While I was watching the game, I noticed that the Heat seemed to be doing pretty poorly from the free throw line.  As it turns out, the Heat did shoot kind of poorly, only hitting 64% of their free throws.  It was also a little unusual that the Pacers got 7 more free throws than the Heat overall; the Heat got to the line a little more than the Pacers during the regular season, and while the Pacers are a tiny bit better at preventing free throw opportunities than the Heat, it’s awfully close.

So, paralleling a post I did two playoffs ago, I decided to see how the Pacers and Heat should have done from the line.  Here are two tables, one for each team, listing how many free throws were taken and made by each player to attempt one along with their regular season FT percentage and how many they would be expected to make.

The expected number is simply their free throw attempts in the game multiplied by their regular season free throw percentage.  For example, David West took 8 free throws and was a 76.8% shooter this season.  Multiply those two and you would expect West to make about 6 free throws on 8 shots.  In last night’s game, though, he only made 4, so West was a little unlucky.  You can do this for each player who took a free throw and then add them up for each team.

As it turns out, the Pacers were pretty much square on.  West was unlucky but Paul George mostly made up for it.  Overall the Pacers would be expected to make just under 25 free throws on their number of attempts and they made 24.  The Heat, on the other hand, would be expected to hit about 19.5 but they only made 16.  Wade was the big offender, although LeBron was a bit off and Ray Allen missed a notable free throw towards the end of regulation (although it’s hard to say that was lucky or unlucky given that he only took two shots).

If you put that together, the Heat should have had an extra two points in the game compared to the Pacers just from hitting their free throws.  The difference is almost entirely from regulation, as both teams made every free throw they took in overtime.  So if you grant the Heat an extra two points at the end of the game, which the Pacers had no chance of affecting since they committed the fouls anyway and just had to watch the Heat shoot, they would have won in regulation despite Paul George’s desperation three pointer.

Of course, you could argue that if the Heat had made a couple extra free throws, the game would have played out differently as the Pacers would have had to foul or take different shots at the end.  And that’s obviously valid, but it also draws attention to the foolishness of blaming Vogel for the last play of overtime.  If any of a number of things had swung a different way, the game would have played out differently and his decision wouldn’t have mattered or might have changed.  I think that focusing on free throws is a relatively clean way of playing ‘what if’, since they would happen regardless and are defense-free, but they still aren’t free of criticism.  But overall, despite how close the game was, I think if you combine the Heat’s poor free throw shooting and George’s lucky three at the end of regulation, it seems obvious that the Pacers were fortunate to get to overtime and it’s kind of petty to blame Vogel for the loss.  Both teams played a tough game, and the Heat got the nod in the end.

As a side note, a “helpful” commenter on that linked post noted how someone at ESPN wrote basically the same analysis as I did and accused me of stealing it.  He failed to notice that my post was up before the ESPN one, but who has time to fact-check before throwing around accusations of plagiarism?  I bring this up only because more time has passed between the game and my post this time, so someone else might have noted the Heat’s poor shooting.  You can only take my word for it that if so, I haven’t seen any articles that mention it.  I do my best to put in links and attributions when my posts are based on or reference someone else’s work, so if I miss anything relevant I appreciate a friendly head’s up.

We’ll start with the Heat and the Pacers.  These two have a nice recent history which you can read about roughly everywhere.  Unfortunately for the Pacers, they simply aren’t as good as the Heat.  They’ll try to win by making the game slow and ugly and hoping that Wade is still beat up and making the rest of the Heat feel that way.  I don’t think it’ll happen though.  With a 3.9 point differential advantage in the regular season, the model has the Heat winning 82% of the time with the Pacers taking one game (narrowly favored over two).  The Smackdown picks for this series aren’t up yet, so I don’t know if I have any ability to make up ground with this pick.

In the West we have the Spurs and Grizzlies back at it as well.  The Smackdown consensus here is a close one despite the generally better play by the Spurs in the regular season.  That leads to a model prediction of the Spurs taking it in 6, winning 73% of the time.  If that happens, I’ll catch up a little bit as two people in the contest took the Grizzlies and the rest (with one exception) took the Spurs in 7.  That being said, I’m pretty much out of the contest as I went the wrong way with the Knicks-Pacers series and the model didn’t know that Russell Westbrook was injured.

