THE BOOK--Playing The Percentages In Baseball

I sometimes get lost in certain-math + baseball-related things, but this seems incredibly straight forward to me. Don’t know how else it can be misconstrued.

Too many other people think otherwise.

Tango - What you are saying is only true if you have determined that the teams that the Mariners played were expected to play at an average rate in those 9 games.  That is to say that they truly represented the average of the teams that the Mariners will face in the other 153 games left in the season.

This is sort of like the following problem.  Suppose you flip a coin and it ends up head 10 times in a row.  What is the probability that it will end up heads on the next flip?  The answer is 1/2.  But I said “sort of”.  In the coin-flipping analogy, we actually know that the probability of heads in any given toss is 1/2, regardless of previous outcomes.  In the M’s case, we really don’t know that their winning probability is .481.  So, while the reasoning that leads to 81 wins is correct, the issue seems to be only of “academic interest.”

Gotta agree with Mike.  This ain’t exactly the Monte Hall/Marilyn Vos Savant question.

Peter, if the Mariners are still a .481 team (is that .481 against the league in general or against their original schedule of teams?) then why does it matter who they played in those 9 games?  Even if they played 9 of the toughest teams, you think that is going to change the WP of the rest of the schedule enough to change the 81 wins to something else?

Let’s say that the entirety of the M’s schedule were a .500 team combined, just for the sake of argument.  Let’s say that the teams they just played 9 games against were .550 teams combined.  That would mean the rest of the schedule were against .4965 teams rather than .500.  That gives them another half win at the most.

As Dave says in the post, the wins are on the board, you can’t take them away.

1. Start with your prior estimate
2. Update the prior with the games played so far - the teams played will come into consideration, but 9 games is such a small sample that it won’t budge the prior any meaningful amount.
3. Extrapolate the new prior over the rest of the season - you can consider the remaining strength of schedule, but with so many games remaining, it’s not significant (not more than a game - see artcle at BPro this week)
4. add estimated future wins to actual past wins

This reminds me of an article by Joe Sheehan on July 21 of last year-- 95 games into the season-- where he insists that he still picks the Braves to win the Wild Card because 95 games isn’t a large enough sample size, he says-- despite the Braves being 9 games out of the Wild Card at the time.

The article:
http://baseballprospectus.com/article.php?articleid=7836
The wild card standings at that time:
http://mlb.mlb.com/mlb/standings/wildcard.jsp?ymd=20080721

MGL - Of course it matters who they played in those games.  The original a priori of .481 was based on the Mariners playing a fixed number of games against a range of teams some of whom were expected to be better and some of whom were expected to be worse than they are.  The probability is higher that they would win the games against the opponents that are expected to be worse then they are, and lower that they would beat the teams that are better than they are.  If they had played Washington all 9 games they would have been expected to win the majority of those games.  They would still have to play against the opponents that had been expected to beat them at the beginning of the season, and those opponents would still be expected to beat them because the neither the Mariners nor their opponents expectations of winning or losing would have changed much due to their performances in the first 9 games.

MGL would be correct if they actually played teams with an (expected) .550 W%, but if instead they played teams with and (expected) .350 pctg, then their remaining schedule would be really tough.  In reality, they probably played three league average teams, and will continue at their actual talent level (be that .481 as expected, or whatever).

The point is that statistically, their expectations don’t change based on record, only (maybe) based on remaining schedule.

I always thought differently.  Say your true talent is .500 (81-81).  Say you play a poor team in a 4 game series and win all four games.  So your sitting at 4-0.  So your saying we need to adjust the orginal 81-81 record to reflect the 4-0 start?

Isn’t it possible that a team with .500 talent wins 4 in a row just by random chance?  Isn’t that the reason for playing 162 games?  To balance out the luck?  I always thought they would still be expected to go 81-81.

Geez, Peter, why are we arguing about something that is as obvious as the noses on our face and is 6th grade mathematics?  Do you honestly think I don’t know that the schedule in those 9 games could affect the composite schedule for the rest of the season?  Did you just read one sentence of my post and then you started hammering on your keyboard?

Let’s say that the entirety of the M’s schedule were a .500 team combined, just for the sake of argument.  Let’s say that the teams they just played 9 games against were .550 teams combined.  That would mean the rest of the schedule were against .4965 teams rather than .500.  That gives them another half win at the most.

Did you not read that sentence?

