## Tim Tebow and the Taxonomy of Clutch

There’s nothing people love more in sports than the appearance of “clutch”ness, probably because the ability to play “up” to a situation implies a sort of super-humanity, and we love our super-heroes. Prior to this last weekend, Tim Tebow had a remarkable streak of games in which he (and his team) played significantly better in crucial 4th-quarter situations than he (or they) did throughout the rest of those contests. Combined with Tebow’s high profile, his extremely public religious conviction, and a “divine intervention” narrative that practically wrote itself, this led to a perfect storm of hype. With the din of that hype dying down a bit (thank you, Bill Belichick), I thought I’d take the chance to explore a few of my thoughts on “clutchness” in general.

This may be a bit of a surprise coming from a statistically-oriented self-professed skeptic, but I’m a complete believer in “clutch.”  In this case, my skepticism is aimed more at those who deny clutch out of hand: The principle that “Clutch does not exist” is treated as something of a sacred tenet by many adherents of the Unconventional Wisdom.

On the other hand, my belief in Clutch doesn’t necessarily mean I believe in mystical athletic superpowers. Rather, I think the “clutch” effect—that is, scenarios where the performance of some teams/players genuinely improves when game outcomes are in the balance—is perfectly rational and empirically supported.  Indeed, the simple fact that winning is a statistically significant predictive variable on top of points scored and points allowed—demonstrably true for each of the 3 major American sports—is very nearly proof enough.

The differences between my views and those of clutch-deniers are sometimes more semantic and sometimes more empirical.  In its broadest sense, I would describe “clutch” as a property inherent in players/teams/coaches who systematically perform better than normal in more important situations. From there, I see two major factors that divide clutch into a number of different types: 1) Whether or not the difference is a product of the individual or team’s own skill, and 2) whether their performance in these important spots is abnormally good relative to their performance (in less important spots), whether it is good relative to the typical performance in those spots, or both.  In the following chart, I’ve listed the most common types of Clutch that I can think of, a couple of examples of each, and how I think they break down w/r/t those factors (click to enlarge):

Here are a few thoughts on each:

### 1. Reverse Clutch

I first discussed the concept of “reverse clutch” in this post in my Dennis Rodman series.  Put simply, it’s a situation where someone has clutch-like performance by virtue of playing badly in less important situations.

While I don’t think this is a particularly common phenomenon, it may be relevant to the Tebow discussion.  During Sunday’s Broncos/Pats game, I tweeted that at least one commentator seemed to be flirting with the idea that maybe Tebow would be better off throwing more interceptions. Noting that, for all of Tebow’s statistical shortcomings, his interception rate is ridiculously low, and then noting that Tebow’s “ugly” passes generally err on the ultra-cautious side, the commentator seemed poised to put the two together—if just for a moment—before his partner steered him back to the mass media-approved narrative.

If you’re not willing to take the risks that sometimes lead to interceptions, you may also have a harder time completing passes, throwing touchdowns, and doing all those things that quarterbacks normally do to win games.  And, for the most part, we know that Tebow is almost religiously (pun intended) committed to avoiding turnovers.  However, in situations where your team is trailing in the 4th quarter, you may have no choice but to let loose and take those risks.  Thus, it is possible that a Tim Tebow who takes risks more optimally is actually a significantly better quarterback than the Q1-Q3 version we’ve seen so far this season, and the 4th quarter pressure situations he has faced have simply brought that out of him.

That may sound farfetched, and I certainly wouldn’t bet my life on it, but it also wouldn’t be unprecedented.  Though perhaps a less extreme example, early in his career Ben Roethlisburger played on a Pittsburgh team that relied mostly on its defense, and was almost painfully conservative in the passing game.  He won a ton, but with superficially unimpressive stats, a fairly low interception rate, and loads of “clutch” performances. His rookie season he passed for only 187 yards a game, yet had SIX 4th quarter comebacks.  Obviously, he eventually became regarded as an elite QB, with statistics to match.

### 2. Not Choking

A lot of professional athletes are *not* clutch, or, more specifically, are anti-clutch. See, e.g., professional kickers.  They succumb under pressure, just as any non-professionals might. While most professionals probably have a much greater capacity for handling pressure situations than amateurs, there are still significant relative imbalances between them.  The athletes who do NOT choke under pressure are thus, by comparison, clutch.

Some athletes may be more “mentally tough” than others.  I love Roger Federer, and think he is among the top two tennis player of all time (Bjorn Borg being the other), and in many ways I even think he is under-appreciated despite all of his accolades.  Yet, he has a pretty crap record in the closest matches, especially late in majors: lifetime, he is 4-7 in 5 set matches in the Quarterfinals or later, including a 2-4 record in his last 6.  For comparison, Nadal is 4-1 in similar situations (2-1 against Federer), and Borg won 5-setters at an 86% clip.

Extremely small sample, sure. But compared to Federer’s normal expectation on a set by set basis over the time-frame (even against tougher competition), the binomial probability of him losing that much without significantly diminished 5th set performance is extremely low:

Thus, as a Bayesian matter, it’s likely that a portion of Rafael Nadal’s apparent “clutchness” can be attributed to Roger Federer.

### 3. Reputational Clutch.

In the finale to my Rodman series, I discussed a fictional player named “Bjordson,” who is my amalgamation of Michael Jordan, Larry Bird, and Magic Johnson, and I noted that this player has a slightly higher Win % differential than Rodman.

Now, I could do a whole separate post (if not a whole separate series) on the issue, but it’s interesting that Bjordson also has an extremely high X-Factor: that is, the average difference between their actual Win % differential and the Win % differential that would be predicted by their Margin of Victory differential is, like Rodman’s, around 10% (around 22.5% vs. 12.5%).  [Note: Though the X-Factors are similar, this is subjectively a bit less surprising than Rodman having such a high W% diff., mostly because I started with W% diff. this time, so some regression to the mean was expected, while in Rodman’s case I started with MOV, so a massively higher W% was a shocker.  But regardless, both results are abnormally high.]

Now, I’m sure that the vast majority of sports fans presented with this fact would probably just shrug and accept that Jordan, Bird and Johnson must have all been uber-clutch, but I doubt it.  Systematically performing super-humanly better than you are normally capable of is extremely difficult, but systematically performing worse than you are normally capable of is pretty easy.  Rodman’s high X-Factor was relatively easy to understand (as Reverse Clutch), but these are a little trickier.

Call it speculation, but I suspect that a major reason for this apparent clutchiness is that being a super-duper-star has its privileges. E.g.:

In other words, ref bias may help super-stars win even more than their super-skills would dictate.

I put Tim Tebow in the chart above as perhaps having a bit of “reputational clutch” as well, though not because of officiating.  Mostly it just seemed that, over the last few weeks, the Tebow media frenzy led to an environment where practically everyone on the field was going out of their minds—one way or the other—any time a game got close late.

### 4. Skills Relevant to Endgame

Numbers 4 and 5 in the chart above are pretty closely related.  The main distinction is that #4 can be role-based and doesn’t necessarily imply any particular advantage.  In fact, you could have a relatively poor player overall who, by virtue of their specific skillset, becomes significantly more valuable in endgame situations.  E.g., closing pitchers in baseball: someone with a comparatively high ERA might still be a good “closing” option if they throw a high percentage of strikeouts (it doesn’t matter how many home runs you normally give up if a single or even a pop-up will lose the game).

Straddling 4 and 5 is one of the most notorious “clutch” athletes of all time: Reggie Miller.  Many years ago, I read an article that examined Reggie’s career and determined that he wasn’t clutch because he hit an relatively normal percentage of 3 point shots in clutch situations. I didn’t even think about it at the time, but I wish I could find the article now, because, if true, it almost certainly proves exactly the opposite of what the authors intended.

The amazing thing about Miller is that his jump shot was so ugly. My theory is that the sheer bizarreness of his shooting motion made his shot extremely hard to defend (think Hideo Nomo in his rookie year).  While this didn’t necessarily make him a great shooter under normal circumstances, he could suddenly become extremely valuable in any situations where there is no time to set up a shot and heavy perimeter defense is a given. Being able to hit ANY shots under those conditions is a “clutch” skill.

### 5. Tactical Superiority (and other endgame skills)

Though other types of skills can fit into this branch of the tree, I think endgame tactics is the area where teams, coaches, and players are most likely to have disparate impacts, thus leading to significant advantages w/r/t winning.  The simple fact is that endgames are very different from the rest of games, and require a whole different mindset. Meanwhile, leagues select for people with a wide variety of skills, leaving some much better at end-game tactics than others.

Win expectation supplants point expectation.  If you’re behind, you have to take more risks, and if you’re ahead, you have to avoid risks—even at the cost of expected value.  If you’re a QB, you need to consider the whole range of outcomes of a play more than just the average outcome or the typical outcome.  If you’re a QB who is losing, you need to throw pride out the window and throw interceptions! There is clock management, knowing when to stay in bounds and when to go down.  As a baseball manager, you may face your most difficult pitching decisions, and as a pitcher, you may have to make unusual pitch decisions.  A batter may have to adjust his style to the situation, and a pitcher needs to anticipate those adjustments.  Etc., etc., ad infinitum.  They may not be as flashy as Reggie Miller 3-ball, but these little things add up, and are probably the most significant source of Clutchness in sports.

### 6. Conditioning

I listed this separately (rather than as an example of 4 or 5) just because I think it’s not as simple and neat as it seems.

While conditioning and fitness are important in every sport, and they tend to be more important later in games, they’re almost too pervasive to be “clutch” as I described it above.  The fact that most major team sports have more or less uniform game lengths means that conditioning issue should manifest similarly basically every night, and should therefore be reflected in most conventional statistics (like minutes played, margin of victory, etc), not just in those directly related to winning.

Ultimately, I think conditioning has the greatest impact on “clutchness” in Tennis, where it is often the deciding factor in close matches

### 7. True Clutch.

And finally, we get to the Holy Grail of Clutch.  This is probably what most “skeptics” are thinking of when they deny the existence of Clutch, though I think that such denials—even with this more limited scope—are generally overstated.  If such a quality exists, it is obviously going to be extremely rare, so the various statistical studies that fail to find it prove very little.

The most likely example in mainstream sports would seem to be pre-scandal Tiger Woods.  In his prime, he had an advantage over the field in nearly every aspect of the game, but golf is a fairly high variance sport, and his scoring average was still only a point or two lower than the competition.  Yet his Sunday prowess is well documented: He has gone 48-4 in PGA tournaments when entering the final round with at least a share of the lead, including an 11-1 record with only a share of the lead.  Also, to go a bit more esoteric, Woods has successfully defended a title 22 times.  So, considering he has 71 career wins, and at least 22 of them had to be first timers, that means his title defense record is closer to 40-45%, depending on how often he won titles many times in a row.  Compare this to his overall win-rate of 27%, and the idea that he was able to elevate his game when it mattered to him the most is even more plausible.