As a side note, the model can provide a little bit of insight into Westbrook’s value.  All the model does is take the difference between two teams’ regular season point differential, combine it with home court advantage, and predict who should win and in how many games.  The Thunder finished the regular season with the league’s best point differential at 9.2.  The Grizzlies finished with a good-but-not-great 4.1.  We can maybe say that Memphis’ number should be a little different since they made the Rudy Gay trade partway through the season.  According to NBA.com, the Grizzlies were a 5.0 team in the back half of the season (the Thunder were a 10.1 team, but we’ll ignore that since I don’t remember them making any particular moves).  According to the model, a 9.2 team with home court playing a 5.0 team should win 83% of the time, and the 5.0 team should win once or twice.  Instead, the Grizzlies won in 5.  Acknowledging that these are probabilistic estimates, and so the Grizzlies winning was in the initial prediction (it should happen 17% of the time even with the Thunder at full strength), how low do we need to knock the Thunder’s point differential to get the Grizzlies winning in 5 as a likely outcome?

If we make them even, the Thunder should still win 57.6% of the time, probably in 6 games.  So let’s say that isn’t close enough.  If we make the Thunder a point worse than Memphis, they win 50% of the time, most likely in 6 games.  To make Memphis favorites, we have to make the Thunder over a point worse than Memphis.  The model doesn’t really like the away team to win in five games (six games is always preferred until a sweep becomes more likely), but it seems fair to say that the Grizzlies would need to be at least three points better for the outcome we saw to be a fairly plausible one out of the options.  That implies that the Thunder played like a team with a differential of only 2 or 3, and the loss of Westbrook cost them 6 or 7 points per game.

That’s a rough estimate; we know that Ibaka had a rough series, for example, and there’s also a trickle-down effect in that not only did Westbrook not play, but it means that his backup played more and his backup’s backup played more.  Those factors make Westbrook look better than the outcome directly suggests.  Westbrook’s value may also differ based on his team; he’s obviously fairly integral to the Thunder’s offense, but somewhere else he may be less valuable.  But all in all it seems fair to say that Westbrook is probably worth at least 4 points per game to the Thunder and maybe as many as 6.  That’s a pretty good number.

With game 1 already in the Penguins’ favor, we might as well start there.  Pittsburgh had a little trouble with the Islanders in the first round, giving up a fair number of goals and losing two games.  Ottawa upset Montreal but made it look easy, moving on in five games and getting two different 6-1 victories.  That being said, the Penguins have an (adjusted for season length) 58 goal advantage over the Senators, so the model sees them winning 74% of the time.  Two wins for the Senators would be reasonable (as the model virtually always says that a series will go six).  With Pittsburgh winning 4-1, the series is off to a predictable start.

The other game going tonight is Kings-Sharks.  After losing their first two games to the Blues, the Kings won four straight to take their first series.  Hockey playoff series tend to be close because of the noise inherent to single NHL games, but these guys took it to heart: the two teams were within a goal of each other or tied for all but five minutes over the course of the six games (including two overtimes).  So say the Kings were the better team if you want, but that sounds awfully even to me.  The Sharks swept the Canucks, which sounds more impressive, but even then two of the games went to overtime.  Regardless, results lead to beliefs which lead to stories, so the Sharks are ‘hot’ right now.  The Kings had the better goal differential in the regular season, so the model expects them to win 55% of the time, victory coming in six games. 

Moving to tomorrow night, the Red Wings will face the Blackhawks.  Detroit made the playoffs in the last days of the season and needed seven games, including a 3-1 record in overtime games, to get to this point.  Anaheim actually outscored them overall.  Chicago had a dominant regular season and took care of the Wild in five games, so they look to be in pretty good shape.  And indeed, Chicago swept the regular season series.  But three of the four games went to overtime, so maybe it’ll be closer than expected?  With a 76 goal advantage though, the model sees Chicago winning 80% of the time with the Wings taking two.

The last series is Boston and New York (and isn’t it always?).  Both teams are coming off of game 7 victories, so momentum is on both sides.  Boston had two wins in overtime while the Rangers only ever lost to Washington in overtime, though, so Boston will have a blatant clutch advantage.  It will probably come into play since Boston was only 7 goals better in the regular season; the two teams will be very evenly matched as far as the model knows.  It sees Boston winning 53% of the time with the Rangers taking two. 

