Fun with Excel #16 – Rigging Live Draws: The Emirates FA Cup

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The Fifth Round Draw of the 2016/17 Emirates FA Cup was rigged.

Bold statement (literally), although that sentence probably meant nothing to anyone who doesn’t follow English Football (re: soccer) and the FA Cup in particular.

A quick introduction to the FA Cup competition, courtesy of Wikipedia (emphasis mine):

The FA Cup, known officially as The Football Association Challenge Cup, is an annual knockout association football competition in men’s domestic English football. First played during the 1871–72 season, it is the oldest association football competition in the world. For sponsorship reasons, from 2015 through to 2018 it is also known as The Emirates FA Cup.

The competition is open to any eligible club down to Levels 10 of the English football league system – all 92 professional clubs in the Premier League and the English Football League (Levels 1 to 4), and several hundred “non-league” teams in Steps 1 to 6 of the National League System (Levels 5 to 10). The tournament consists of 12 randomly drawn rounds followed by the semi-finals and the final. Entrants are not seeded, although a system of byes based on league level ensures higher ranked teams enter in later rounds – the minimum number of games needed to win the competition ranges from six to fourteen.

In the modern era, only one non-league team has ever reached the quarter finals, and teams below Level 2 have never reached the final. As a result, as well as who wins, significant focus is given to those “minnows” (smaller teams) who progress furthest, especially if they achieve an unlikely “giant-killing” victory.

It’s no secret that when it comes to the FA Cup, “giant-killing” victories are more exciting to the average viewer, and therefore better for TV ratings. Therefore, the tournament organizers are incentivized to create as many “minnow-giant” match-ups as possible. Specifically, this means matching up teams from the top level of the English football league system (more commonly known as the English Premier League, or EPL) with teams from lower levels (2nd Tier = Championship, 3rd Tier = League One, 4th Tier = League Two, 5th Tier = National League, etc.) While match-ups in the first 12 rounds of the tournament are determined using “randomly drawn” balls, it has been shown that such live draw events can be effectively rigged by cooling or freezing certain balls.

This year’s FA Cup Fifth Round Draw provided an interesting case study to test the rigging hypothesis, because out of the 16 teams going into the Fifth Round, 8 of them were from the EPL (Tier 1), while the remaining 8 were all from lower divisions. Coincidentally, the 8 EPL teams just happened to get drawn against the 8 non-EPL teams, conveniently leading to the maximum number of 8 “minnow-giant” match-ups. This result should seem suspicious even if you are not familiar with probability theory, but to illustrate just how unlikely such a result is, I will walk through the math.

In order to calculate the probability of the aforementioned result, we first need to figure out the total number of match-ups (i.e. pairs) that can be arranged among a group of 16 teams. As with most problems in mathematics, there is more than one solution, but perhaps the most intuitive one is this: Take one of the 16 teams at random. That first team can be paired up with 15 possible other teams. After a pair is made, 14 teams will remain. Again, we take one of the 14 teams at random. This team can be paired up with 13 possible other teams. By repeating this logic, we see that there are a total of 15x13x11x9x7x5x3x2x1=2,027,025 unique pairs. It turns out that mathematicians already have a function that simplifies this exact result: the double factorial (expressed as n!!). Therefore, we can generalize that for any group of n objects, the number of unique pairings is equal to (n-1)!!

To calculate the total number of ways to draw exactly 8 “minnow-giant” match-ups, we can imagine putting all 8 of the EPL teams in a line. Since we are looking to match the EPL teams one-to-one with the non-EPL teams, the question becomes: how many different ways can we line up the non-EPL teams so that they are paired up with the EPL teams? The answer to that is simply 8x7x6x5x4x3x2x1=8!=40,320. It is important to understand why we keep the order of the EPL teams unchanged while we only change the order of the non-EPL teams; otherwise, we would be grossly over-counting!

The probability of drawing exactly 8 “minnow-giant” match-ups is therefore 40,320/2,027,025=1.99%, or just a tad under 2%! To verify this, I ran a Monte Carlo simulation involving 50,000 trials, of which 961 trials ended up with exactly 8 “minnow-giant” match-ups, or 1.92%. The below table and chart also show the theoretical probabilities of drawing n “minnow-giant” match-ups, for 0 ≤ n ≤ 8. (Bonus Question: Can you convince yourself why it’s impossible to draw an odd number of “minnow-giant” pairs among a group of 16 teams?)