Of course, I still contend that the most clutch thing I have ever seen is Packattack’s final jump onto the .1 wire in his legendary A11 run.  Tim Tebow, eat your heart out!

## Bayes’ Theorem, Small Samples, and WTF is Up With NBA Finals Markets?

Seriously, I am dying to post about something non-NBA related, and I should have my Open-era tennis ELO ratings by surface out in the next day or so.  But last night I finally got around to checking the betting markets to see how the NBA Finals—and thus my chances of winning the Smackdown—were shaping up, and I was shocked by what I found.  Anyway, I tossed a few numbers around, and thought you all might find them interesting.  Plus, there’s a nice little object-lesson about the usefulness of small sample size information for making Bayesian inferences.  This is actually one area where I think the normal stat geek vs. public dichotomy gets turned on its head:  Most statistically-oriented people reflexively dismiss any empirical evidence without a giant data-set.  But in certain cases—particularly those with a wide range of coherent possibilities—I think the general public may even be a little too conservative about the implications of seemingly minor statistical anomalies.

# Freaky Finals Odds:

First, I found that most books seem to see the series as a tossup at this point.  Here’s an example from a European sports-betting market:

Intuitively, this seemed off to me.  Dallas needs to win 1 out of the 2 remaining games in Miami.  Assuming the odds for both games are identical (admittedly, this could be a dubious assumption), here’s a plot of Dallas’s chances of winning the series relative to Miami’s expected winrate per home game:

So for the series to be a tossup, Miami needs to be about a 71% favorite per game.  Even at home in the playoffs, this is extremely high.  Depending on what dataset you use, the home team wins around 60-65% of the time in the NBA regular season and about 65%-70% of the time in the postseason.  But that latter number is a bit deceptive, since the playoffs are structured so that more games are played in the homes of the better teams: aside from the 2-3-2 Finals, any series that ends in an odd number of games gives the higher-seeded team (who is often much better) an extra game at home.  In fact, while I haven’t looked into the issue, that extra 5% could theoretically be less than the typical skill-disparity between home and away teams in the playoffs, which would actually make home court less advantageous than in the regular season.

Now, Miami has won only 73% of their home games this season, and it was against below-average competition (overall, they had one of the weakest schedules in the league).  Counting the playoffs, at this point Dallas actually has a better record than Miami (by one game), and they played an above-average schedule.  More importantly, the Mavs won 68% of their games on the road (compare to the league average of 35-40%).  Not to mention, Dallas is 5-2 against the Heat overall, and 2-1 against them at home (more on that later).

So how does the market tilt so heavily to this side?  Honestly, I have no idea. Many people are much more willing to dismiss seemingly incongruent market outcomes than I am.  While I obviously think the market can be beaten, when my analytical results diverge wildly from what the money says, my first inclination is to wonder what I’m doing wrong, as the odds of a massive market failure are probably lower than the odds that I made a mistake. But, in this case, with comparatively few variables, I don’t really get it.

It is a well-known phenomenon in sports-betting that huge games often have the juiciest (i.e., least efficient) lines.  This is because the smart money that normally keeps the market somewhat efficient can literally start to run out.  But why on earth would there be a massive, irrational rush to bet on the Heat?  I thought everyone hated them!

# Fun With Meta-Analysis:

So, for amusement’s sake, let’s imagine a few different lines of reasoning (I’ll call them “scenarios”) that might lead us to a range of different conclusions about the present state of the series:

1. Miami won at Home ~73% of the time while Dallas won on the road (a fairly stunning) 68% of the time.  If these values are taken at face value, a generic Miami Home team would be roughly 5% better than a generic Dallas road team, making Miami a 52.5% favorite in each game.
2. The average home team in the NBA wins about 63% of the time.  Miami and Dallas seem pretty evenly matched, so Miami should win each game ~63% of the time as well.
3. Let’s go with the very generous end of broader statistical models (discounting early-season performance, giving Miami credit for championship experience, best player, and other factors), and assume that Miami is about 5-10% better than Dallas on a neutral site.  The exact math on this is complicated (since winning is a logistic function), but, ballpark, this would translate into about a 65.5% chance at home.
4. Markets rule!  Approximate Market Price for a Miami series win is ~50%, translating into the 71% chance mentioned above above.

Here’s a scatter-plot of the chances of Dallas winning the series based on those per-game estimates:

Ignore the red dots for now—we’ll get back to those.  The blue dots are the probability of Dallas winning at least one of the next two games (using the same binomial formula as the function above).  Now, hypothetically, let’s assume you thought each of these analyses were equally plausible, your overall probability for Dallas winning the title would simply be the average of the four scenario’s results, or right around 60%.  Note: I am NOT endorsing any of these lines of reasoning or any actual conclusions about this series here—it’s just a thought experiment.

# A Little Bayesian Inference:

As I mentioned above, the Mavericks are 5-2 against the Heat this season, including 2-1 against them in Miami.  Let’s focus on the second stat: Sticking with the assumption that you found each of these 4 lines of reasoning equally plausible prior to knowing Dallas’s record in Miami, how should your newly-acquired knowledge that they were 2-1 affect your assessment?

Well, wow! 3 games is such a miniscule sample, it can’t possibly be relevant, right?  I think most people—stat geek and layperson alike—would find this statistical event pretty unremarkable.  In the abstract, they’re right: certainly you wouldn’t let such a thing invalidate a method or process built on an entire season’s worth of data. Yet, sometimes these little details can be more important than they seem.  Which brings us to perhaps the most ubiquitously useful tool discovered by man since the wheel: Bayes’ Theorem.

Bayes’ Theorem, at it’s heart, is a fairly simple conceptual tool that allows you to do probability backwards:  Garden-variety probability involves taking a number of probabilistic variables and using them to calculate the likelihood of a particular result.  But sometimes you have the result, and would like to know how it affects the probabilities of your conditions: Bayesian analysis makes this possible.

So, in this case, instead of looking at the games or series directly, we’re going to look at the odds of Dallas pulling off their 2-1 record in Miami under each of our scenarios above, and then use that information to adjust the probabilities of each.  I’ll go into the detail in a moment, but the relevant Bayesian concept is that, given a result, the new probability of each precondition will be adjusted proportionally to its prior probability of producing that result.  Looking at the red dots above (which are technically the cumulative binomial probability of Miami winning 0 or 1 out of 3 games), you should see that Dallas is far more likely to go 2-1 or better on Miami’s turf if they are an even match than if Miami is a huge favorite—over twice as likely, in fact.  Thus, we should expect that scenarios suggesting the former will become much more likely, and scenarios suggesting the latter will become much less so.

In its simplest form, Bayes’ Theorem states that the probability of A given B is equal to the probability of B given A times the prior probability of A (probability before our new information), divided by the prior probability of B:

$P(A|B)= \frac{P(B|A)*P(A)} {P(B)}$

Though our case looks a little different from this, it is actually a very simple example.  First, I’ll treat the belief that the four analyses are equally likely to be correct as a “discrete uniform distribution” of a single variable.  That sounds complicated, but it simply means that there are 4 separate options, one of which is actually correct, and each of which is equally likely. Thus, the odds of any given scenario are expressed exactly as above (B is the 2-1 outcome):

$P(S_x)= \frac{P(B|S_x)*P(S_x)} {P(B)}$

The prior probability for Sx is .25.  The prior probability of our result (the denominator) is simply the sum of the probabilities of each scenario producing that result, weighted by each scenario’s original probability.  But since these are our only options and they are all equal, that element will factor out, as follows:

$P(B)= P(S_x)*(P(B|S_1)+P(B|S_2)+P(B|S_3)+P(B|S_4))$

Since P(Sx) appears in both the numerator and the denominator, it cancels out, leaving our probability for each scenario as follows:

$P(S_x)= \frac{P(B|S_x)} {P(B|S_1)+P(B|S_2)+P(B|S_3)+P(B|S_4)}$

The calculations of P(B|Sx) are the binomial probability of Dallas winning exactly 2 out of 3 games in each case (note this is slightly different from above, so that Dallas is sufficiently punished for not winning all 3), and Excel’s binom.dist() function makes this easy.  Plugging those calculations in with everything else, we get the following adjusted probabilities for each scenario:

Note that the most dramatic changes are in our most extreme scenarios, which should make sense both mathematically and intuitively: going 2-1 is much more meaningful if you’re a big dog.

Our new weighted average is about 62%, meaning the 2-1 record improves our estimate of Dallas’s chances by 2%, making the gap between the two 4%: 62-38 (24% difference) instead of 60-40. That may not sound like much, but a few percentage points of edge aren’t that easy to come by.  For example, to a gambler, that 4% could be pretty huge: you normally need a 5% edge to beat the house (i.e., you have to win 52.5% of the time), so imagine you were the only person in the world who knew of Dallas’s miniature triumph—in this case, that info alone could get you 80% of the way to profit-land.

# Making Use:

I should note that, yes, this analysis makes some massively oversimplifying assumption—in reality, there can be gradients of truths between the various scenarios, with a variety of interactions and hidden variables, etc.—but you’d probably be surprised by how similar the results are whether you do it the more complicated way or not. One of the things that makes Bayesian inference so powerful is that it often reveals trends and effects that are relatively insulated from incidental design decisions.  I.e., the results of extremely simplified models are fairly good approximations of those produced by arbitrarily more robust calculations.  Consequently, once you get used to it, you will find that you can make quick, accurate, and incredibly useful inferences and estimates in a broad range of practical contexts.  The only downside is that, once you get started on this path, it’s a bit like getting Tetrisized: you start seeing Bayesian implications everywhere you look, and you can’t turn it off.

Of course, you also have to be careful: despite the flexibility Bayesian analysis provides, using it in abstract situations—like a meta-analysis of nebulous hypotheses based on very little new information—is very tricky business, requiring good logical instincts, a fair capacity for introspection, and much practice.  And I can’t stress enough that this is a very different beast from the typical talking head that uses small samples to invalidate massive amounts of data in support of some bold, eye-catching and usually preposterous pronouncement.

Finally, while I’m not explicitly endorsing any of the actual results of the hypo I presented above, I definitely think there are real-life equivalents where even stronger conclusions can be drawn from similarly thin data.  E.g., one situation that I’ve tested both analytically and empirically is when one team pulls off a freakishly unlikely upset in the playoffs: it can significantly improve the chances that they are better than even our most accurate models (all of which have significant error margins) would indicate.