So there you go.  If everything goes ‘as expected’, which it never does, we’ll have Chicago and L.A. in the West Finals to determine who gets the winner of Boston-Pittsburgh in the Stanley Cup Finals.  But if Detroit, San Jose, New York, or Ottawa make it, don’t be surprised.

In the first round I was 100% correct on the Heat sweep, as was pretty much everyone else in the Smackdown (the lone exception was Benjamin Morris).  I was also right about the Knicks, Pacers, Thunder, and Spurs.  Everyone else also made the same picks, although Henry’s mom had the right number of games for the Spurs; she, Haberstroh, and the Crowd had the right number for the Pacers; and she, Ilardi, and the Crowd had the right number for the Knicks while Ma actually took the Celtics.  So it’s fair to say that those series were never really in doubt.  Those correct picks gave me 27 points in the first round.

Things went wrong with the Warriors-Nuggets series, where the Warriors pulled the upset.  It’s fair to call it an upset for a variety of reasons, but the fact that everyone in the Smackdown picked the Nuggets seems to verify it.  I also missed the Grizzlies’ victory, which only three people picked (Haberstroh, Ma, and Henry’s mom), and the Bulls’ victory, which half the field picked (the Crowd, Ma, Ilardi, and Arturo).  The Grizzlies’ win was unexpected, I would say, but certainly helped by Griffin getting injured.  That’s part of why these are probabilities and not certainties.  The Bulls’ win, I think, was less expected in part because they were dealing with injuries as well as being on the road.  But put it all together and my 27 points are about one series behind the average of 32.3; if the Nets had held on or the Clippers had played to form, I’d probably be right in there.

Let’s see if I can catch up in the next few rounds.  One place that won’t happen is with the Heat-Bulls series.  The Heat have a 7.6 point differential advantage, which my model translates into a 94% chance of victory.  I have the most likely number of games as a sweep, although one Bulls win isn’t unlikely.  Everyone in the Smackdown has the Heat in 5, so this is really a question of if anyone gets the bonus two points or not.

The other series in the East is Knicks-Pacers.  This is an interesting one for a number of reasons (the teams are a good study in contrasts if nothing else), but in particular the Knicks only out-scored the Pacers by .2 over the course of the season.  That’s pretty close, although their home court advantage will also help.  I give the Knicks a 60% chance of winning, with the Pacers taking two.  The Smackdown field is split here, with half taking the Pacers in six and the other half taking the Knicks (three in seven games, Ilardi in five).  This will be one of the series that decides who wins the contest.

Moving to the West, we have the Spurs and the Warriors.  The Spurs were probably the third-best team in the league this year, while the Warriors are playing this year’s version of the spunky team that everyone hopes has a good run.  I think some people will want to give the Warriors a chance because they are much younger and more athletic than the Spurs, and Curry always has a chance of shooting them into a game, but the fact is that the Spurs are simply a much better team.  I have them winning 89% of the time with the Warriors winning one (barely edging out a sweep).  Spurs in 5 is a popular pick, as the only other choices in the Smackdown are Haberstroh taking the Spurs in 7 and Henry’s mom taking the Warriors in 7.  Henry’s mom is inexplicably winning so far, so this will either be where she blows her shot or really puts things away.

Finally we have perhaps the most interesting series of the second round, the Grizzlies and Thunder.  Just looking at the regular season, that wouldn’t be the case; the Thunder’s 5.1 better differential gives them a 87% chance of winning.  But as we all know, part of that other 13% is crazy stuff like your second-best player injuring his knee in the first round.  So the model is taking the Thunder in 5 because it doesn’t know any better, but it’s hard to imagine that the Thunder should really be favored to that extent (if at all).  In fact, the Thunder were only 3.5 point favorites today, which suggests they should be about even with the Grizzlies, and six of the eight Smackdowners are taking Memphis to win in 6.  The exceptions are Henry’s mom and Stahlhut, taking the Thunder in 6 and 7 respectively.

So there it is: Heat in 4, Knicks in 6, Spurs in 5, and Thunder in 5 (with an obvious caveat).  The Heat, Spurs, and Grizzlies will have little effect on the Smackdown overall, as they are popular choices overall, but Knicks-Pacers will play a larger role and someone might make a big move if the Warriors or Thunder manage to win their series.

