. C(3, 0) is used for a four-game series, and is equal to 1--the only string of wins and losses that can produce a four-game series is WWWW. The formula for combinations is:

C(n, x) = n!/(x!(n-x)!)

So C(4, 1), the number of different combinations that can produce a five-game series, is 4!/(1!(4-1)!) = 4*3*2*1/(1*(3*2*1)) = 4. You can confirm this, as there are four possible strings (LWWWW, WLWWW, WWLWW, and WWWLW) that produce a five-game series. In fact, you can logically work out all the combinations fairly easily without the math for this application since we are only dealing with a seven-game series.

In a five-game series, the fifth game must be a win (same for the sixth and seventh games of six and seven-game series, respectively). So the victor can lose game 1, game 2, game 3, or game 4.

In a six-game series, the victor can lose games:
12, 13, 14, 15, 23, 24, 25, 34, 35, 45 = 10 combinations

And in a seven-game series:
123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456 = 20 combinations

Anyway, doing all the math (and then doubling since we have only considered this from the perspective of one team), the theoretical probability of a given series length is:
4 = 12.5%
5 = 25%
6 = 31.25%
7 = 31.25%

So theoretically (since WS home field sites are on a 12-345-56 pattern), in 43.75% of World Series, the number of home games will be equal. 25% of the time, the team with the home field disadvantage on paper will actually play more home games, and 31.25% of the time the team with home field advantage on paper will get to benefit from it--if and only if there is a game seven.

We'll get back to some theoretical stuff later, but let's look at the actual empirical World Series results. I considered all World Series from 1922-2008 (1922 is when the seven-game series returned permanently) with the following exceptions:

* 1922 and 1923--both Giants/Yankees series, in 1922 they shared the Polo Grounds, and in 1923 they didn't follow the 12-345-67 pattern
* 1943-45--in the war years, a 123-4567 format was used to cut down on travel (and in 1944, the Cardinals and Browns shared Sportsman's Park, which would have made it unusual in any case)

First, let's look at the empirical proportions of series by length:


As you can see, the empirical and theoretical don't actually track particularly well. I'm not going to discuss this phenomenon in-depth here, but it is something to keep in mind when we delve back into theoretical stuff at the end of the post. The assumptions are all faulty to some degree or another--the teams are not evenly matched, the results of the games are not truly independent (Even if you start with the premise that this is largely true during the regular season, one could conjecture that it is less true in a short series as behavior will be highly influenced by the series status--teams down 3-1 behave a lot differently than teams up 3-1 or tied 2-2. This is a classic case of what Bill James called the law of competitive balance.), we have not considered home field advantage, etc. For some more reading on this topic, check out Phil Birnbaum's post at Sabermetric Research and the Baseball Research Journal piece referenced there ("Relative Team Strengths in the World Series" by Alexander E. Cassuto and Franklin Lowenthal, BRJ #35).

Getting back to the actual data, we see what I will call a reverse home field advantage (a 5-game series, in which the "road" team actually hosts 3 games and plays two on the road) 20% of the time, no home field advantage (4 or 6 game series) 41% of the time, and a true home field advantage (7-game series) 40% of the time.

How often does the team with paper home field advantage actually win the Series? Let's break it down by series length:


This is pretty interesting, IMO. The paper home team wins 57% of the series, which seems impressive, but their strongest advantage comes when there is no home field advantage (61%), followed by reverse home fields (56%), and just 53% when there is a true home field.

Of course, the sample sizes aren't great when it's broken down like this, and it is unsurprising that the proportion of series won is less in seven games. What is interesting, though, is that the on-paper home team has such an advantage, and even in series in which they don't really benefit from it in the raw count. Are the first two games at home that much of an advantage, or is there something else going on here?

I'll leave that as a rhetorical question. There are a lot of factors in play here--the sample sizes aren't that large, we have not accounted for the quality of specific teams (which is tough to do in any case because of the fact they play in different leagues which were until recently truly separate in the regular season), etc.--and I don't really want to speculate about the influence of these myriad factors.

I did take a look at the regular season W% of the World Series participants, but as I just said, that's not a particularly telling measure, as it is possible that the leagues were unbalanced in any given year and that a lower W% in one could actually be indicative of a higher-quality team. I checked it anyway, and found that, for the group of series defined throughout this post, the winners had a mean W% of .616 with a median of .616, while the losers had a mean W% of .612 with a median of .610.

Teams with on-paper home field advantage had a mean W% of .615 and a median of .611; teams without on-paper home field advantage had a mean W% of .613 and a median of .610. There's no evidence of any sort of fluky quality difference, at least to the extent that W% captures quality. In terms of W%, the World Series winners, losers, on-paper home teams, and on-paper road teams are all essentially equal.

Let's also break down the series outcomes by on-paper home field advantage coupled with which team had a superior record. These figures will exclude the 1949 and 1958 series as the participants had identical regular season records:


So the team with the worse record has actually triumphed in one more series than their higher W% opponents (for reference, the mean W% for teams with the better record is .635 with a median of .636; the mean W% for teams with the lesser record is .593 with a median of .597, again excluding 1949 and 1958). Teams with home field advantage have been very successful, but those with worse records and home field even more so than teams which had both advantages.

Let's break down the home field W% by each game in the series:


As you can see, games 1, 2, and 6, which are home games for the team with on paper HFA, are the ones with the highest home W%. In game 7, the home field advantage is not particularly large. Those who make a big deal out of WS HFA are fond of pointing out that the home team has won the last eight game 7s, but they were just 2-6 in the previous eight, and I doubt there is anything significant going on. (Although I should point out that the period does correspond to the introduction of the designated hitter in WS play, even if I don't believe that has a significant effect (**)) Between 1952 and 1979 (which includes the 2-6 period mentioned above), road teams were 13-3 in game sevens.