## Game Theory in Practice: Smackdown Meta-Strategy

Going into the final round of ESPN’s Stat Geek Smackdown, I found myself 4 points behind leader Stephen Ilardi, with only 7 points left on the table: 5 for picking the final series correctly, and a bonus 2 for also picking the correct number of games.  The bottom line being, the only way I could win is if the two of us picked opposite sides.  Thus, with Miami being a clear (though not insurmountable) favorite in the Finals, I picked Dallas.  As noted in the ESPN write-up”

“The Heat,” says Morris, “have a better record, home-court advantage, a better MOV [margin of victory], better SRS [simple rating system], more star power, more championship experience, and had a tougher road to the Finals. Plus Miami’s poor early-season performance can be fairly discounted, and it has important players back from injury. Thus, my model heavily favors Miami in five or six games.

But I’m sure Ilardi knows all this, so, since I’m playing to win, I’ll take Dallas. Of course, I’m gambling that Ilardi will play it safe and stick with Miami himself since I’m the only person close enough to catch him. If he assumes I will switch, he could also switch to Dallas and sew this thing up right now. Game-theoretically, there’s a mixed-strategy Nash equilibrium solution to the situation, but without knowing any more about the guy, I have to assume he’ll play it like most people would. If he’s tricky enough to level me, congrats.

Since I actually bothered to work out the equilibrium solution, I thought some of you might be interested in seeing it. Also, the situation is well-suited to illustrate a couple of practical points about how and when you should incorporate game-theoretic strategies in real life (or at least in real games).

# Some Game Theory Basics

Certainly many of my readers are intimately familiar with game theory already (some probably much more than I am), but for those who are less so, I thought I should explain what a “mixed-strategy Nash equilibrium solution” is, before getting into the details on the Smackdown version (really, it’s not as complicated as it sounds).

A set of strategies and outcomes for a game is an “equilibrium” (often called a “Nash equilibrium”) if no player has any reason to deviate from it.  One of the most basic and most famous examples is the “prisoner’s dilemma” (I won’t get into the details, but if you’re not familiar with it already, you can read more at the link): the incentive structure of that game sets up an equilibrium where both prisoners rat on each other, even though it would be better for them overall if they both kept quiet.  “Rat/Rat” is an equilibrium because an individual deviating from it will only hurt themselves.  Bother prisoners staying silent is NOT an equilibrium, because either can improve their situation by switching strategies (note that games can also have multiple equilibriums, such as the “Which Side of the Road To Drive On” game: both “everybody drives on the left” and “everybody drives on the right” are perfectly good solutions).

But many games aren’t so simple.  Take “Rock-Paper-Scissors”:  If you pick “rock,” your opponent should pick “paper,” and if he picks “paper,” you should take “scissors,” and if you take “scissors,” he should take “rock,” etc, etc—at no point does the cycle stop with everyone happy.  Such games have equilibriums as well, but they involve “mixed” (as opposed to “pure”) strategies (trivia note: John Nash didn’t actually discover or invent the equilibrium named after him: his main contribution was proving that at least one existed for every game, using his own proposed definitions for “strategy,” “game,” etc).  Of course, the equilibrium solution to R-P-S is for each player to pick completely at random.

If you play the equilibrium strategy, it is impossible for opponents to gain any edge on you, and there is nothing they can do to improve their chances—even if they know exactly what you are going to do.  Thus, such a strategy is often called “unexploitable.”  The downside, however, is that you will also fail to punish your opponents for any “exploitable” strategies they may employ: For example, they can pick “rock” every time, and will win just as often.

# The Smackdown Game

The situation between Ilardi and I going into our final Smackdown picks is just such a game: If Ilardi picked Miami, I should take Dallas, but if I picked Dallas, he should take Dallas, in which case I should take Miami, etc.  When you find yourself in one of these “loops,” generally it means that the equilibrium solution is a mixed strategy.

Again, the equilibrium solution is the set of strategies where neither of us has any incentive to deviate.  While finding such a thing may sound difficult in theory, for 2-player games it’s actually pretty simple intuitively, and only requires basic algebra to compute.

First, you start with one player, and find their “break-even” point: that is, the strategy their opponent would have to employ for them to be indifferent between their own strategic options.  In this case, this meant: How often would I have to pick Miami for Miami and Dallas to be equally good options for Ilardi, and vice versa.

So let’s formalize it a bit:  “EV” is the function “Expected Value.”  Let’s call Ilardi or I picking Miami “iM” and “bM,” and Ilardi or I picking Dallas “iD” and “bD,” respectively.   Ilardi will be indifferent between picking Miami and Dallas when the following is true:

$EV(iM)=EV(iD)$

Let’s say “WM” = the odds of the Heat winning the series.  So now we need to find EV(iM) in terms of bM and WM.  If Ilardi picks Miami, he wins every time I pick Miami, and every time Miami wins when I pick Dallas.  Thus his expected value for picking Miami is as follows:

$EV(iM)=1*bM+WM*(1-bM)$

When he picks Dallas, he wins every time I don’t pick Miami, and every time Miami loses when I do:

$EV(iD)=1*(1-bM)+(1-WM)*bM$

Setting these two equations equal to each other, the point of indifference can be expressed as follows:

$1*bM+WM*(1-bM)=1*(1-bM)+(1-WM)*bM$

Solving for bM, we get:

$bM=(1-WM)$

What this tells us is MY equilibrium strategy.  In other words, if I pick Miami exactly as often as we expect Miami to lose, it doesn’t matter whether Ilardi picks Miami or Dallas, he will win just as often either way.

Now, to find HIS equilibrium strategy, we repeat the process to find the point where I would be indifferent between picking Miami or Dallas:

$EV(bM)=EV(bD)$

$EV(bM)=MW*(1-iM)$

$EV(bD)=(1-MW)*iM$

$MW*(1-iM)=(1-MW)*iM$

$iM=WM$

In other words, if Ilardi picks Miami exactly as often as they are expected to win, it doesn’t matter which team I pick.

Note the elegance of the solution: Ilardi should pick each team exactly as often as they are expected to win, and I should pick each team exactly as often as they are expected to lose.  There are actually a lot of theorems and such that you’d learn in a Game Theory class that make identifying that kind of situation much easier, but I’m pretty rusty on that stuff myself.

So how often would each of us win in the equilibrium solution?  To find this, we can just solve any of the EV equations above, substituting the opposing player’s optimal strategy for the variable representing the same.  So let’s use the EV(iM) equation, substituting (1-WM) anywhere bM appears:

$EV(iEq)=1*(1-WM)+WM*(1-(1-WM))$

Simplify:

$EV(iEq)=1 - WM +WM^2$

Here’s a graph of the function:

Obviously, it doesn’t matter which team is favored: Ilardi’s edge is weakest when the series is a tossup, where he should win 75% of the time.  The bigger a favorite one team is, the bigger the leader’s advantage.

Now let’s Assume Miami was expected to win 63% of the time (approximately the consensus), the Nash Equilibrium strategy would give Ilardi a 76.7% chance of winning, which is obviously considerably better than the 63% chance that he ended up with by choosing Miami to my Dallas—so the actual picks were a favorable outcome for me. Of course, that’s not to say his decision was wrong from his perspective: Either of us could have other preferences that come into play—for example, we might intrinsically value picking the Finals correctly, or someone in my spot (though probably not me) might care more about securing their 2nd-place finish than about having a chance to overtake the leader, or Ilardi might want to avoid looking bad if he “outsmarted himself” by picking Dallas while I played straight-up and stuck with Miami.

But even assuming we both wanted to maximize our chances of winning the competition, picking Miami may still have been Ilardi’s best strategy given when he knew at the time, and it would have been a fairly common outcome if we had both played game-theoretically anyway.  Which brings me to the main purpose for this post:

# A Little Meta-Strategy

In reality, neither of us played our equilibrium strategies.  I believed Ilardi would pick Miami more than 63% of the time, and thus the correct choice for me was to pick Dallas.  Assuming Ilardi believed I would pick Dallas less than 63% of the time, his best choice was to pick Miami.  Indeed, it might seem almost foolhardy to actually play a mixed strategy: what are the chances that your opponent ever actually makes a certain choice exactly 37% of the time?  Whatever your estimation, you should go with whichever gives you the better expected value, right?

This is a conundrum that should be familiar to any serious poker players out there. E.g., at the end of the hand, you will frequently find yourself in an “is he bluffing or not?” (or “should I bluff or not?”) situation.  You can work out the game-theoretically optimal calling (or bluffing) rate and then roll a die in your head.  But really, what are the chances that your opponent is bluffing exactly the correct percentage of the time?  To maximize your expected value, you gauge your opponent’s chances of bluffing, and if you have the correct pot odds, you call or fold (or raise) as appropriate.

So why would you ever play the game-theoretical strategy, rather than just making your best guess about what your opponent is doing and responding to that?  There are a couple of answers to this. First, in a repeating game, there can be strategic advantages to having your opponent know (or at least believe) that you are playing such a strategy.  But the slightly trickier—and for most people, more important—answer is that your estimation might be wrong: playing the “unexploitable” strategy is a defensive maneuver that ensures your opponent isn’t outsmarting you.

The key is that playing any “exploiting” strategy opens you up to be exploited yourself.  Think again of Rock-Paper-Scissors:  If you’re pretty sure your opponent is playing “rock” too often, you can try to exploit them by playing “paper” instead of randomizing—but this opens you up for the deadly “scissors” counterattack.  And if your opponent is a step ahead of you (or a level above you), he may have anticipated (or even set up) your new strategy, and has already prepared to take advantage.

Though it may be a bit of an oversimplification, I think a good meta-strategy for this kind of situation—where you have an equilibrium or “unexploitable” strategy available, but are tempted to play an exploiting but vulnerable strategy instead—is to step back and ask yourself the following question:  For this particular spot, if you get into a leveling contest with your opponent, who is more likely to win? If you believe you are, then, by all means, exploit away.  But if you’re unsure about his approach, and there’s a decent chance he may anticipate yours—that is, if he’s more likely to be inside your head than you are to be inside his—your best choice may be to avoid the leveling game altogether.  There’s no shame in falling back on the “unexploitable” solution, confident that he can’t possibly gain an advantage on you.

Back in Smackdown-land: Given the consensus view of the series, again, the equilibrium strategy would have given Ilardi about a 77% chance of winning.  And he could have announced this strategy to the world—it wouldn’t matter, as there’s nothing I could have done about it.  As noted above, when the actual picks came out, his new probability (63%) was significantly lower.  Of course, we shouldn’t read too much into this: it’s only a single result, and doesn’t prove that either one of us had an advantage.  On the other hand, I did make that pick in part because I felt that Ilardi was unlikely to “outlevel” me.  To be clear, this was not based on any specific assessment about Ilardi personally, but based my general beliefs about people’s tendencies in that kind of situation.