As a side note, all goal differentials were multiplied by about 1.7 to square them with a usual 82-game season.  So the +53 Chicago Blackhawks are input as a +91 team, which makes them the third-best team in my database (behind the 2006 Senators and Wings).

Speaking of the Black Hawks, they’re playing the Wild, whose adjusted differential of -12 is the third-worst in my database.  The third best playing the third worst means this is the second-biggest mismatch in my set behind the 2006 Ottawa-Tampa Bay series.  Ottawa won that in 5, which I’m sure the Hawks would be happy with.  However, the thoroughly out-classed Oilers beat the Red Wings in 6 that year, and in 2010 Montreal beat Washington in 7 to fill out all the examples I have of series with a 90+ goal differential difference.  So it’s not like being a big favorite is doing anyone any favors here.  But the model says that Chicago should win 86% of the time and the Wild will win 2 games.

The Red Wings help illustrate the noise in such a short season.  Seven days ago they were out of the playoffs and had a -3 goal differential.  Then they went 4-0-0 to finish the season and end up with a +9 differential.  A 12 goal swing would be big for any team at any point, but it would even out a lot more if they were to play another 30-plus games.  But as it stands, the Wings managed to jump to 7th and draw the Anaheim Ducks instead of Chicago.  The Ducks are favored to win 60% of the time and the Wings should win 2.  Practically every NHL series is predicted to go 6, but to illustrate how close the numbers are, the Wings are expected to be swept 8.5% of the time, win one 16.8%, win two 20%, and win 3 14.5%.  Of course they win about 40% of the time.  So it isn’t like the six game series is anything like a lock.

The Vancouver Canucks get the dubious distinction (also to be shared in the next match-up) of being the home team despite having a worse goal differential than their opponent.  The Sharks were a bit better on that count but had fewer points on the record.  Since my model can’t find a significant benefit of home ice (the effect is not significant but actually slightly negative), that means the Sharks are favored but they are close enough that it’s basically a coin flip.  The Sharks win 52% of the time and the Canucks win two games.

The Blues were also behind their opponents, the Kings, but only by a single goal.  Match-ups between then 4 and 5 seed tend to be close in any sport, but this is kind of silly.  The Kings get the tiny nod with a 51% chance of winning the series and the Blues can expect to win twice.

Moving to the East we start with the Penguins and Islanders.  The Penguins might have the most obvious excuse for being underrated since Sidney Crosby missed time with a broken jaw.  He was out for 12 regular season games, but that’s a quarter of the season this year.  So when he comes back, it’s reasonable to think the Pens will be even better than the second-best differential in the league.  If that happens against the Islanders, the Pens might improve on their 80% chance to win the series (the Islanders are, of course, expected to win two).

Canadian fans have to be happy that three of their teams made the playoffs, but they’re probably less happy that three others didn’t and two have to play each other in the first round.  The Canadiens get the Ottawa Senators, but I guess on the bright side that means that at least one Canadian team will reach the second round.  Montreal is favored to win 58% of the time and Ottawa takes two.

Continuing a streak in a relatively recent rivalry, the Washington Capitals will host the NY Rangers.  The Caps got the third seed by winning their comparatively weak division (they were five points behind the 4th-seed Bruins), but they were still just ahead of the Rangers in terms of points and goal differential.  By which I mean one point and one goal.  Washington moves on 50.4% of the time and the Rangers win two.

Finally we have the Boston Bruins hosting the Toronto Maple Leafs.  The Leafs haven’t made the playoffs since the year before the lockout (you know, the big lockout), but I’m sure they won’t just be happy to be there.  They have a good chance to move on as the Bruins are expected to win 57% of the time with Toronto taking two.

So there you go.  As you can see by the percentages, and comparing them to the NBA predictions, there is a lot of uncertainty.  Only two series, the 1-8 match-ups, break 80% for the favored team whereas only two NBA series were under 80%.  Both of those were still over 70% while every other NHL series was 60% or less.  If you enjoy March Madness for the ‘anything can happen’ appeal, maybe you should try some hockey.