But wait, it gets even better. Out of the 8 non-EPL teams, 4 teams were from the Championship (2nd Tier league), 2 teams were from League One (3rd Tier), and 2 teams were from the National League (5th Tier). Arsenal, which has been sponsored by Emirates since 2006, ended up drawing Sutton United, one of only two teams (the other being Lincoln City) from the National League (5th Tier). Now, what are the chances that the team that shares a sponsor with the competition itself ends up drawing one of the two easiest (in theory) match-ups available?

The number of ways for Arsenal to draw a National League (5th Tier) team (i.e. either Sutton United or Lincoln City), without any restrictions on how the other match-ups are drawn, is 270,270. We arrive at this number by first assuming Arsenal and Sutton United are already paired off, thus leaving 14 teams reaming. The 14 teams can be paired off in 13!!=135,135 ways without restriction. We can repeat the same reasoning for an Arsenal/Lincoln City pair. Therefore, we double 135,135 to arrive at 270,270. This yields a theoretical probability of 270,270/2,027,025=13.33% (Monte Carlo resulted in 6,620/50,000=13.24%), which is almost 1 in 6. However, this is only the probability of Arsenal drawing a 5th Tier team with no other match-up restrictions. In reality, there were already 8 “minnow-giant” match-ups drawn in the first place.

Therefore, the question becomes: what is the probability that 8 “minnow-giant” match-ups are drawn AND Arsenal draws a 5th Tier team? We already know there are 40,320 possible match-ups for the first part of the requirement. Satisfying both parts of the requirement must result in a number smaller than 40,320. Think of it like this: we start off with the fact that the 8 EPL teams are matched up one-to-one with the 8 non-EPL teams. There are 2 different ways to pair Arsenal with a 5th Tier team (since there are only 2 such teams). Of the remaining teams, there are 7!=5,040 ways to pair them off such that the EPL and non-EPL teams are still matched one-to-one. Therefore, the total number of match-ups satisfying both requirements is 2×7!=10,080. This yields a theoretical probability of 10,080/2,027,025=0.50% (Monte Carlo resulted in 250/50,000=0.50%).

In conclusion, there was only a 0.50% chance that the 2016/17 Emirates FA Cup Fifth Round Draw would lead to exactly 8 “minnow-giant” match-ups AND Arsenal drawing 1 of the 2 National League (5th Tier) teams. The fact that it happened anyway suggests that the drawing process may not have been 100% random.

As always, you can find my back up here. Please note, however, that I had to change all of the Monte Carlo formulas to values and save the file as .xlsb instead of .xlsx, as the file was way too large before (71 MB).

I would also like to give credit to the Chelsea subreddit for inspiring me to explore this topic.

Fun with Excel #15 – Fantasy Football: The Value of Optimal Play

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Another year, another season, another Fantasy Football championship. Well, almost. We made it to the Finals for the second year in a row, but sadly lost to the 12-1 team in our league by a score of 144.7 to 156.6. Suddenly, my last post on match-ups now seems very perspicacious:

Everyone has had that one week where they score 150 points, only for their opponent to somehow put up 160).

Did I just quote myself so that I could use “perspicacious” in a sentence? Yes, yes I did. Interestingly enough, the eventual winner also won our league two years ago, meaning that the same two teams have combined for 3 out of the last 3 Championships and 4 out of the last 6 Finals appearances. Coincidence?

In this post, I will examine the importance of optimal play in Fantasy Football.

Defining Optimal Play

In the strictest sense, playing optimally means that a manager is maximizing his or her total points scored on a weekly basis. In a perfect world, this would entail picking up the best free agents by position and starting them throughout the course of the season (e.g. acquiring Jack Doyle (TE) from the waivers and starting him in Week 1 when he got 18.5 points, the highest point total for all TEs that week). While such a strategy is theoretically possible, the large amount of roster churn required as well as the significant amount of risk involved make this strategy almost impossible to implement in any sort of standard league.

A more practical interpretation of optimal play is a manager’s ability to choose the best starting lineup given his or her current roster for that week. By defining optimal play like this, we split the manager’s abilities into two distinct buckets: (1) all the actions required to arrive at his or her current roster (i.e. drafting, waiver wire acquisitions, trading) and (2) setting the best starting lineup. Isolating the impact of (2) is a relatively easy exercise to do in retrospect (since all the data is available), and that’s exactly what I did over the 13-week Regular Season.

Of course, it should go without saying that the subsequent analysis does not account for the inherent randomness in Fantasy Football (i.e. variability in player performance, match-ups/schedule, etc.)