Was I right? The outcome and reasoning given in the final “picking game” has given me no reason to believe otherwise, though I think that the reciprocal lack of information this time around was a major part of that advantage.  If Ilardi and I find ourselves in a similar spot in the future (perhaps in next year’s Smackdown), I’d guess the considerations on both sides would be quite different.

## The Case for Dennis Rodman, Part 4/4(b): The Finale (Or, “Rodman v. Jordan 2”)

[ADDED: Unsurpisingly, this post has been getting a lot of traffic, which I assume includes a number of new readers who are unfamiliar with my “Case For Dennis Rodman.” So, for the uninitiated, I’d like to (at least temporarily) repeat a few of my late-comer intro points from Part 4(a): “The main things you need to know about this series are that it’s 1) extremely long (sprawling over 13 sections in 4 parts), 2) ridiculously (almost comically) detailed, and 3) only partly about Dennis Rodman.  There is a lot going on, so to help new and old readers alike, I have a newly-updated “Rodman Series Guide,” which includes a broken down list of articles, a sampling of some of the most important graphs and visuals, and a giant table summarizing the entire series by post, including the main points on both sides of the analysis.”]

So it comes down to this: With Rodman securely in the Hall of Fame, and his positive impact conclusively demonstrated by the most skeptical standards of proof I can muster, what more is there to say? Repeatedly, my research on Rodman has led to unexpectedly extreme discoveries: Rodman was not just a great rebounder, but the greatest of all time—bar none. And despite playing mostly for championship contenders, his differential impact on winning was still the greatest measured of any player with data even remotely as reliable as his. The least generous interpretation of the evidence still places Rodman’s value well within the realm of the league’s elite, and in Part 4(a) I explored some compelling reasons why the more generous interpretation may be the most plausible.

Yet even that more generous position has its limitations. Though the pool of players I compared with Rodman was broadly representative of the NBA talent pool on the whole, it lacked a few of the all-time greats—in particular, the consensus greatest: Michael Jordan. Due to that conspicuous absence, as well as to the considerable uncertainty of a process that is better suited to proving broad value than providing precise individual ratings, I have repeatedly reminded my readers that, even though Rodman kept topping these lists and metrics, I did NOT mean to suggest that Rodman was actually greater than the greatest of them all. In this final post of this series, I will consider the opposite position: that there is a plausible argument (with evidence to back it up) that Rodman’s astounding win differentials—even taken completely at face value—may still understate his true value by a potentially game-changing margin.

# A Dialogue:

First off, this argument was supposed to be an afterthought. Just a week ago—when I thought I could have it out the next morning—it was a few paragraphs of amusing speculation. But, as often seems to be the case with Dennis Rodman-related research, my digging uncovered a bit more than I expected.

The main idea has its roots in a conversation I had (over bruschetta) with a friend last summer. This friend is not a huge sports fan, nor even a huge stats geek, but he has an extremely sharp analytical mind, and loves, loves to tear apart arguments—and I mean that literally: He has a Ph.D. in Rhetoric. In law school, he was the guy who annoyed everyone by challenging almost everything the profs ever said—and though I wouldn’t say he was usually right, I would say he was usually onto something.

That night, I was explaining my then-brand new “Case for Dennis Rodman” project, which he was naturally delighted to dissect and criticize. After painstakingly laying out most of The Case—of course having to defend and explain many propositions that I had been taking for granted and needing to come up with new examples and explanations on the fly, just to avoid sounding like an idiot (seriously, talking to this guy can be intense)—I decided to try out this rhetorical flourish that made a lot of sense to me intuitively, but which had never really worked for anyone previously:

“Let me put it this way: Rodman was by far the best third-best player in NBA History.”

As I explained, “third best” in this case is sort of a term of art, not referring to quality, but to a player’s role on his team. I.e., not the player a team is built around (1st best), or even the supporting player in a “dynamic duo” (like HOF 2nd-besters Scotty Pippen or John Stockton), but the guy who does the dirty work, who mostly gets mentioned in contexts like, “Oh yeah, who else was on that [championship] team? Oh that’s right, Dennis Rodman”).

“Ah, so how valuable is the best third-best player?”

At the time, I hadn’t completely worked out all of the win percentage differentials and other fancy stats that I would later on, but I had done enough to have a decent sense of it:

“Well, it’s tough to say when it’s hard to even define ‘third-best’ player, but [blah blah, ramble ramble, inarticulate nonsense] I guess I’d say he easily had 1st-best player value, which [blah blah, something about diminishing returns, blah blah] . . . which makes him the best 3rd-best player by a wide margin”.

“How wide?”

“Well, it’s not like he’s as valuable as Michael Jordan, but he’s the best 3rd-best player by a wider margin than Jordan was the best 1st-best player.”

“So you’re saying he was better than Michael Jordan.”

“No, I’m not saying that. Michael Jordan was clearly better.”

“OK, take a team with Michael Jordan and Dennis Rodman on it. Which would hurt them more, replacing Michael Jordan with the next-best primary scoring option in NBA history, or replacing Rodman with the next-best defender/rebounder in NBA history?”

“I’m not sure, but probably Rodman.”

“So you’re saying a team should dump Michael Jordan before it should dump Dennis Rodman?”

“Well, I don’t know for sure, I’m not sure exactly how valuable other defender-rebounders are, but regardless, it would be weird to base the whole argument on who happens to be the 2nd-best player. I mean, what if there were two Michael Jordan’s, would that make him the least valuable starter on an All-Time team?”

“Well OK, how common are primary scoring options that are in Jordan’s league value-wise?”

“There are none, I’m pretty sure he has the most value.”

“BALLPARK.”

“I dunno, there are probably between 0 and 2 in the league at any given time.”

“And how common are defender/rebounder/dirty workers that are in Rodman’s league value-wise?”

“There are none.”

“BALLPARK.”

“There are none. Ballpark.”

“So, basically, if a team had Michael Jordan and Dennis Rodman on it, and they could replace either with some random player ‘in the ballpark’ of the next-best player for their role, they should dump Jordan before they dump Rodman?”

“Maybe. Um. Yeah, probably.”

“And I assume that this holds for anyone other than Jordan?”

“I guess.”

“So say you’re head-to-head with me and we’re drafting NBA All-Time teams, you win the toss, you have first pick, who do you take?”

“I don’t know, good question.”

“No, it’s an easy question. The answer is: YOU TAKE RODMAN. You just said so.”

“Wait, I didn’t say that.”

“O.K., fine, I get the first pick. I’ll take Rodman. . . Because YOU JUST TOLD ME TO.”

“I don’t know, I’d have to think about it. It’s possible.”

Up to this point, I confess, I’ve had to reconstruct the conversation to some extent, but these last two lines are about as close to verbatim as my memory ever gets:

“So there you go, Dennis Rodman is the single most valuable player in NBA History. There’s your argument.”

“Dude, I’m not going to make that argument. I’d be crucified. Maybe, like, in the last post. When anyone still reading has already made up their mind about me.”

And that’s it. Simple enough, at first, but I’ve thought about this question a lot between last summer and last night, and it still confounds me: Could being the best “3rd-best” player in NBA history actually make Rodman the best player in NBA history? For starters, what does “3rd-best” even mean? The argument is a semantic nightmare in its own right, and an even worse nightmare to formalize well enough to investigate. So before going there, let’s take a step back:

# The Case Against Dennis Rodman:

At the time of that conversation, I hadn’t yet done my league-wide study of differential statistics, so I didn’t know that Rodman would end up having the highest I could find. In fact, I pretty much assumed (as common sense would dictate) that most star-caliber #1 players with a sufficient sample size would rank higher: after all, they have a greater number of responsibilities, they handle the ball more often, and should thus have many more opportunities for their reciprocal advantage over other players to accumulate. Similarly, if a featured player can’t play—potentially the centerpiece of his team, with an entire offense designed around him and a roster built to supplement him—you would think it would leave a gaping hole (at least in the short-run) that would be reflected heavily in his differentials. Thus, I assumed that Rodman probably wouldn’t even “stat out” as the best Power Forward in the field, making this argument even harder to sell. But as the results revealed, it turns out feature players are replaceable after all, and Rodman does just fine on his own. However, there are a couple of caveats to this outcome:

First, without much larger sample sizes, I wouldn’t say that game-by-game win differentials are precise enough to settle disputes between players of similar value. For example, the standard deviation for Rodman’s 22% adjusted win differential is still 5% (putting him less than a full standard deviation above some of the competition). This is fine for concluding that he was extremely valuable, but it certainly isn’t extreme enough to outright prove the seemingly farfetched proposition that he was actually the most valuable player overall. The more unlikely you believe that proposition to be, the less you should find this evidence compelling—this is a completely rational application of Bayes’ Theorem—and I’m sure most of you, ex ante, find the proposition very very unlikely. Thus, to make any kind of argument for Rodman’s superiority that anyone but the biggest Rodman devotees would find compelling, we clearly need more than win differentials.

Second, it really is a shame that a number of the very best players didn’t qualify for the study—particularly the ultimate Big Three: Michael Jordan, Magic Johnson, and Larry Bird (who, in maybe my favorite stat ever, never had a losing month in his entire career). As these three are generally considered to be in a league of their own, I got the idea: if we treated them as one player, would their combined sample be big enough to make an adequate comparison? Well, I had to make a slight exception to my standard filters to allow Magic Johnson’s 1987 season into the mix, but here are the results:

Adjusted Win percentage differential is Rodman’s most dominant value stat, and here, finally, Herr Bjordson edges him. Plus this may not fully represent these players’ true strength: the two qualifying Jordan seasons are from his abrupt return in 1994 and his first year with the Wizards in 2001, and both of Bird’s qualifying seasons are from the last two of his career, when his play may have been hampered by a chronic back injury. Of course, just about any more-conventional player valuation system would rank these players above (or way above) Rodman, and even my own proprietary direct “all-in-one” metric puts these three in their own tier with a reasonable amount of daylight between them and the next pack (which includes Rodman) below. So despite having a stronger starting position in this race than I would have originally imagined, I think it’s fair to say that Rodman is still starting with a considerable disadvantage.

So let’s assume that at least a few players offer more direct value than Dennis Rodman. But building a Champion involves more than putting together a bunch of valuable players: to maximize your chances of success, you must efficiently allocate a variety of scare resources, to obtain as much realized value as possible, through a massively complicated set of internal and external constraints.