Starting in the West, we’ll have the Thunder hosting the Rockets.  The Rockets are a pretty entertaining team with a wide-open offense, but the Thunder are both entertaining and good.  They led the league in point differential pretty handily and won the second-most games in the league.  These two have played three times this year, with the Rockets winning one close game and the Thunder winning two blow-outs.  Seems about right to me.  With home court and a 5.7 point differential advantage, I have the Thunder winning the series 90% of the time, and the Rockets probably win one game although a sweep is the second-most-likely option.

The Spurs get the Lakers thanks to L.A. pulling out an overtime win on the last night of the season.  Had they lost and Utah won, the Lakers wouldn’t have made it at all.  Instead they get to face the third-best record (and differential) in the league without Kobe and who knows who else.  The Spurs and Lakers have played three times this year, although one of those games was just a few days ago and the Lakers may not have been fully focused on winning.  The Spurs won the other two games, although both were close.  Despite that close record, the model has the Spurs winning the series 89% of the time and the Lakers win one game.

The Nuggets and Warriors seem like an interesting match on paper.  Both are young-ish teams that have gotten some hype in recent years as teams on the move.  The Nuggets get the extra attention of apparently not having a ‘superstar’, which makes it seem impossible that they would win 57 games.  Yet they have, and although their point differential isn’t as good as the rest of the top of the West, they managed to draw the team with the worst differential on their side of the bracket.  The model likes the Nuggets to win 85% of the time with the Warriors winning one.  That matches up with the season series, which the Nuggets took 3-1 with the Warriors winning once at home by a point.

Our last match-up in the West is the Clippers hosting the Grizzlies.  This was one of my toss-ups last year, where I took the Grizzlies with home court but the Clippers moved on in 7.  This year it’s a little easier since the Clippers have home court and a better differential.  They also took the season series 3-1.  I have the Clippers moving on 76% of the time and the Grizzlies winning two games (a very narrow favorite over one game).

On to the East!  We start with probable championship favorites the Heat hosting the never-exciting Bucks.  There really isn’t much to say here besides this is the most lopsided match-up in the playoffs; the Heat don’t have the best differential in the league but it’s still 9.4 points better than the Bucks’.  I’m taking the sweep.

Next we have some old-timey basketball as the Knicks host the Celtics.  Storied franchises, etc etc.  Unfortunately for the Celtics, they just aren’t the same team as they have been recently.  They made the playoffs while playing .500 basketball and have the point differential to match (actually slightly negative; they were outscored on the season).  The Knicks have actually been good though.  I have them winning 86% of the time with the Celtics probably winning a game.

In the 3 versus 6 spot we have the Pacers hosting the Hawks.  The Pacers actually have the same record as the 4-seed Nets, but they have a much better differential.  The Hawks won more games than last year, but have a much worse differential.  Otherwise I can’t see this as a very exciting series.  The Pacers are a quality squad but mostly do it with defense; hopefully they’ll at least play at the Hawks’ pace so the games have some up-and-down to them.  I have the Pacers winning 82% of the time with the Hawks winning one.

Finally we have the Jay-Z-less Nets hosting the Derrick Rose-less Bulls.  The Bulls got the wrong end of that completely artificial comparison as Jay-Z doesn’t play basketball but Rose is almost certainly better than Hinrich and Robinson.  Even still, the Bulls have a decent chance.  The Nets are only kind of good, and the Bulls still played a bit over .500.  The Bulls even managed to win the season series 3-1, although 3 of those games were close.  And there’s always the chance that Rose will decide to play after all.  But the model knows what the model knows, so I’m taking the Nets to win in 6; overall they should win about 70% of the time.

So there it is.  The first round predictions are pretty chalky, which is typical.  Only the two 4-5 series have a decent shot at producing upsets, although you never know.  Winning 90% of the time means losing 10% of the time.  Here’s hoping for an entertaining first round, if not necessarily a surprising one.

Let’s stick with the coin flipping for a second.  Let’s say I told you I was going to flip a coin 50 times, and I wanted you to predict each flip.  How would you do it?  You would probably try to randomly guess heads or tails each time, and in the end you’d have a sequence of heads and tails in about equal proportion.