Results

I reviewed the Regular Season performance (i.e. Weeks 1-13) for all 14 teams in our league and calculated each team’s hypothetical optimal score every week, as well as the number of points below optimal (PBO) that each team actually scored. The smaller the PBO, the better.

Here’s a sample of what that looked like in Excel:

The following tables show a summary of the actual, optimal, and PBO results for each team over the Regular Season:

The average PBO per week was 13.0 in our league, with a standard deviation of 3.4. Out of the 14 teams, 11 teams performed within one standard deviation of the mean. Team 8 had an average PBO per week of only 7.3, while Teams 7 and 9 were on the other end of the spectrum, boasting PBO values of 17.9 and 21.0 (more than two SDs out!), respectively. But did these outliers perform significantly better or worse in terms of Regular Season record?

Although the sample size is quite small, the answer appears to be “not really.” As we learned in my last post, Regular Season rank has a much stronger correlation with Total Points For than any other factor, and that is apparent in the scatter plot below. There was a -69.4% correlation between Regular Season rank and Total Points For (remember, it’s negative because a smaller rank denotes a better overall record), compared to only a 2.7% correlation between Regular Season rank and Total Points Below Optimal.

To further hit the point home, the league average Total Points For was 1,456.0. The six teams that made the Playoffs had an average of 1,512.3, while the eight teams that didn’t make the Playoffs had an average of 1,413.7. When it came to Total Points Below Optimal, however, there was virtually no difference between teams that made the Playoffs and those that didn’t: the league average Total PBO was 168.7, while Playoff teams averaged 168.5 and non-Playoff teams averaged 168.9.

The below table summarizes our league’s Regular Season rankings. In addition to each team’s Total PBO, I also calculated the Total PBO for each team’s opponents throughout the Regular Season. Lastly, I ran an alternate scenario where I assumed each team had the same schedule but played optimally throughout the entire Regular Season.

Concluding Remarks

While playing optimally may not be directly correlated with each team’s Regular Season record, it turns out that Total PBO for each team’s opponents was correlated -37.2% with that team’s record. Playoff teams’ average Opponent PBO was 187.8, compared to non-Playoff teams’ average of 154.5 (remember, overall average of 168.7). In other words, teams with better records “capitalized” on their opponents’ mistakes, and this had a non-trivial impact on the final Regular Season standings.

In addition, the optimal Regular Season scenario (last three columns of the table above) yielded some interesting results. If every team played optimally every week, then the top 10 teams over the Regular Season would have remain unchanged. However, due to the overall competitiveness of the league (six teams were either 8-5 or 7-6 in the Regular Season), an extra game won or lost in this scenario led to Teams 12 and 1 dropping out of the Playoffs and Teams 4 and 7 making it instead.

The “all optimal” scenario also indirectly highlights (once again) the painfulness of unlucky match-ups. Team 6, for example, had the second highest Total Points For but only ended up 5-8 and in 10th place. In the “all optimal” scenario, Team 6 once again boasted the second highest Total Points For but remained unchanged at 5-8 and 10th place. This is yet another argument for having more flexible Playoff seeding, as mentioned in my previous post:

Another alternative would be to reserve the last seed in the Playoffs for the team that made the top 6 in points scored during the Regular Season (assuming a 6 team Playoff format) but did not make the top 6 in win-loss record.

Interestingly, Team 8, which had the lowest PBO in the league, also benefited from having the second lowest Opponent PBO. Under the “all optimal” scenario, Team 8 would have dropped from having a 4-9 record and holding 11th place to just winning one game and being in dead last.

Before I sign off, let’s revisit our earlier decision to

split the manager’s abilities into two distinct buckets: (1) all the actions required to arrive at his or her current roster (i.e. drafting, waiver wire acquisitions, trading) and (2) setting the best starting lineup.

Stated simply, Bucket (1) measures a manager’s ability to maximize his or her Total Points For under an “all optimal” scenario. Inherent randomness aside, I would argue that this is largely a test of skill. Bucket (2) can be further split into two aspects: (A) the manager’s ability to set the optimal lineup every week (largely a test of skill) and (B) the ability of the manager’s opponents to play optimally (more a test of luck, as schedules/match-ups are in play).

As always, you can find my work here.

Fun with Excel #14 – Fantasy Football Match-ups Matter!