For example, league rules may affect how much money you can spend and how many players you can carry on your roster. Game rules dictate that you only have so many players on the floor at any given time, and thus only have so many minutes to distribute. Strategic realities require that certain roles and responsibilities be filled: normally, this means you must have a balance of talented players who play different positions—but more broadly, if you hope to be successful, your team must have the ability to score, to defend, to rebound, to run set plays, to make smart tactical maneuvers, and to do whatever else that goes into winning. All of these little things that your team has to do can also be thought of as a limited resource: in the course of a game, you have a certain number of things to be done, such as taking shots, going after loose balls, setting up a screens, contesting rebounds, etc. Maybe there are 500 of these things, maybe 1000, who knows, but there are only so many to go around—and just as with any other scarce resource, the better teams will be the ones that squeeze the most value out of each opportunity.

Obviously, some players are better at some things than others, and may contribute more in some areas than others—but there will always be trade-offs. No matter how good you are, you will always occupy a slot on the roster and a spot on the floor, every shot you take or every rebound you get means that someone else can’t take that shot or get that rebound, and every dollar your team spends on you is a dollar they can’t spend on someone else. Thus, there are two sides to a player’s contribution: how much surplus value he provides, and how much of his team’s scarce resources he consumes.

The key is this: While most of the direct value a player provides is observable, either directly (through box scores, efficiency ratings, etc.) or indirectly (Adjusted +/-, Win Differentials), many of his costs are concealed.

## Visible v. Invisible Effects

Two players may provide seemingly identical value, but at different costs. In very limited contexts this can be extremely clear: thought it took a while to catch on, by now all basketball analysts realize that scoring 25 points per game on 20 shots is better than scoring 30 points a game on 40 shots. But in broader contexts, it can be much trickier. For example, with a large enough sample size, Win Differentials should catch almost anything: everything good that a player does will increase his team’s chances of winning when he’s on the floor, and everything bad that he does will decrease his team’s chances of losing when he’s not. Shooting efficiency, defense, average minutes played, psychological impact, hustle, toughness, intimidation—no matter how abstract the skill, it should still be reflected in the aggregate.

No matter how hard the particular skill (or weakness) is to identify or understand, if its consequences would eventually impact a player’s win differentials, (for these purposes) its effects are visible.

But there are other sources of value (or lack thereof) which won’t impact a player’s win differentials—these I will call “invisible.” Some are obvious, and some are more subtle:

### Example 1: Money

“Return on Investment” is the prototypical example of invisible value, particularly in a salary-cap environment, where every dollar you spend on one player is a dollar you can’t spend on another. No matter how good a player is, if you give up more to get him than you get from him in return, your team suffers. Similarly, if you can sign a player for much less than he is worth, he may help your team more than other (or even better) players who would cost more money.

This value is generally “invisible” because the benefit that the player provides will only be realized when he plays, but the cost (in terms of limiting salary resources) will affect his team whether he is in the lineup or not. And Dennis Rodman was basically always underpaid (likely because the value of his unique skillset wasn’t fully appreciated at the time):

Note: For a fair comparison, this graph (and the similar one below) includes only the 8 qualifying Shaq seasons from before he began to decline.

Aside from the obvious, there are actually a couple of interesting things going on in this graph that I’ll return to later. But I don’t really consider this a primary candidate for the “invisible value” that Rodman would need to jump ahead of Jordan, primarily for two reasons:

First, return on investment isn’t quite as important in the NBA as it is in some other sports: For example, in the NFL, with 1) so many players on each team, 2) a relatively hard salary cap (when it’s in place, anyway), and 3) no maximum player salaries, ROI is perhaps the single most important consideration for the vast majority of personnel decisions.  For this reason, great NFL teams can be built on the backs of many underpaid good-but-not-great players (see my extended discussion of fiscal strategy in major sports here).

Second, as a subjective matter, when we judge a player’s quality, we don’t typically consider factors that are external to their actual athletic attributes. For example, a great NFL quarterback could objectively hurt his team if he is paid too much, but we still consider him great. When we ask “who’s the best point guard in the NBA,” we don’t say, “IDK, how much more does Chris Paul get paid than Jason Kidd?” Note this is basically a social preference: It’s conceivable that in some economically-obsessed culture, this sort of thing really would be the primary metric for player evaluation. But personally, and for the purposes of my argument, I prefer our more traditional values on this one.

### Example 2: Position

In the “perfect timing” department, a commenter “Siddy Hall” recently raised a hypothetical very similar to my friend’s:

You get 8 people in a room, all posing as GM’s. We’re allowed to select 5 players each from the entire history of the NBA. Then we’ll have a tournament. At PF, I would grab Rodman. And I’m confident that I’d win because he’s on my team. He’d dominate the glass and harass and shutdown a superstar. I think he’s the finest PF to ever play the game.

Of course, you need to surround him with some scorers, but when is that ever a problem?

The commenter only explicitly goes so far as to say that Rodman would be the most valuable power forward. Yet he says he is “confident” that he would win, with the only caveat being that his team gets other scorers (which is a certainty). So, he thinks Rodman is the best PF by a wide enough margin that his team would be a favorite against the team that got Michael Jordan. Let me play the role of my friend above: whether he means to or not, he’s basically saying that Rodman is more valuable than Jordan.

In this example, “position” is the scarce resource. Just as a player can be valuable for the amount of money the team must spend on him, he can also be valuable for his position. But this value can be visible, invisible, or both.

This is probably easiest to illustrate in the NFL, where positions and responsibilities are extremely rigid. An example I used in response to the commenter is that an NFL kicker who could get you 2 extra wins per season could be incredibly valuable. These two extra wins obviously have visible value: By definition, this is a player for whom we would expect to observe a 2 game per season win differential. But there’s another, very important way in which this player’s value would be much greater. As I said in response to the commenter, a +2 kicker could even be more valuable than a +4 quarterback.

In order to play the 2 win kicker, the only cost is your kicker slot, which could probably only get you a fraction of a win even if you had one of the best in the league on your team (relevant background note: kickers normally don’t contribute much, particularly since bad kickers likely influence their teams to make better tactical decisions, and vice-versa). But to play a 4-win quarterback, the cost is your quarterback slot. While the average QB and the average kicker are both worth approximately 0 games, good quarterbacks are often worth much more, and good kickers are worth very little.

Put most simply, because there are no other +2 kickers, that kicker could get 2 wins for virtually ANY team. The +4 QB would only provide 2 wins for teams who would be unable to acquire a +2 quarterback by other means. Or you can think about it conversely: Team A signs the kicker, and Team B signs the QB. For the moment, Team B might appear better, but the most value they will ever be able to get out of their QB/Kicker tandem is +4 games plus epsilon. Team A, on the other hand, can get more value out of their QB/kicker combo than Team B simply by signing any QB worth +2 or greater, who are relatively common.

Why does this matter? Well, in professional sports, we care about one thing more than any other: championships. Teams that win championships do so by having the best roster with the most value. Players like our special kicker provide unique avenues to surplus value that even great other players can’t.

To generalize a bit, you could say that value vs. a replacement player is generally visible, as it will be represented in win differentials no matter who you play for. But a player’s value relative to the entire distribution of players at his position can lead to substantial invisible benefits, as it can substantially improve his team’s ability to build a championship contender.

## Formalizing “I-Factor”

Unfortunately, in basketball, such distinctions are much more nebulous. Sure, there are “positions,” but the spot where you line up on the floor is very different from the role you play. E.g., your primary scoring responsibilities can come from any position. And even then “roles” are dynamic and loosely defined (if at all)—some roles that are crucial to certain teams don’t even exist on others. Plus, teams win in different ways: you can do it by having 5 options on offense with 5 guys that can do everything (OK, this doesn’t happen very often, but the Pistons did it in 03-04), or you can be highly specialized and try to exploit the comparative advantages between your players (this seems to be the more popular model of late).

Rodman was a specialist. He played on teams that, for the most part, didn’t ask him to do more than what he was best at—and that probably helped him fully leverage his talents. But the truly amazing part is how much of a consistent impact he could have, on such a variety of different teams, and with seemingly so few responsibilities.

So let’s posit a particular type of invisible value and call it “I-Factor,” with the following elements:

1. It improves your team’s chances of building a championship contender.
2. It wouldn’t be reflected in your game-to-game win differential.
3. It stems from some athletic or competitive skill or attribute.

In the dialogue above, I suggested that Rodman had an inordinate positive impact for a “3rd-best” player, and my friend suggested (insisted really) that this alone should vault him above great but more ordinary “1st-best” players, even if they had significantly more observable impact. Putting these two statements together, we have an examinable hypothesis: That Dennis Rodman’s value relative to his role constituted a very large “I-Factor.”

# Evaluating the Hypothesis:

Because the value we’re looking for is (by definition) invisible, its existence is ridiculously hard—if not impossible—to prove empirically (which is why this argument is the dessert instead of the main course of this series).

However, there could be certain signs and indicators we can look for that would make the proposition more likely: specifically, things that would seem unusual or unlikely if the hypothesis were false, but which could be explainable either as causes or effects of the hypothesis being true.

Since the hypothesis posits both an effect (very large I-Factor), and a cause (unusually high value for his role), we should primarily be on the lookout for two things: 1) any interesting or unusual patterns that could be explainable as a consequence of Rodman having a large I-Factor, and 2) any interesting or unusual anomalies that could help indicate that Rodman had an excessive amount of value for his role.

## Evidence of Effect:

To lighten the mood a bit, let’s start this section off with a riddle:

Q. What do you get for the team that has everything?

A. Dennis Rodman.

Our hypothetical Rodman I-Factor is much like that of our hypothetical super-kicker in the NFL example above. The reason that kicker was even more valuable than the 2 wins per season he could get you is that he could get those 2 wins for anyone. Normally, if you have a bunch of good players and you add more good players, the whole is less than the sum of its parts. In the sports analytics community, this is generally referred to as “diminishing returns.” An extremely simple example goes like this: Having a great quarterback on your team is great. Having a second great quarterback is maybe mildly convenient. Having a third great quarterback is a complete waste of space. But if you’re the only kicker in the league who is worth anywhere near 2 wins, your returns will basically never be diminished. In basketball, roles and responsibilities aren’t nearly as wed to positions as they are in football, but the principle is the same. There is only one ball, and there are only so many responsibilities: If the source of one player’s value overlaps the source of another’s, they will both have less impact. Thus, if Rodman’s hypothetical I-Factor were real, one thing we might expect to find is a similar lack of diminishing returns—in other words, an unusual degree of consistency.

And indeed, Rodman’s impact was remarkably consistent. His adjusted win differential held at between 17% and 23% for 4 different teams, all of whom were championship contenders to one extent or another. Obviously the Bulls and Pistons each won multiple championships. The two years that Rodman spent with the pre-Tim-Duncan-era Spurs, they won 55 and 62 games respectively (the latter led the league that season, though the Spurs were eliminated by eventual-champion Houston in the Western Conference Finals). In 1999, Rodman spent roughly half of the strike-shortened season on the Lakers; in that time the Lakers went 17-6, matching San Antonio’s league-leading winning percentage. But, in a move that was somewhat controversial with the Lakers players at the time, Rodman was released before the playoffs began, and the Lakers fell in the 2nd round—to the eventual-champion Spurs.