You can mock this up in Excel pretty easily instead of actually flipping coins (and I encourage you do this for the next part, when you’ll need to convince yourself of something).  In the first column, type =rand() in a cell, and then copy it down for 100 cells or so.  You should see a number between 0 and 1 in each cell.  Then do the same thing in the next column.  What we’re going to pretend is that each of those numbers is a probability.  We’ll say that the first column is the coin, and every cell that has a number under .5 is a flip that came up heads.  The second column is our guess, and every time that has a number under .5 we guessed heads.  To make this a bit more clear, in the third column type =if(A2<.5,”heads”,”tails”)   (assuming your first coin random number is in A2) and in the fourth column type =if(B2<.5,”heads”,”tails”)  (assuming your first guess random number is in B2), and then copy those down for however many flips and guesses you created.  As a comparison, make a fifth column that just says heads in every cell.  Then we’ll check to see if we were right; make a sixth column that starts with the cell =if(C2=D2,1,0) and copy it down.  If the coin result and your guess result are in C and D, this will mark a 1 every time you guessed the flip right and a 0 every time you were wrong.  Do the same thing in a seventh column for that comparison column.  Finally, pick two random cells off to the side and type =average(F2:F101) in one and =average(G2:G101) in the other  (assuming you did 100 cells under a label row like I did, and made the columns I described).  These numbers tell you how often your guess was correct and how often guessing ‘heads’ every time was correct.  Even with 100 flips there’s going to be a little noise, but both averages should be around .5, or 50%.

Ok, so what did we figure out?  If you guess coin flips and probability match (say heads or tails equally often), you’ll be at 50%.  Not surprising.  But if you made that comparison column where you guessed heads every time, you’ll also be at 50%.  What’s the big deal?  Well, this is a special case where those are both equally good strategies.  What if we had a rigged coin that came up heads 75% of the time?  We can model this by going to our flip and guess columns (columns C and D) and making them <.75 instead of <.5.  Now the coin will come up heads 75% of the time, and we’ll probability match and guess heads 75% of the time.  What you should find now is that our guesses are never correct as often as guessing heads every time.  What gives?  The problem is that we’re guessing randomly.  If there’s a 75% chance of the coin coming up heads, and a 75% chance of us guessing heads, but those two events are uncorrelated, than we’ll only say heads when the coin is heads .75*.75 = 56.25% of the time.  Similarly, we’ll get the tails call correct .25*.25 = 6.25% of the time.  Put them together and we’re right 62.5% of the time.  In contrast, by guessing heads every time we know we’re going to hit on the 75% of the time the coin comes up heads.  When you get away from 50/50 events, probability matching starts to become a bad idea.

What does this have to do with your bracket?  I think that people intuitively want to probability match to pick their games.  One of the ‘rules’ I hear every year is that there’s always one 5-12 upset, so you should pick one 12 seed to get out of the first round.  But here’s the thing; there are four 5-12 games.  Which one do you pick?  If those four games are equally likely to provide the upset, then we’re in the coin flipping situation where heads comes up 75% of the time, but instead of heads we have the 5 seed advancing.  In the long run, you’re going to be correct more often by picking every 5 seed to advance as opposed to picking one 12 seed.  The same thing goes for 4-11 and 6-10 games and so on.  The exception would be the 8 versus 9 seed; those are roughly 50/50 and as we saw before, it doesn’t matter much if you go with all 8s or mix it up.

Of course, the bracket adds a little something extra to the situation.  If all you had to do was be correct as often as possible, it would make sense to pick every favorite.  In the long run, that would give you the best outcome.  But if your goal is to win a bracket in a particular year, this no longer makes sense.  Imagine that everyone uses that strategy; everyone would end up tied.  If you can make an unusual pick that happens to be correct, you would get those points that no one else does and win the bracket.  You’re less likely to be right, but you have increased your chances of being an individual winner.  So game theory plays a bit of a role in picking brackets.  Most brackets also have rules the make different games worth different numbers of points; for example, there’s usually an ‘upset bonus’.  In the bracket I play, you get points for the seed of the team that advances: if you pick a 5 seed and they win, you get 5 points.  If you picked the 12 seed and they win, you get 12.  What does this do to our pick strategy?  Well, say we pick all four 5 seeds to advance.  We expect three to do so on average, so in an average bracket we’ll get 5*3 = 15 points.  Say we probability match and pick one 12 seed.  If we get them right, we’ll get 5*3+12 = 27 points, but that may not happen that often.  All four 5 seeds might actually win, in which case we’re back to 15 points.  But maybe a 12 seed wins, but not the one we picked.  In that case, we only get 10 points for the two 5 seeds we picked correctly.  Since we’ll only pick the right upset 6.25% of the time, we’re unlikely to hit that big 27 point round.  And if you go the other way and pick all four 12 seeds to advance, to try and make sure you hit the bonus?  If only one actually wins you only get 12 points, and miss out on the expected 15 from picking all four 5 seeds.  You can envision a payoff where it would make sense to pick upsets more often, but the payoff alone in this scenario (ignoring the game theory idea above) doesn’t cut it.  And this ignores the potential points later in the bracket; a 12 seed is unlikely to get 2 upsets in a row, but a 5 seed might win and then upset a 4 seed (Wisconsin if you’re pro-Big Ten?  VCU if you’re anti-Big Ten?), which you can never pick if you take all the 12s to advance.