Obviously if they played 9 games against .650 teams or .350 teams, then the rest of the schedule would be affected a little bit, but who plays 9 games against .350 teams?  There isn’t even such thing as a .350 team, at least not this year.  It takes 9 games of playing a .620 team or a .380 team just to make a 1 win difference in the rest of the schedule!  So yes, it is technically true that who they played “matters” (that is trivially obvious), however, for all practical purposes it does not matter.

John, you COULD adjust your “prior” estimate of .500 based on 4 wins.  If you did, you would adjust a tiny bit of course, because as you said, the chance that a .510 team wins 4 in a row is around the same as the chance that a .480 team wins 5 in a row.  You might come up with a “revised” wp of .501 at the most (of course if you won those 4 games against a poor team, you would revise just a scoche less than if you won those 4 games against an average or better team).
Anyway, that is the teams’ strength and expected win percentage for all remaining games.  So with 158 games remaining, they are expected to win half of them, or 79 games, for a grand total of 83 wins at the end of the season.

In the article cited, David specifically said, “going on the assumption that we don’t change our estimate of the Mariner’s true talent for the rest of the season.”

If you had a good projection before the season started, and nothing much about the team has changed, personnel or injury-wise, there really isn’t much of a reason to revise your estimate of their true WP no matter what their record is.  Now, if the team is hitting and/or pitching (and defense and base running of course) better than your projections indicated, even if all the players you anticipated playing are pretty much playing, then of course, you would have to revise your projections for all players to include their current season stats, which would cause you to revise your overall estimate of the team’s WP. But, after 8 or 10 games, even if the team is playing real well, offensively and/or defensively, it ain’t gonna cause your individual player projections to change very much at all, and hence your team projection won’t change much either.

Of course, in general if a team is playing better or worse than your projection thought it would, then there might be other factors which you missed in your original analysis.  However, it’s gonna take 30 or 40 games (obviously there is no magic number) to convince me that my projections are “missing” something.

http://nymag.com/news/sports/53975/index2.html

So I called my mom. She suggested I work on being “Obama Cool.”

-- Will Leitch, on advice he got from his mother in debating in an unfriendly environment

***

The post by John/10 is exactly the reason that I posted this thread.

The nuances being discussed by others (the remaining schedule impacting the number of wins, the potential that your prior needs to be updated after 9 games, etc) is beside the main point that I was trying to make.  However, I am enjoying the discussion of those aspects, since they are more fruitful for discussion.

Going back to John/10 (and others who are thinking like him but haven’t posted to that effect): if you flip a coin, and every time you get heads, you get $1, and every time you get tails, you get 0$, and you flip the coin 162 times, you are expected to have (on average) 81$.  Now, let’s start the game over, and you flipped the coin 12 times already, each time you got a head.  That’s 12$ in your pocket.  Presuming the coin is not weighted (your prior remains 50/50), you have 150 coin flips to go.  How much money will you make on those 150 coin flips?  75$.  That’s (an average of) 75$ future dollars, plus the 12$ already in your pocket, for a total of 87$.  So, to Dave’s original point (which Brian/6 broke down as a step-by-step), you add up the gains already in hand to the expected future gains, with no expectation that the future gains are impacted by the gains already in hand.  Just like real-life betting.

***

Now, back to the other fruitful discussion, regarding the changing of the prior: the team has the same players on April 1 as they did at the start of the 10th game.  There are no rookies really to speak of whose talent level we have to revise.  Basically, no surprises.  Endy is hitting very well, but with over 2000 career PA already, adding another 30 PA to that won’t make a smidge of difference (unless we think his talent level has changed because he’s become a smarter hitter for whatever reason).  Gutierrez is the real deal fielding-wise, but the Fans Scouting Report already told us that, as did UZR.  etc, etc, etc.  So, even if they went 9-0, with 50 runs scored and 25 runs allowed, we won’t change our estimate other than a smidge.

The changing schedule is a bit more problematic, but at this point, they’ve basically played a bit better than average teams, meaning they have a smidge under average teams to play (relative to the average of their schedule, natch).  As noted earlier, even if they played .600 teams (which is really the maximum you should expect) for 9 games, that makes their remaining schedule .494.  So, the impact is .006*153 = 1 more or less win, had they played an extreme schedule already.

Furthermore, had all 9 games been played with in-division rivals, this gives them a doubly-whammy to increase their chance of winning the division, since they not only have 9 wins in hand, but there are 9 losses spread out to those 3 teams as well.