But consistency should only be evidence of invisible value if it is unusual—that is, if it exists where we wouldn’t expect it to. So let’s look at Rodman’s consistency from a couple of different angles:

### Angle 1: Money (again)

The following graph is similar to my ROI graph above, except instead of mapping the player’s salary to his win differential, I’m mapping the rest of the team’s salary to his win differential:

Note: Though obviously it’s only one data point and doesn’t mean anything, I find it amusing that the one time Shaq played for a team that had a full salary-cap’s worth of players without him, his win differential dropped to the floor.

So, basically, whether Rodman’s teams were broke or flush, his impact remained fairly constant. This is consistent with unusually low diminishing returns.

### Angle 2: Position (again)

A potential objection I’ve actually heard a couple of times is that perhaps Rodman was able to have the impact he did because the circumstances he played in were particularly well-suited to never duplicating his skill-set: E.g., both Detroit and Chicago lacked dominant big men. Indeed, it’s plausible that part of his value came from providing the defense/rebounding of a dominant center, maximally leveraging his skill-set, and freeing up his teams to go with smaller, more versatile, and more offense-minded players at other positions (which could help explain why he had a greater impact on offensive efficiency than on defensive efficiency). However, all of this value would be visible. Moreover, the assumption that Rodman only played in these situations is false. Not only did Rodman play on very different teams with very different playing styles, he actually played on teams with every possible combination of featured players (or “1st and 2nd-best” players, if you prefer):

As we saw above, Rodman’s impact on all 4 teams was roughly the same. This too is consistent with an unusual lack of diminishing returns.

## Evidence of Cause:

As I’ve said earlier, “role” can be very hard to define in the NBA relative to other sports. But to find meaningful evidence that Rodman provided an inordinate amount of value for his role, we don’t necessarily need to solve this intractable problem: we can instead look for “partial” or “imperfect” proxies. If some plausibly related proxy were to provide an unusual enough result, its actual relationship to the posited scenario could be self-reinforced—that is, the most likely explanation for the extremely unlikely result could be that it IS related to our hypothesis AND that our hypothesis is true.

So one scarce resource that is plausibly related to role is “usage.” Usage Rate is the percentage of team possessions that a player “uses” by taking a shot or committing a turnover. Shooters obviously have higher usage rates than defender/rebounders, and usage generally has little correlation with impact. But let’s take a look at a scatter-plot of qualifying players from my initial differential study (limited to just those who have positive raw win differentials):

The red dot is obviously Dennis Rodman. Bonus points to anyone who said “Holy Crap” in their heads when they saw this graph: Rodman has both the highest win differential and the lowest Usage Rate, once again taking up residence in Outlier Land.

Let’s look at it another way: Treating possessions as the scarce resource, we might be interested in how much win differential we get for every possession that a player uses:

Let me say this in case any of you forgot to think it this time:

“Holy Crap!”

Yes, the red dot is Dennis Rodman. Oh, if you didn’t see it, don’t follow the blue line, it won’t help.

This chart isn’t doctored, manipulated, or tailored in any way to produce that result, and it includes all qualifying players with positive win differentials. If you’re interested, the Standard Deviation on the non-Rodman players in the pool is .19. Yes, that’s right, Dennis Rodman is nearly 4.5 standard deviations above the NEXT HIGHEST player. Hopefully, you see the picture of what could be going on here emerging:  If value per possession is any kind of proxy (even an imperfect one) for value relative to role, it goes a long way toward explaining how Rodman was able to have such incredible impacts on so many teams with so many different characteristics.

The irony here is that the very aspect of Rodman’s game that frequently causes people to discount his value (“oh, he only does one thing”) may be exactly the quality that makes him a strong contender for first pick on the all-time NBA playground.

# Conclusions:

Though the evidence is entirely circumstantial, I find the hypothesis very plausible, which in itself should be shocking. While I may not be ready to conclude that, yes, in fact, Rodman would actually be a more valuable asset to a potential championship contender than Michael freaking Jordan, I don’t think the opposite view is any stronger: That is, when you call that position crazy, conjectural, speculative, or naïve—as some of you inevitably will—I am fairly confident that, in light of the evidence, the default position is really no less so.

In fact, even if this hypothesis isn’t exactly true, I don’t think the next-most-likely explanation is that it’s completely false, and these outlandish outcomes were just some freakishly bizarre coincidence—it would be more likely that there is some alternate explanation that may be even more meaningful. Indeed, on some level, some of the freakish statistical results associated with Rodman are so extreme that it actually makes me doubt that the best explanation could actually stem from his athletic abilities. That is, he’s just a guy, how could he be so unusually good in such an unusual way? Maybe it actually IS more likely that the groupthink mentality of NBA coaches and execs accidentally DID leave a giant exploitable loophole in conventional NBA strategy; a loophole that Rodman fortuitously stumbled upon by having such a strong aversion to doing any of the things that he wasn’t the best at. If that is the case, however, the implications of this series could be even more severe than I intended.

# Series Afterword:

Despite having spent time in law school, I’m not a lawyer. Indeed, one of the reasons I chose not to be one is because I get icky at the thought of picking sides first, and building arguments later.

In this case, I had strong intuitions about Rodman based on a variety of beliefs I had been developing about basketball value, combined with a number of seemingly-related statistical anomalies in Rodman’s record. Though I am naturally happy that my research has backed up those intuitions—even beyond my wildest expectations—I felt prepared for it to go the other way. But, of course, no matter how hard we try, we are all susceptible to bias.

Moreover, inevitably, certain non-material choices (style, structure, editorial, etc.) have to be made which emphasize the side of the argument that you are trying to defend. This too makes me slightly queasy, though I recognize it as a necessary evil in the discipline of rhetoric. My point is this: though I am definitely presenting a “case,” and it often appears one-sided, I have tried to conduct my research as neutrally as possible. If there is any area where you think I’ve failed in this regard, please don’t hesitate to let me know. I am willing to correct myself, beef up my research, or present compelling opposing arguments alongside my own; and though I’ve published this series in blog form, I consider this Case to be an ongoing project.

If you have any other questions, suggestions, or concerns, please bring them up in the comments (preferably) or email me and I will do my best to address them.

Finally, I would like to thank Nate Meyvis, Leo Wolpert, Brandon Wall, James Stuart, Dana Powers, and Aaron Nathan for the invaluable help they provided me by analyzing, criticizing, and/or ridiculing my ideas throughout this process. I’d also like to thank Jeff Bennett for putting me on this path, Scott Carder for helping me stay sane, and of course my wife Emilia for her constant encouragement.

## The Aesthetic Case Against 18 Games

By most accounts, the NFL’s plan to expand the regular season from 16 to 18 games is a done deal.  Indulge me for a moment as I take off my Bill-James-Wannabe cap and put on my dusty old Aristotle-Wannabe kausia:  In addition to various practical drawbacks, moving to 18 games risks disturbing the aesthetic harmony—grounded in powerful mathematics—inherent in the 16 game season.
Analytically, it is easy to appreciate the convenience of having the season break down cleanly into 8-game halves and 4-game quarters.  Powers of 2 like this are useful and aesthetically attractive: after all, we are symmetrical creatures who appreciate divisibility.  But we have a possibly even more powerful aesthetic attachment to certain types of asymmetrical relationships:  Mozart’s piano concertos aren’t divided into equally-sized beginnings, middles and ends.  Rather, they are broken into exposition, development, and recapitulation—each progressively shorter than the last.

Similarly, the 16 game season can fairly cleanly be broken into 3 or 4 progressively shorter but more important sections.  Using roughly the same proportions that Mozart would, the first 10 games (“exposition”) would set the stage and reveal who we should be paying attention to; the next 3-4 games (“development”) would be where the race for playoff positioning really begins in earnest, and the final 2-3 weeks (“recapitulation”) are where hopes are realized and hearts are broken—including the final weekend when post-season fates are settled.  Now, let’s represent the season as a rectangle with sides 16 (length of the season) and 10 (length of the “exposition”), broken down into consecutively smaller squares representing each section:

Note: The “last” game gets the leftover space, though if the season were longer we could obviously keep going.

At this point many of you probably know where this is going: The ratio between each square to all of the smaller pieces is roughly equal, corresponding to the “divine proportion,” which is practically ubiquitous in classical music, as well as in everything from book and movie plots to art and architecture to fractal geometry to unifying theories of “all animate and inanimate systems.”  Here it is again (incredibly clumsily-sketched) in the more recognizable spiral form:

The golden ratio is represented in mathematics by the irrational constant phi, which is:

1.6180339887…

Which, when divided into 1 gets you:

.6180339887…

Beautiful, right? So the roughly 10/4/1/1 breakdown above is really just 16 multiplied by 1/phi, with the remainder multiplied by 1/phi, etc—9.9, 3.8, 1.4, .9—rounded to the nearest game.  Whether this corresponds to your thinking about the relative significance of each portion of the season is admittedly subjective.  But this is an inescapably powerful force in aesthetics (along with symmetricality and symbols of virility and fertility), and can be found in places most people would never suspect, including in professional sports.  Let’s consider some anecdotal supporting evidence:

• The length of a Major League Baseball season is 162 games.  Not 160, but 162.  That should look familiar.
• Both NBA basketball and NHL hockey have 82-game seasons, or roughly half-phi.  Note 81 games would be impractical, because of need for equal number of home and road games (but bonus points if you’ve ever felt like the NBA season was exactly 1 game too long).
• The “exposition” portion of a half-phi season would be 50 games.  The NHL and NBA All-Star breaks both take place right around game 50, or a little later, each year.
• Though still solidly in between 1/2 and 2/3 of the way through the season, MLB’s “Summer Classic” usually takes place slightly earlier, around game 90 (though I might submit that the postseason crunch doesn’t really start until after teams build a post-All Star record for people to talk about).
• The NFL bye weeks typically end after week 10.
• Fans and even professional sports analysts are typically inclined to value “clutch” players—i.e., those who make their bones in the “Last” quadrant above—way more than a non-aesthetic analytical approach would warrant.