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The vast majority of Fantasy Football leagues use head-to-head (H2H) scoring, where a (random) schedule is determined at the beginning of the season and mirrors how the sport is played in real life. During the Regular Season (typically Weeks 1-13), each team will usually play every other team at least once, with each team’s win-less record determining its Playoff seeding. Therefore, the objective for each team is to simply maximize its points total every week. While this is a straightforward task, the variability of individual player scoring can get frustrating, especially when coupled with the randomness of H2H match-ups. Everyone has had that one week where they score 150 points, only for their opponent to somehow put up 160. Other times, it feels like the Fantasy Gods are conspiring against you as you finish the season top 3 in Points For, but first in Points Against by a long shot.

So how much additional variability does H2H scheduling really introduce, and are there more equitable scoring formats? To explore this, I looked no further than my previous Fantasy season (shameless plug about winning the Championship in my first year here).

To start off, here are the 14 teams from my league last year and their scoring through the Regular Season (first 13 weeks, ESPN PPR scoring):

chart-0

Here is the corresponding standings table (yes, there was a tie game…our league has since switched to fractional scoring):

chart-1

The correlation between final ranking (i.e. Playoff seeding) and total points scored over the Regular Season was 87.6% (technically negative 87.6%, since smaller numbers correspond to higher/better rankings). To further complicate things, my league was structured with two divisions (2D) last season (Division A and Division B with 7 teams each). So rather than teams being ranked solely based on win-loss record, the top team (by record) from Division A and the top team from Division B were awarded the #1 and #2 seeds, while the top 3 teams from each division were given the top 6 seeds (and the only spots in the Playoffs). This resulted in Team 10 (9-4) finishing as the #2 team despite having a worse Regular Season record than Team 11 (10-3), which finished #3. As you can see in the table below, switching to a more standard single division (1D) scoring format led to the swap of Teams 10 and 11, which is arguably more “fair.” The correlation between the rankings under the 1D case and total points scored was 86.1%, very similar to that of the 2D case.

chart-2

The third scoring format I explored was simply total Points For (PF). As you can see in the table above, this led to a fairly decent shakeup in the overall rankings. Under the PF scoring format, Teams 1, 8, and 11 would have ranked 3 places lower than under the 1D scoring format, suggesting that these teams benefited from “lucky” H2H match-ups (under 1D). On the flip side, Teams 4 and 12 would have ranked 4 and 3 places higher, respectively, than under 1D, suggesting that these two teams were hurt by “unlucky” H2H match-ups. Notably, Team 4 finished 10th under 1D scoring (well outside the Playoffs) but would have finished 6th under PF scoring (securing the last Playoff spot).

Lastly, I ran a Monte Carlo simulation consisting of 1,000 trials, randomizing the schedules of every team over the entire Regular Season. Each individual trial was scored under the 1D format, but my goal was to measure the average ranking of each team over a large number of repetitions, and to compare the results with both the 1D (Base Case) and PF formats.

The results of the simulation were similar to those under the PF scoring format. Once again, Teams 1, 8, and 11 would have ranked 3 places lower than under the 1D scoring format, suggesting that these teams benefited from “lucky” H2H match-ups. In contrast, Teams 4 and 5 would have ranked 3 and 4 places higher, respectively, than under 1D, suggesting that these two teams were hurt by “unlucky” H2H match-ups. The correlation between the rankings under the PF and MC cases and total points scored was 97.4% and 96.2%, respectively. This makes intuitive sense because the MC case minimizes the impact of additional variance introduced by H2H scheduling, while the PF case eliminates such variance completely.

In addition to looking at the correlation between Playoff seeding and total points scored, I also explored the impact of team volatility (i.e. the standard deviation of each team’s weekly score over the course of the 13 Regular Season games) on the final rankings of the teams. I came up with a “Sharpe Ratio“, which took each team’s average points scored per week and divided it by the standard deviation of each team’s weekly score. I hypothesized that teams with higher Sharpe Ratios would generally be more successful, although I was curious whether this would be a stronger indicator of success than simply looking at total points scored. As you can see in the table below, the correlation between ranking and Sharpe Ratio was in fact significantly lower than the correlation between ranking and total points scored, coming in at roughly 41% under the 2D, 1D, and PF cases and 52.4% under the MC case.