I’ll just add that, while I agree that the strength of teams faced to date is likely fairly comparable to those they will face over the rest of the year, at least in aggregate, when talking about the anticipated winning percentage of the opponents, we are not talking about the team’s actual winning percentage, but their anticipated winning percentage against the M’s.

For instance, while we might anticipate the Red Sox to have a .550 winning percentage against the league, if we believe that the M’s are a sub-.500 team, we might expect the Red Sox to have a higher anticipated winning percentage against the M’s.  In this particular instance, it probably makes little or no difference, but that does not mean that Dave was absolutely correct in the way he formulated his post, though I appreciate his enthusiasm.  The M’s are awful fun to watch this year, especially when the ball is in the air.

MGL is right that the argument over whether the Mariners will win 78 or 79 or 80 or 81 games given what they had done so far is trivial.  But I took the discussion as Tango has explained it - that it is important to illustrate a point on which many people are confused.  Since neither Dave nor Tango had mentioned the quality of the opposition prior to my first post I thought that it was important to do so.  MGL’s example of playing much better than average teams in his post (which I had read thoroughly) although mathematically correct was slightly misleading from my point of view.  What I was trying to point out was if the Mariners had played teams projected to be worse than .481.  The opposition wouldn’t have to be real bad (my choice of Washington was also a poor one), they would only have had to have been projected at .480 for the the Mariners to have been expected to have one 5 games out of 9 instead of 4 and for the current expectation for wins for the season to be 80 games instead of 81.

The flawed belief system actually has a name.  You can Google “Gambler’s Fallacy” and read all about it.

The opposition wouldn’t have to be real bad (my choice of Washington was also a poor one), they would only have had to have been projected at .480 for the the Mariners to have been expected to have one 5 games out of 9 instead of 4 and for the current expectation for wins for the season to be 80 games instead of 81.

That is another way to look at the “strength of schedule adjustment.” It will give you the same answer of course as if you adjusted the strength of the remaining schedule.

Sometimes when I say, “It won’t make a difference, I mean the difference is so small as to have no practical significance and sometimes I mean, “It WON’T make a difference,” as in exactly zero.  I thought I made that distinction clear by showing how it WOULD make a difference in the section I quoted in my last post, but that difference would be small. Just like when Tango says that 9 games of data from Endy or anyone else on the M’s won’t change our expectations for them, he doesn’t mean, “Won’t change” as in exactly zero, he means, “The difference is so small that you can ignore it...”

But Peter did the right thing in pointing out to the less statistically literate readers on our site that the rigorous version of the solution includes adjusting the rest of the season’s expected w/l by the strength of the remaining schedule as compared to the anticipated strength before the season started (if they were expected to play an entire season of .490 teams, and the first 9 games were against .490 teams - below average - there would be no need to do any adjustments), as well as slightly adjusting your estimate of the team’s true WP based on their performance so far, either the w-l record if that is all you know (in which case, your re-estimate of the team’s WP is tiny), or their RS/RA, or the team’s underlying offensive and defensive performance.  How much you adjust that team WP not only is a function of how much more data you have (obviously) but also the granularity of that data (are you just using the team w/l record so far or are you using player OPS and ERA or FIP?) as well as the “confidence” you have in your pre-season projection, as well as (this is the final “as well as") knowledge you have of changes in personnel (compared with your initial estimates), playing time, injuries, changes in true talent by players, and that sort of thing.

Tango/12- How many games would a team have to play, so that we could re-estimate their prior projected winning percentage?  I agree that 9 games isn’t enough to say that the Mariners are actually a better team than we expected, but if they continue to play at a high level for another 10-15 games, do we adjust the prior winning percentage?

Also, the fact the Bedard is pitching so well and is healthy, may adjust their prior projected record.  If Dave only projected him to pitch 150 innings with a 3.50 FIP, and he now looks like he could pitch 200 innings with a 3.00 FIP, that would add nearly 3 wins to their projected record.  So, by simply adjusting the prior to fit the fact that Bedard looks like he will outperform his projection by a lot, would make them almost a .500 team instead of a .481 team.  In that case, given their hot start, they could be projected to win 84 games.

Basically, a lot of people have trouble differentiating between regression towards the mean, and gambler’s fallacy.

Here is how it actually works:
--Joe Blow is a “true” .300 hitter.  He is currently hitting .400.  Chances are, from now on he will hit .300 and as the sample increases he will have hit .300 for longer than he hit for .400 and his seasonal average will, thus, be closer to .300.