So, in general it’s a bad idea to probability match if you expect non-50/50 results.  Another relevant example would be weather forecasts; if there’s a 70% chance of rain, do you take your umbrella 70% of the time?  You’ll end up wet more often than if you took the umbrella every time.  But there are some situations where game theory says not to make the same guess every time, and there are pay-off structures that might make it worthwhile to move closer to probability matching or even guessing unlikely outcomes.  The best idea would be to figure out what situation you’re in and adjust your guessing strategy appropriately.  Good luck with your bracket!

With sports, I think the most important of this article is the part about generalizability.  If you look at any of the NBA all-in-one statistics (PER, WP, RAPM, etc), they aim to give you a summary of what a player is ‘worth’ to a team.  We might imagine that such a measure will generalize; we take a sample (the preseason, the first X games of a season, one year out of a career) and assume/hope it applies to other situations (the regular season, the other 82-X games in the season, next year).  However, we don’t have a sample of the NBA; we only have what actually happens.  This means that when the circumstances change for a team or a player, what we think we know can change dramatically.  Take, for example, the year-to-year correlation chart near the end of this post.  It shows that the correlation in Wins Produced for players who change teams is much lower than for players who stayed on the same team.  Players who change teams are those who are most likely to change their role on a team or their circumstances in general (perhaps they moved on because they were injured or over/under-producing).  I haven’t seen a similar plot for other metrics, but I imagine similar results would come out.

You don’t have to change teams to have such a change though.  Imagine that the Heat switched LeBron and Ray Allen’s duties.  LeBron would run off screens and be more of a catch-and-shoot guy while Allen would handle the ball a lot, drive to the rim, and (try to) become an opponent-crushing engine of basketball destruction.  LeBron would certainly still be valuable in his new role; he’s a decent enough shooter.  Allen may or may not be good in his new role; he used to be the focal point of an offense, but he was much younger then.  I think all of our intuitions would agree that even if they were still productive players with these new responsibilities, both LeBron and Allen’s productivity would change.  But we don’t really know how much and it’s hard to say what the effect on the Heat would be as a team, since it’s likely that Wade would pick up some extra duties to help Ray out.  That is, it would be tough to generalize.  And that’s with a pretty drastic change in team strategy; what if the Heat just decided to run their offense a little differently by adding more LeBron post sets, or more Wade isolations?  We can guess at the consequences, but they would certainly be guesses.

Another way to say this is that I think it’s important to assume that a lot of what we think we know about players is context-dependent.  Related to that, a lot of what we think we know about how a player would do in a different context is complete speculation.  Four years ago, if you had asked me about J.J. Redick’s ability to lead a team, I would have been doubtful; he’s basically a catch-and-shoot guy that comes off the bench.  But the last two years seem to have demonstrated that he had (or learned) some of what it takes to be more of a focal point: his minutes and starts are up, but most of his numbers have stayed steady despite an uptick in usage.  He apparently isn’t completely limited on offense.  When people talk about Tyson Chandler’s limitations, I sort of assume that they’re right and he can’t shoot jumpers.  But maybe his coaches and teammates won’t let him, because he’s so much better at playing near the rim?  Or who’s to say that if there had been the opportunity, he could’ve learned to shoot jumpers?  We don’t know, and we can’t know because the information is never in our ‘sample’ because it isn’t really a sample.