chart-3

So what does all of this mean for Fantasy managers? The name of the game has always been points maximization, and the work that we’ve done in this Post confirms that. In the case of my Fantasy league last season, ranking teams based on one particular (i.e. random) H2H schedule reduced the correlation between overall ranking and total points scored by roughly 10%, thus introducing additional “randomness” to the game. While simply awarding Playoff seeding based on total points scored over the Regular Season may be the fairest scoring format, it certainly takes away from the drama of H2H match-ups that makes Fantasy Football so fun in the first place. One potential compromise is to let the Regular Season run its usual course, but then re-seed the Playoffs according to a comprehensive Monte Carlo simulation. This would minimize the variability introduced by H2H scheduling and ensure that teams are not being helped or hurt by “lucky” or “unlucky” schedules. Another alternative would be to reserve the last seed in the Playoffs for the team that made the top 6 in points scored during the Regular Season (assuming a 6 team Playoff format) but did not make the top 6 in win-loss record. Under this format, Team 4 would have made the Playoffs as the #6 seed last season, displacing Team 13. However, it is worth noting that while different scoring formats would have led to different rankings, the variations on average were still relatively minor. Indeed, the top 5 teams in my league last season (Teams 1, 10, 11, 12, and 5) would have finished in the top 5 regardless of the scoring format used.

chart-4

Before I sign off, I will leave you with one final chart, which is a box-and-whisker plot of the Monte Carlo simulation results. As you can see, the combination of volatility at the team level and variance introduced by H2H scheduling results in a fairly wide range of potential outcomes for every team (with some interesting results, such as Teams 2, 4, and 13 all potentially finishing anywhere between #1 and #14, inclusive). In general, however, the chart still provides an effective visualization of the relative ranking of each team, which I found quite elegant.

chart-5

As always, you can find my backup data here.

Fun with Excel #13 – The Laws of Attraction III: Two Inches Taller

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An interesting discussion came up at work the other day, when a co-worker mentioned that a friend of hers (a guy) had opted to undergo an expensive and painful surgery to gain a few inches in height. While the consensus seemed to be that such an operation would ultimately not be worth it, I must admit that I entertained the idea briefly…or at least long enough to run a few numbers for my own curiosity.

Most people have probably heard of the phrase “tall, dark, and handsome” being used to describe the physical attributes of an attractive male. While it’s certainly a generalization, there’s little doubt that when it comes to traditional heterosexual relationships, women tend to prefer taller men. Although attraction is often the result of a variety of factors, I wanted to focus on just one attribute (height) and quantify the impact of being two inches taller on a man’s romantic prospects.

Assumptions

I started with a sample population of 200 million people (100 million men and 100 million women). I then assumed that the heights for both men and women were normally distributed: “adult male heights are on average 70 inches  (5’10”) with a standard deviation of 4 inches. Adult women are on average a bit shorter and less variable in height with a mean height of 65  inches (5’5”) and standard deviation of 3.5 inches.” Although relationship matching is a two-sided problem, I assumed for the purposes of this exercise that only women’s preferences counted (and that every woman shared the same preferences), while men were assumed to be indifferent.

For example, if a 5’5” woman preferred men who were at least her height but not more than 12 inches taller than her, then her potential prospects would include all men between 5’5” and 6’5” (85,429,107 men, or 85.4% of the male population). Conversely, a 5’10” man’s prospects would include all women between 4’10” and 5’10” (90,068,614 women, or 90.1% of the female population).

As always, my data is available here.

From a Woman’s Perspective…

I first examined the impact of women’s preferences on the number of potential men that they would be attracted to. The bold black and red lines represent the number of males and females (left y-axis) distributed by height (x-axis). The dotted blue, green, and purple lines represent the number of potential male prospects (right y-axis) for a given female’s height. The three colors represent three different preference cases:

  • Case A: All females prefer between her height (0” lower bound) and 12 inches taller (+12” upper bound). This is our base case
  • Case B: All females prefer between 3 inches shorter (-3”) and 12 inches taller (+12”)
  • Case C: All females prefer between 6 inches shorter (-6”) and 12 inches taller (+12”)

Of course, there are a multitude of different preferences that we could test, but I mainly wanted to see what would happen if women became more receptive to dating shorter men.

chart-1

The total number of potential men increases from Case A to Case B to Case C. This should surprise no one, as the total preference range is expanding in each case. What should also be intuitive is that while every woman is better off (which we will define as having a greater number of potential male prospects), how much each woman benefits is dependent on her height. Because we are pushing out the lower bound of women’s preferences from 0” to -3” to -6” in the three cases, it makes sense that doing so disproportionately benefits taller women. This is represented graphically by the right tails of the distributions shifting out (from blue to green to purple) while the left tails remain anchored.