And here is how some people THINK it works:
--Joe Blow is a “true” .300 hitter.  He is currently hitting .400.  Chances are, he will probably go hitless in his next few games so that his seasonal average is closer to .300.

That is what we are seeing with the confusion on USSMariner, IMO.

Nick, there is no magical number of games in which you “can” start to re-adjust your prior estimate.  You can do it anytime you want.  Obviously the more reliable and the more granular the new data is, the more you can use it to adjust your priors. Same with the amount of data, everything else being equal. 

As far using new information about Bedard or whatever, you can use anything you want.  In fact, it doesn’t really make any sense to update your prior estimate of a team’s WP by adjusting that estimate.  Just come up with a new estimate given the information you now have.

The article in question, of course, was simply assuming that you were keeping your initial estimate of the team’s WP.

If all you know is a pre-season estimate of a team’s WP, and you have X number of new games to work with, how do you update their true RP?  Again, it is not “when” do you update it (after 9 games, 15 games, etc.), it is “how” do you update it.

You need to know the uncertainty of your initial estimate.  Once you do, it is trivial (if you know how to do it) to do the update.  Here is a simple example:

Let’s say that your initial estimate was .481 but you thought or new there was a 50% chance that they were (and still are) really a .461 team and 50% that they were a .501 team.

Now, let’s say that they go 7-2 in the first 9 games.  You use a binomial formula to figure out the chances of a .461 team going exactly 7-2. 

The probability of exactly x successes in N trials in a binomial probability distribution is:

p to the x power times 1-p to the n-x power.

So .481 to the 7th power times .519 to the 5th power or .000243199

Do the same thing for a .501 team and you get .0002451132.

So the chance that they are a .501 team is now defined by the ratio of the two probabilities weighted by the initial probabilities, which were .5 and .5.  That comes out to .5221562.  That is the probability that they are a .501 team now.  And of course the probability that they are now a .481 team is one minus that.

So our new mean estimate of their true WP is now .5221562 * .501 + .4778438 * .461, or .4818862.  That corresponds to a difference of .1356 wins at the end of the year (for the remaining 153 games).

So, in summary, if you started out with a .481 WP for the team, and you kept that assumption after they go 7-2, they are expected to win another 73.593 games.  If that .481 represented a 50% chance they were really a .501 team and a 50% chance they were a .461 team, then after a 7-2 record, you would update your estimate of their true WP by almost another “point” (.481 to .482) and that would mean they would be expected to win 73.73 games rather than 73.593.

Obviously an “extreme” record after 15 games would impact that new estimated WP even more.  20 games, even more.  Of course 7-2 after 9 games is going to impact about as much as 15-10 after 25 games, or something like that…

I’m reminded of the article in the 1985 Bill James Baseball Abstract about the 1984 Tigers. He talks about direct significance--which I think is what Tom and Dave were talking about at the top of thread (going 7-2 simply means the team is +3) and indicative significance, which I interpret as similar to the discussion of updating your prior estimate.

For example, James wrote, “The Tigers hot getaway… was early evidence of an outstanding team. A 9-0 start is not directly significant; if the team you have to beat is 5-4, a four-game lead with 153 to play isn’t anything. Its significance lies in this fact: The chance that a .650 baseball team will win nine straight games is ten times as great as the chance that a .500 team will win nine straight games.”

James then did a nice study comparing teams final records to their records after various numbers of games played.  His conclusion was, “the only meaningful thing that happens in the first few weeks of the season (other than a Dickie Thon-type injury) is that an occasional team will come out of the gate with murder in its heart. If a team starts out 1-4, 3-1, 0-5, that really makes no difference; it’s just four or five games with a long time to catch up. But when a team comes out like the A’s in 1981, like the Braves in ‘82, like the Tigers in ‘84, playing as if they had a message for the league, then that is something that should be looked at very carefully.”

It would be interesting to try to do an analysis of data like those studied by James, but doing it more formally with the updating of priors (that is, using Bayes Theorem).

Seattle is now 8-3. Last year they went on five 8-3 runs:

Record  Count (2009) Count (2001)
 0-11        2            0
 1-10        8            0
  2-9       16            0
  3-8       40            0
  4-7       32            1
  5-6       20            3
  6-5       15           17
  7-4       14           38
  8-3        5           51
  9-2        0           26
 10-1        0           11
 11-0        0            5
TOTAL      152          152

Thought I’d throw the 2001 team in for comparison. Based on last year’s distribution, there’s a 5/152 = 3% chance that this year’s team is no different than last year’s. A start of 9-2 or better could be evidence that something is different this year, because it’s a run they weren’t able to achieve last year.