Etc.
So fine, say you accept this argument about how people observe sports, your next question may be: well, what’s wrong with 18 games? any number of games can be divided into phi-sized quadrants, right?  Well, the answer is basically yes, it can, but it’s not pretty:

The numbers 162, 82, and 16 all share a couple of nice qualities: first they are all roughly divisible by 4, so you have nice clean quarter-seasons.  Second, they each have aesthetically pleasing “exposition” periods: 100 games in MLB, 50 in the NBA and NHL, and 10 in the NFL.  The “exposition” period in an 18-game season would be 11 games.  Yuck!  These season-lengths balance our competing aesthetic desires for the harmony of symmetry and excitement of asymmetry.  We like our numbers round, but not too round.  We want them dynamic, but workable.

Finally, as to why the NFL should care about vague aesthetic concerns that it takes a mathematician to identify, I can only say: I don’t think these patterns would be so pervasive in science, art, and in broader culture if they weren’t really important to us, whether we know it or not.  Human beings are symmetrical down the middle, but as some guy in Italy noticed, golden rectangles are not only woven into our design, but into the design of the things we love.  Please, NFL, don’t take that away from us.

## The Case for Dennis Rodman, Part 2/4 (a)(i)—Player Valuation and Conventional Wisdom

Dennis Rodman is a – perhaps the – classic hard case for serious basketball valuation analysis.  The more you study him, the more you are forced to engage in meta-analysis: that is, examining the advantages and limitations of the various tools in the collective analytical repertoire.  Indeed, it’s even more than a hard case, it’s an extremely important one: it is just these conspicuously difficult situations where reliable analytical insight could be most useful, yet depending on which metric you choose, Rodman is either a below-average NBA player or one of the greatest of all time.  Moreover, while Rodman may be an “extreme” of sorts, this isn’t Newtonian Physics: the problems with player valuation modeling that his case helps reveal – in both conventional and unconventional forms – apply very broadly.

This section will use Dennis Rodman as a case study for my broader critique of both conventional and unconventional player valuation methods.  Sub-section (i) introduces my criticism and deals with conventional wisdom, and sub-section (ii) deals with unconventional wisdom and beyond.  Section (b) will then examine how valuable Rodman was specifically, and why.  Background here, here, here, here, and here.

# First – A Quick Meta-Critique:

Why is it that so many sports-fans pooh-pooh advanced statistical analysis, yet, when making their own arguments, spout nothing but statistics?

• [So-and-so] scored 25 points per game last season, solidifying their position in the NBA elite.
• [Random QB] had ten 3000-yard passing seasons, he is sooo underrated.
• [Player x]’s batting average is down 50 points, [team y] should trade him while they still can.

Indeed, the vast majority of people are virtually incapable of making sports arguments that aren’t stats-based in one way or another.  Whether he knows it or not, Joe Average is constantly learning and refining his preferred models, which he then applies to various problems, for a variety of purposes — not entirely unlike Joe Academic.  Yet chances are he remains skeptical of the crazy-talk he hears from the so-called “statistical experts” — and there is truth to this skepticism: a typical “fan” model is extremely flexible, takes many more variables from much more diverse data into account, and ultimately employs a very powerful neural network to arrive at its conclusions.  Conversely, the “advanced” models are generally rigid, naïve, over-reaching, hubristic, prove much less than their creators believe, and claim even more.  Models are to academics like screenplays are to Hollywood waiters: everyone has one, everyone thinks theirs is the best, and most of them are garbage.  The broad reliability of “common sense” over time has earned it the benefit of the doubt, despite its high susceptibility to bias and its abundance of easily-provable errors.

The key is this: While finding and demonstrating such error is easy enough, successfully doing so should not – as it so often does – lead one (or even many) to presume that it qualifies them to replace that wisdom, in its entirety, with their own.

I believe something like this happened in the basketball analytic community:  reacting to the manifest error in conventional player valuation, the statisticians have failed to recognize the main problem – one which I will show actually limits their usefulness – and instead have developed an “unconventional” wisdom that ultimately makes many of the same mistakes.

# Conventional Wisdom – Points, Points, Points:

The standard line among sports writers and commentators today is that Dennis Rodman’s accomplishments “on the court” would easily be sufficient to land him in the Hall of Fame, but that his antics “off the court” may give the voters pause.  This may itself be true, but it is only half the story:  If, in addition to his other accomplishments, Rodman had scored 15 points a game, I don’t think we’d be having this discussion, or really even close to having this discussion (note, this would be true whether or not those 15 points actually helped his teams in any way).  This is because the Hall of Fame reflects the long-standing conventional wisdom about player valuation: that points (especially per game) are the most important measure of a player’s (per game) contribution.
Whether most people would explicitly endorse this proposition or not, it is still reflected in systematic bias.  The story goes something like this:  People watch games to see the players do cool things, like throw a ball from a long distance through a tiny hoop, and experience pleasure when it happens.  Thus, because pleasure is good, they begin to believe that those players must be the best players, which is then reinforced by media coverage that focuses on point totals, best dunks plays of the night, scoring streaks, scoring records, etc.  This emphasis makes them think these must also be the most important players, and when they learn about statistics, that’s where they devote their attention.  Everyone knows about Kobe’s 81 points in a game, but how many people know about Scott Skiles’s 30 assists? or Charles Oakley’s 35 rebounds? or Rodman’s 18 offensive boards? or Shaq’s 15 blocks?  Many fans even know that Mark Price is the all-time leader in free throw percentage, or that Steve Kerr is the all-time leader in 3 point percentage, but most have never even heard of rebound percentage, much less assist percentage or block percentage.  And, yes, for those who vote for the Hall of Fame, it is also reflected in their choices.  Thus, before dealing with any fall-out for his off-court “antics,” the much bigger hurdle to Dennis Rodman’s induction looks like this:

This list is the bottom-10 per-game scorers (of players inducted within 25 years of their retirement).  If Rodman were inducted, he would be the single lowest point-scorer in HoF history.  And looking at the bigger picture, it may even be worse than that.  Here’s a visual of all 89 Hall of Famers with stats (regardless of induction time), sorted from most points to fewest:

So not only would he be the lowest point scorer, he would actually have significantly fewer points than a (linear) trend-line would predict the lowest point scorer to have (and most of the smaller bars just to the left of Rodman were Veteran’s Committee selections).  Thus, if historical trends reflect the current mood of the HoF electorate, resistance is to be expected.

The flip-side, of course, is the following:

Note: this graphic only contains the players for whom this stat is available, though, as I demonstrated previously, there is no reason to believe that earlier players were any better.
Clearly, my first thought when looking at this data was, “Who the hell is this guy with a TRB% of only 3.4?”  That’s only 1 out of every *30* rebounds!* The league average is (obviously) 1 out of 10.  Muggsy Bogues — the shortest player in the history of the NBA (5’3”) — managed to pull in 5.1%, about 1 out of every 20.  On the other side, of course, Rodman would pace the field by a wide margin – wider, even, than the gap between Jordan/Chamberlain and the field for scoring (above).  Of course, the Hall of Fame traditionally doesn’t care that much about rebounding percentages:

So, of eligible players, 24 of the top 25 leaders in points per game are presently in the Hall (including the top 19 overall), while only 9 of the top 25 leaders in total rebound percentage can say the same.  This would be perfectly rational if, say, PPG was way way more important to winning than TRB%.  But this seems unlikely to me, for at least two reasons: 1) As a rate stat, TRB% shouldn’t be affected significantly by game or team pace, as PPG is; and 2) TRB% has consequences on both offense and defense, whereas PPG is silent about the number of points the player/team has given up.  To examine this question, I set up a basic correlation of team stats to team winning percentage for the set of every team season since the introduction of the 3-point shot.  Lo and behold, it’s not really close:

Yes, correlation does not equal causation, and team scoring and rebounding are not the same as individual scoring and rebounding.  This test isn’t meant to prove conclusively that rebounding is more important than scoring, or even gross scoring — though, at the very least, I do think it strongly undermines the necessity of the opposite: that is, the assumption that excellence in gross point-scoring is indisputably more significant than other statistical accomplishments.
Though I don’t presently have the data to confirm, I would hypothesize (or, less charitably, guess) that individual TRB% probably has a more causative effect on team TRB% than individual PPG does on team PPG [see addendum] (note, to avoid any possible misunderstanding, I mean this only w/r/t PPG, not points-per-possession, or anything having to do with shooting percentages, true or otherwise).  Even with the proper data, this could be a fairly difficult hypothesis to test, since it can be hard to tell (directly) whether a player scoring a lot of points causes his team to score a lot of points, or vice versa.  However, that hypothesis seems to be at least partially supported by studies that others have conducted on rebound rates – especially on the offensive side (where Rodman obviously excelled).

The conventional wisdom regarding the importance of gross points is demonstrably flawed on at least two counts: gross, and points.  In sub-section (ii), I will look at how the analytical community attempted to deal with these problems, as well as at how they repeated them.
*(It’s Tiny Archibald)

I posted this as a Graph of the Day a while back, and thought I should add it here:

More info in the original post, but the upshot is that my hypothesis that “individual TRB% probably has a more causative effect on team TRB% than individual PPG does on team PPG” appears to be confirmed (the key word is “differential”).

## Applied Epistemology in Politics and the Playoffs

Two nights ago, as I was watching cable news and reading various online articles and blog posts about Christine O’Donnell’s upset win over Michael Castle in Delaware’s Republican Senate primary, the hasty, almost ferocious emergence of consensus among the punditocracy – to wit, that the GOP now has virtually zero chance of picking up that seat in November – reminded me of an issue that I’ve wanted to blog about since long before I began blogging in earnest: NFL playoff prediction models.

Specifically, I have been critical of those models that project the likelihood of each surviving team winning the Super Bowl by applying a logistic regression model (i.e., “odds of winning based on past performance”) to each remaining game.  In January, I posted a number of comments to this article on Advanced NFL Stats, in which I found it absurd that, with 8 teams left, his model predicted that the Dallas Cowboys had about the same chance of winning the Super Bowl as the Jets, Ravens, Vikings, and Cardinals combined. In the brief discussion, I gave two reasons (in addition to my intuition): first, that these predictions were wildly out of whack with contract prices in sports-betting markets, and second, that I didn’t believe the model sufficiently accounted for “variance in the underlying statistics.”  Burke suggested that the first point is explained by a massive epidemic of conjunction-fallacyitis among sports bettors.  On its face, I think this is a ridiculous explanation: i.e., does he really believe that the market-movers in sports betting — people who put up hundreds of thousands (if not millions) of dollars of their own money — have never considered multiplying the odds of several games together?  Regardless, in this post I will put forth a much better explanation for this disparity than either of us proffered at the time, hopefully mooting that discussion.  On my second point, he was more dismissive, though I was being rather opaque (and somehow misspelled “beat” in one reply), so I don’t blame him.  However, I do think Burke’s intellectual hubris regarding his model (aka “model hubris”) is notable – not because I have any reason to think Burke is a particularly hubristic individual, but because I think it is indicative of a massive epidemic of model-hubrisitis among sports bloggers.