One important caveat on comparing the game-by-game numbers is that as the series extends past the minimum of four games, we should expect to see less of a difference as mismatched teams are eliminated. It doesn't explain why the home field advantages are much smaller in games 3, 4, and 5, though, as there's no reason to suspect that the on-paper road teams are of substantially different quality than the on-paper home teams.

The overall World Series home W% is .573, high compared to the regular season average which is generally somewhere in the neighborhood of .540. Let's use this figure in place of a default assumption of a 50% outcome in each game to model the outcome of a series. Using the combinations detailed above, we can find the probability of any series outcome given these assumptions. For example, the probability of a 4-2 series in which the home team wins games 1, 2, 4, 5, and 6 would be .573^5*.427 (five home wins and one road win). Under these assumptions, we get these probabilities for the possible series outcomes (in this table, "home" refers to the teams with on-paper HFA and "road" to their opponents):


Even using the sample home W% of .573, we only expect the team with HFA to win 52.3% of the time. In fact, teams with HFA have won 56.8% of the series (46 of 79). What is the probability that this could have happened by chance, assuming that 52.3% is the true probability and that each series is independent of the others? It's 12.1%. Even if we assume that there is no true home field advantage at all, and each team will win 50% of the time, there is still a 5.7% chance that 46 out of 79 would be observed.

How about the individual game results (home teams are 268-200, .573)? If the true home field W% was .540 as it generally is for the regular season (and given all the other necessary assumptions for use of the binomial distribution), the probability of 268 successes in 468 trials is 7.1%.

So I am decidedly uncomfortable drawing any conclusions about the strength of home field advantage (on the series or game level) in the World Series from the sample data. The actual results show a stronger home field advantage than we might have expected, but not to such an extent that we must conclude that regular season assumptions about home field advantage do not apply.

It's certainly a good thing to have home field advantage for the World Series, or any game for that matter, and I'm not going to try to argue that basing home field on which league won the All-Star Game is anything but a gimmick. However, given that the previous method of determining home field was simply to alternate it yearly between the leagues, I don't think there's any real harm being done by this approach. If you really wanted to reward the stronger league, the overall interleague record would be far more likely to successfully identify the stronger league, but I don't consider the whole matter worth getting exercised over.

I have posted a Google Spreadsheet with the sequence of games in each series if you are interested. The first group of columns marked G1 through G7 indicate whether the eventual WS champion won the game (W) or lost (L). The second group of columns indicate whether the home team in that particular game won (H) or whether the road team won (R).

Finally, I'll close with some useless trivia. You probably know that there have been three series in which the home team won each game (1987 Twins over Cardinals, 1991 Twins over Braves, and 2001 Diamondbacks over Yankees). The most road games ever won in a series (that I considered for this study) is five, which has happened seven times--1926 Cardinals over Yankees, 1934 Cardinals over Tigers, 1952 Yankees over Dodgers, 1968 Tigers over Cardinals, 1972 A's over Reds, 1979 Pirates over Orioles, and 1996 Yankees over Braves.

P.S. After I wrote this post, but before I published it, Sky Andrecheck published a piece on the importance of World Seires HFA at Baseball Analysts. It addresses an interesting question that I will paraphrase as "Since the Dodgers have such a large lead in the playoff race, is the single most important regular season game left on their schedule (with regards to winning the World Series) the All-Star Game?"

I'll let you read Andrecheck's article to find the answer, but there's one minor point which overlaps with this post worth commenting on. Andrecheck notes that the playoff HFA has been higher than the regular season historically, and reasons that this has to do with the home team being the better team more often than not. While this is true for the league playoffs, there's no reason to suspect it to be true for the World Series in which home field alternated between leagues (even if the All-Star result method of determining home field has the effect of giving on-paper home field to a better team more often than not, home field has not been decided by that rule nearly often enough to have any impact on the results, and the amount of noise involved would be incredible in any event). I don't disagree with the notion that we can't say with any certainty that the World Series HFA is of different magnitude than the regular season HFA, but the better team having more home games leaves a lot to be desired as an explanation (again, for the World Series, not the league playoffs).

He also gives the probability of the team with home field winning as 51.26%, assuming that the home W% in the World Series is 54%. I didn't provide this figure in my post, as I approached the question from the standpoint of "Even if .570 is the true HW%...", but I am in agreement with it (naturally, as it is true by definition given the assumptions we both made).

In the comments to Andrecheck's article, there was a link to Cyril Morong's look at WS HFA, published in 2006, which means that I pretty much repeated here what he had done. However, we disagree on the probability of the on-paper home field team winning the series in six games (and thus of course we also disagree on the probability of them winning the series period). I am pretty sure that this is due to a faulty six-game series sequence he used.

(*) Sorry, I can't help it. I SHOULD take the high ground, but the sniveling "It's not REAL baseball" is way too much for me to handle. I'm weak like that.

(**) There was no DH in the World Series until 1978, at which point it was introduced on an alternating year basis. So in 1978, 1980, 1982, etc. the DH was used in all World Series games, and was not used at all in 1979, 1981, 1983, etc. Starting in 1986, the home team's rules were used.

So while the run of Game Seven home wins begins with the Cardinals in 1982 and also includes the Royals in 1985, in those series the road team's rule was being used in Game 7. All of the game sevens that follow, of course, used the home team's rule.