Dackle/21

They did not go on 5 8-3 runs.  Five times during the season, they were 8-3 in their last 11 games.  I’m guessing those 5 times were all in the same week, meaning a bunch of those 5 8-3 ‘runs’ include THE SAME wins.  The runs aren’t independent.  If you break the season into independent 11 game chunks, I doubt the 2008 Mariners went 8-3 more than once.

If you overlap the runs then you get more granular data. But, if you do want to break the season into 11-game chunks, then last year the Ms went on two distinct runs of 8-3 or better:

In late June and early July: 9-3
In late August: 8-2 (tack on a loss to make it 8-3)

So ... 162 games divided by 11 = 14.7 “11-game chunks”.

Two runs of 8-3 or better divided by 14.7 = 13.6% chance—even higher than the more granular result.

But ... the binomial distribution says the odds of a 61-101 team going 8-3 or better is 1.983%. Multiply by 152 11-game runs, and theoretically the Ms should’ve gone 8-3 or better about three times.

Anything can happen in 11 games.

I would just like to question the theory behind this.  I understand the math and why it works like this, but as Dackle mentioned it is possible that the M’s are just on a random hot streak that should be expected by x number win team.  The projections aren’t for 1 game, they project an entire season.  So is it right to assume that the M’s are x winning percentage true talent level team, and that they should expect to win that percentage of whatever amount of games they play?  I am not trying to be argumentative, I would just like to know why it is like that.

I think one very interesting aspect of Dave’s post is that he specifically states that you shouldn’t have changed your mind about the strength of the M’s _ROSTER_ because 9 games is to small a sample size.

But a team projection is reasonably not just the projected stats of its players, but those projections combined with a set of assumptions about how those players will be used. And nine games may very well be enough to see paterns in the usage of the players on a team to cause a need to revise the assumptions about playing time.

Now I don’t follow the M’s in detail so I have no idea if there is anything like this going on in this case, I am just speaking about in general.

Things such as platooning, bating order and reliver usage might prove itself different enough from what presumed prior to the start of the season to adjust the estimated win% enough to be worth plus/minus a win or two…

You guys are on top of everything.  Pretty cool.

Nick/24: I’m not following your question.  Can you restate?

Nick, it’s been shown that winning streaks don’t imply a losing streak is coming.  And vice versa.

One could test this whole independence thing:  Take the first ten (or whatever) games from previous seasons and that season’s projected win total (by PECOTA, CHONE, whatever).  Compare the initial 10-game record to the record the rest of the season to the projected record.  I think it’s been done before, I just can’t find where.

While I understand that part of it is for ease of calculation, is there anyone who actually takes the strength of the starting pitching into effect when evaluating strength of schedule?

After all, facing Bedard and Felix as opposed to Silva and Washburn is going to have a large impact on projected runs allowed, which in turn would greatly change our projection on wins.

Since there are always injuries, and it’s almost impossible to project who the starters will be each time in a season matchup, it is very difficult to add to projections, but I would still like to see strength of schedule based not on the relative strength of a team’s overall record, but specifically on the lineups faced and the projected runs scored/allowed in those matchups.

Does anyone know of anyone who’s done that?

Sal, you can’t do that. Creating strength of schedule adjustments based on opponents’ overall wp already includes their starting pitching and “lineups” of course.  You have no idea who is going to pitch against whom after the first week or two in the season.  About the only thing you can do, which I used to do, is to make an adjustment for righty/lefty starters.  IOW you can estimate how many righty and lefty starters each team is going to face and if that team happens to be better versus LH or RH pitchers (based on their projections of course) that might change your strength of schedule adjustment a little.  It is probably not worth the effort though.  Obviously after the fact, if you want to know how many games a team has under or over-performed their expectation, then you want to include the pitching they faced.  But that doesn’t tell you anything about the rest of the season other than influencing your updated projections (for example, if you thought Griffey was a true .750 OPS hitter and he went .850 through the first 10 games, that might raise your projection just a hair, but if that .850 was against 10 crappy RH pitchers, then it probably wouldn’t change your projection).  Again, not really worth the effort…

But you can adjust strength of schedule for projected runs scored/against for the current roster. In other words, the true talent of the Angels may have been X with Lackey, Escobar, etc. But certainly, any team playing them in the near future is going to get a boost to projected runs scored on account of injuries.

Sure it doesn’t help to project September, but it certainly helps to project April/May.