In Section 1 of this post, I will discuss what I personally mean by “applied epistemology” (with apologies to any actual applied epistemologists out there) and what I think some of its more-important implications are.  In Section 2, I will try to apply these concepts by taking a more detailed look at my problems with the above-mentioned playoff prediction models.

# Section 1: Applied Epistemology Explained, Sort Of

For those who might not know, “epistemology” is essentially a fancy word for the “philosophical study of knowledge,” which mostly involves philosophers trying to define the word “knowledge” and/or trying to figure out what we know (if anything), and/or how we came to know it (if we do).  For important background, read my Complete History of Epistemology (abridged), which can be found here: In Plato’s Theaetetus, Socrates suggests that knowledge is something like “justified true belief.”  Agreement ensues.  In 1963, Edmund Gettier suggests that a person could be justified in believing something, but it could be true for the wrong reasons.  Debate ensues.  The End.

A “hot” topic in the field recently has been dealing with the implications of elaborate thought experiments similar to the following:

*begin experiment*

*end experiment*

In reality, the fact that you might be wrong, even when you’re so sure you’re right, is more than a philosophical curiosity, it is a mathematical certainty.  The processes that lead you to form beliefs, even extremely strong ones, are imperfect.  And when you are 100% certain that a belief-generating process is reliable, the process that led you to that belief is likely imperfect.  This line of thinking is sometimes referred to as skepticism — which would be fine if it weren’t usually meant as a pejorative.

When push comes to shove, people will usually admit that there is at least some chance they are wrong, yet they massively underestimate just what those chances are.  In political debates, for example, people may admit that there is some miniscule possibility that their position is ill-informed or empirically unsound, but they will almost never say that they are more likely to be wrong than to be right.  Yet, when two populations hold diametrically opposed views, either one population is wrong or both are – all else being equal, the correct assessment in such scenarios is that no-one is likely to have it right.

When dealing with beliefs about probabilities, the complications get even trickier:  Obviously many people believe some things are close to 100% likely to be true, when the real probability may be some-much if not much-much lower.  But in addition to the extremes, people hold a whole range of poorly-calibrated probabilistic beliefs, like believing something is 60% likely when it is actually 50% or 70%.  (Note: Some Philosophically trained readers may balk at this idea, suggesting that determinism entails everything having either a 0 or 100% probability of being true.  While this argument may be sound in classroom discussions, it is highly unpragmatic: If I believe that I will win a coin flip 60% of the time, it may be theoretically true that the universe has already determined whether the coin will turn up heads or tails, but for all intents and purposes, I am only wrong by 10%).

But knowing that we are wrong so much of the time doesn’t tell us much by itself: it’s very hard to be right, and we do the best we can.  We develop heuristics that tend towards the right answers, or — more importantly for my purposes — that allow the consequences of being wrong in both directions even out over time.  You may reasonably believe that the probability of something is 30%, when, in reality, the probability is either 20% or 40%.  If the two possibilities are equally likely, then your 30% belief may be functionally equivalent under many circumstances, but they are not the same, as I will demonstrate in Section 2 (note to the philosophers: you may have noticed that this is a bit like the Gettier examples: you might be “right,” but for the wrong reasons).

There is a science to being wrong, and it doesn’t mean you have to mope in your study, or act in bad faith when you’re out of it.  “Applied Epistemology” (at least as this armchair philosopher defines it) is the study of the processes that lead to knowledge and beliefs, and of the practical implications of their limitations.

## Part 2:  NFL Playoff Prediction Models

Now, let’s finally return to the Advanced NFL Stats playoff prediction model.  Burke’s methodology is simple: using a logistic regression based on various statistical indicators, the model estimates a probability for each team to win their first round matchup.  It then repeats the process for all possible second round matchups, weighting each by its likelihood of occurring (as determined by the first round projections) and so on through the championship.  With those results in hand, a team’s chances of winning the tournament is simply the product of their chances of winning in each round.  With 8 teams remaining in the divisional stage, the model’s predictions looked like this:

Burke states that the individual game prediction model has a “history of accuracy” and is well “calibrated,” meaning that, historically, of the teams it has predicted to win 30% of the time, close to 30% of them have won, and so on.  For a number of reasons, I remain somewhat skeptical of this claim, especially when it comes to “extreme value” games where the model predicts very heavy favorites or underdogs.  (E.g’s:  What validation safeguards do they deploy to avoid over-fitting?  How did they account for the thinness of data available for extreme values in their calibration method?)  But for now, let’s assume this claim is correct, and that the model is calibrated perfectly:  The fact that teams predicted to win 30% of the time actually won 30% of the time does NOT mean that each team actually had a 30% chance of winning.

That 30% number is just an average.  If you believe that the model perfectly nails the actual expectation for every team, you are crazy.  Since there is a large and reasonably measurable amount of variance in the very small sample of underlying statistics that the predictive model relies on, it necessarily follows that many teams will have significantly under or over-performed statistically relative to their true strength, which will be reflected in the model’s predictions.  The “perfect calibration” of the model only means that the error is well-hidden.

This doesn’t mean that it’s a bad model: like any heuristic, the model may be completely adequate for its intended context.  For example, if you’re going to bet on an individual game, barring any other information, the average of a team’s potential chances should be functionally equivalent to their actual chances.  But if you’re planning to bet on the end-result of a series of games — such as in the divisional round of the NFL playoffs — failing to understand the distribution of error could be very costly.

For example, let’s look at what happens to Minnesota and Arizona’s Super Bowl chances if we assume that the error in their winrates is uniformly distributed in the neighborhood of their predicted winrate:

For Minnesota, I created a pool of 11 possible expectations that includes the actual prediction plus teams that were 5% to 25% better or worse.  I did the same for Arizona, but with half the deviation.  The average win prediction for each game remains constant, but the overall chances of winning the Super Bowl change dramatically.  To some of you, the difference between 2% and 1% may not seem like much, but if you could find a casino that would regularly offer you 100-1 on something that is actually a 50-1 shot, you could become very rich very quickly.  Of course, this uniform distribution is a crude one of many conceivable ways that the “hidden error” could be distributed, and I have no particular reason to think it is more accurate than any other.  But one thing should be abundantly clear: the winrate model on which this whole system rests tells us nothing about this distribution either.

The exact structure of this particular error distribution is mostly an empirical matter that can and should invite further study.  But for the purposes of this essay, speculation may suffice.  For example, here is an ad hoc distribution that I thought seemed a little more plausible than a uniform distribution:

This table shows the chances of winning the Super Bowl for a generic divisional round playoff team with an average predicted winrate of 35% for each game.  In this scenario, there is a 30% chance (3/10) that the prediction gets it right on the money, a 40% chance that the team is around half as good as predicted (the bottom 4 values), a 10% chance that the team is slightly better, a 10% chance that it is significantly better, and a 10% chance that the model’s prediction is completely off its rocker.  These possibilities still produce a 35% average winrate, yet, as above, the overall chances of winning the Super Bowl increase significantly (this time by almost double).  Of course, 2 random hypothetical distributions don’t yet indicate a trend, so let’s look at a family of distributions to see if we can find any patterns:

This chart compares the chances of a team with a given predicted winrate to win the Super Bowl based on uniform error distributions of various sizes.  So the percentages in column 1 are the odds of the team winning the Super Bowl if the predicted winrate is exactly equal to their actual winrate.  Then each subsequent column is the chances of them winning the Superbowl if you increase the “pool” of potential actual winrates by one on each side.  Thus, the second number after 35% is the odds of winning the Super Bowl if the team is equally likely to be have a 30%, 35%, or 40% chance in reality, etc.  The maximum possible change in Super Bowl winning chances for each starting prediction is contained in the light yellow box at the end of each row.  I should note that I chose this family of distributions for its ease of cross-comparison, not its precision.  I also experimented with many other models that produced a variety of interesting results, yet in every even remotely plausible one of them, two trends – both highly germane to my initial criticism of Burke’s model – endured:
1.  Lower predicted game odds lead to greater disparity between predicted and actual chances.
To further illustrate this, here’s a vertical slice of the data, containing the net change for each possible prediction, given a discreet uniform error distribution of size 7:

2.  Greater error ranges in the underlying distribution lead to greater disparity between predicted and actual chances.

To further illustrate this, here’s a horizontal slice of the data, containing the net change for each possible error range, given an initial winrate prediction of 35%:

Of course these underlying error distributions can and should be examined further, but even at this early stage of inquiry, we “know” enough (at least with a high degree of probability) to begin drawing conclusions.  I.e., We know there is considerable variance in the statistics that Burke’s model relies on, which strongly suggests that there is a considerable amount of “hidden error” in its predictions.  We know greater “hidden error” leads to greater disparity in predicted Super Bowl winning chances, and that this disparity is greatest for underdogs.  Therefore, it is highly likely that this model significantly under-represents the chances of underdog teams at the divisional stage of the playoffs going on to win the Superbowl.  Q.E.D.

This doesn’t mean that these problems aren’t fixable: the nature of the error distribution of the individual game-predicting model could be investigated and modeled itself, and the results could be used to adjust Burke’s playoff predictions accordingly.  Alternatively, if you want to avoid the sticky business of characterizing all that hidden error, a Super-Bowl prediction model could be built that deals with that problem heuristically: say, by running a logistical regression that uses the available data to predict each team’s chances of winning the Super Bowl directly.

Finally, I believe this evidence both directly and indirectly supports my intuition that the large disparity between Burke’s predictions and the corresponding contract prices was more likely to be the result of model error than market error.  The direct support should be obvious, but the indirect support is also interesting:  Though markets can get it wrong just as much or more than any other process, I think that people who “put their money where their mouth is” (especially those with the most influence on the markets) tend to be more reliably skeptical and less dogmatic about making their investments than bloggers, analysts or even academics are about publishing their opinions.  Moreover, by its nature, the market takes a much more pluralistic approach to addressing controversies than do most individuals.  While this may leave it susceptible to being marginally outperformed (on balance) by more directly focused individual models or persons, I think it will also be more likely to avoid pitfalls like the one above.

## Conclusions, and My Broader Agenda

The general purpose of post is to demonstrate both the importance and difficulty of understanding and characterizing the ways in which our beliefs – and the processes we use to form them — can get it wrong.  This is, at its heart, a delicate but extremely pragmatic endeavor.  It involves being appropriately skeptical of various conclusions — even when they seem right to you – and recognizing the implications of the multitude of ways that such error can manifest.

I have a whole slew of ideas about how to apply these principles when evaluating the various pronouncements made by the political commentariat, but the blogosphere already has a Nate Silver (and Mr. Silver is smarter than me anyway), so I’ll leave that for you to consider as you see fit.