I’ve used the NBA as an example, but the ideas really apply to all sports.  What we know about soccer players probably depends on their teams and teammates.  And the ideas apply to different levels of a single sport as well; I wouldn’t assume that what we know about the NBA applies to the WNBA or college game outside of a few basic things.  Even then I would be hesitant; I would assume that shooting threes is better than long twos at any level, for example, but I could be wrong.  There could be a league somewhere where accuracy drops dramatically enough at the three point line that it wipes out the value of the extra point.  And if such a league existed, you’d be foolish to apply what you know about it to the NBA.  You should always consider the conditions under which your data are collected before you get in a rush to talk about how they apply to other situations.

The WoW article weaves around a bit, but I think I can sum up the plus/minus part pretty succinctly: they think it does a bad job of telling you which players did well.  Not all of their points are right on-target, but the critical ones (with a tiny bit of my own expansion) are that: a) with plus/minus you can’t tell which player was specifically responsible for the team doing well in that time period; b) if you move to adjusted plus/minus to attempt to correct for that, you run into huge issues with sample size and colinearity; and c) the final results don’t seem to do a very good job of describing player quality.

Now, it’s important to note at this point that I don’t think many basketball stats guys would disagree with those three points.  Taking my usual reading habits (ESPN, WoW, APBR, a couple other things here and there) as representative, plus/minus and adjusted plus/minus simply aren’t used a lot.  When they are used, it’s almost always with a caveat, only to get a sense of defense (which isn’t necessarily captured well by the box score), or with other all-in-one metrics to get a more rounded view of a player.  A previous post of mine fleshes out some of the issues with APM and includes a number of links to other people describing and/or discussing it if you’re a curious reader.

So how about Tango?  Tango says that he likes APM for hockey (there isn’t enough scoring) but loves it for basketball.  As an aside, I imagine that if you ran your hockey APM regression using one of a logistic/mulitnomial/Poisson link, it would probably clean things up a bit.  But you would still have terrible colinearity issues, so I don’t think it would be a big help.  Moving on, plus/minus has a great intuitive hook in that it’s a with you/without you comparison; you typically run it with scoring but you could look at rebounding, fielding in baseball, whatever.  If a team has more of a good thing (or less of a bad thing) when a certain player is playing than when he isn’t playing, that’s a sign that the guy is doing something good.  Then it just becomes an issue of trying to tease out all the confounding info – the other guys on the field, who comes in to replace that player when he’s out, who the opponent has playing when he’s in versus out, and so on.  That’s why Tango says that you live for sample size: you need enough circumstances where a guy was both in and out with enough other stuff going on (various teammates, various opponents, etc) that you get a good sense of any given situation.

I think both camps here are right and there’s just a bit of miscommunication.  I wholeheartedly agree with Tango’s title: plus/minus is a GREAT concept.  And I also agree that if you have a big enough sample, it would probably do just fine.  The link he has to his catcher study sounds great.  The issue (and where I agree with the WoW article) is that, at least in the NBA, we rarely have the sample sizes needed to get APM to behave well.  And if you gather enough data to get it to behave well, now the interpretation of the number has changed quite a bit.  You can’t run a regression and scale things a bit to say “Kobe Bryant was worth 2 points a game to the Lakers in 2010″ with any confidence, but you might be able to say “Kobe Bryant was worth 2 points a game to the Lakers on average from 2005 to 2010″.  And then how useful is that?  If you wanted to know who the best player was over, say, a five year period, then I would tell you to fire up your APM calculations and have at it.  But that is rarely a question worth asking.

I want to know how good a player is now, or is likely to be next year, so that I can sign him to my team (whether I’m a real GM or a fantasy GM).  Even more helpful would be knowing why a player is good so that I can tell if he’ll mesh well with my system and the other guys on my team (if I’m a real GM; thank god that isn’t a fantasy issue).  APM doesn’t tell me that at all, although I suppose I could run regressions for any facet of the game I could think of (adjusted rebounding, adjusted assists, adjusted shooting, etc) and try to use that to create a picture of what a player contributes.  But the numbers APM gives me in any of those circumstances are going to be poor representations of the truth.  They won’t be useless; lots of stats out there have some value.  Player points per game tells you something about winning.  But there are better stats out there than APM, at least when basketball is concerned.