It’s worth noting that even a woman of average height (5’5”) would experience notable benefits from the expansion of lower bound preferences. In Case A, her potential prospects would include 85.4% of the male population, but that figure jumps to 93.7% in Case B and 95.7% in Case C. However, the same shifts in lower bound preferences would be a complete game-changer for taller women. A 5’8.5” woman (1 SD above the mean) would see her potential prospects increase from 64.2% in Case A to 86.5% in Case B and 96.5% in Case C. Lastly, the “ideal height” (i.e. the height at which a woman would have the most potential prospects) in each of the 3 scenarios would be 5’4” in Case A (86.6% of male population), 5’5.5” in Case B (93.9%), and 5’7” in Case C (97.6%).

From a Man’s Perspective…

The bold black and red lines remain unchanged, but the dotted blue, green, and purple lines now represent the number of potential females prospects for a given male’s height in Case A, Case B, and Case C, respectively.

chart-2

The dotted lines are a mirror image of the ones in the first chart. The total number of potential women increases from Case A to Case B to Case C. Everyone man is better off, but shorter men are benefiting disproportionately. This is represented graphically by the left tails of the distributions shifting out (from blue to green to purple) while the right tails remain anchored.

A man of average height (5’10”) would experience benefits from the women expanding their lower bound preferences, but these benefits are not as significant as the gains that the average woman would experience. This is due to the fact that men are on average taller than women and that women tend to prefer taller men. In Case A, the average man’s potential prospects would include 90.1% of the female population (already very high), and that figure increases to 96.6% in Case B and 97.6% in Case C. The relative gains from Case A to Case B to Case C (6.5% and 1.0%) are indeed less than the gains of the average woman (8.3% and 2.0%).

For the same reasons, women changing their lower bound preferences would be an even greater game-changer for shorter men than it would be for taller women. A 5’6” man (1 SD below the mean) would see his potential prospects increase from 61.2% in Case A to 87.3% in Case B and 97.6% in Case C (gains of 26.1% and 10.3%, versus 22.3% and 10.0% for the tall women 1 SD above the mean). Lastly, the “ideal height” (i.e. the height at which a man would have the most potential prospects) in each of the 3 scenarios would be 5’11” in Case A (91.4% of female population), 5’9.5” in Case B (96.8%), and 5’8” in Case C (99.0%). Just to hammer the point home, notice how these peak percentages are higher than those of their female counterparts (86.6%, 93.9%, and 97.6%). Lastly, I’d like to point out that in our base case preference scenario (Case A), the ideal heights are 5’4” for women and 5’11” for men, which is very close to their average heights of 5’5” and 5’10”.

The Impact of Two Inches of Height

All this brings us to our main question: as a man, how much is it worth to be two inches taller? At this point, it should surprise no one that the answer is “it depends” (doesn’t it always?). Given everything that we’ve learned so far, we should expect shorter men to benefit more from a potential height increase than taller men.

We see that this is in fact the case in the chart below.

chart-3

The dashed lines measure the percentage increase (right y-axis) in the number of potential female prospects for a man of given height (x-axis). These dashed distributions resemble exponential decay, with extremely short men benefiting significantly from being two inches taller, but with those benefits rapidly diminishing as men approach 5’6” and taller. The shapes of these distributions are also impacted by the female preference case. Case A shows the steepest decline while Case C shows the shallowest. This should make intuitive sense, because Case A is the least tolerant preference range while Case C is the most tolerant.

Lastly, if we zoom in on the impact of being two inches taller on only men who are 5′ and taller, then we can get a clearer sense of where the breakeven height is for each preference case. Here, we define breakeven as the height where a man receives no benefit from being two inches taller. Any man taller than the breakeven height would in fact be worse off if he received the height increase of two inches.

chart-4

In the base case preference case (Case A), the breakeven male height is 5’10”, which is exactly the average male height! In other words, a 5’10” man and a 6′ man have the same number of female prospects, which ties in beautifully to our earlier conclusion that 5’11” is the ideal male height in Case A (5’10” and 6′ are simply symmetric points on a distribution where 5’11” is the peak). By the same analysis, the breakeven male height is 5’8.5” in Case B and 5’7” in Case C.

The fact that the breakeven male height decreases as the female preference range increases (from Case A to Case B to Case C) is significant because it very clearly shows that there are two ways to solve the attraction disadvantage that shorter men face: (1) make them taller or (2) get women to change their preferences. The former is a solution that comes at a steep physical and financial cost (to an individual), while the latter is more of a mental and cultural challenge (that may in fact be more difficult to achieve as a society).

As a 5’7” man, I find it amusing that I would boost my potential prospects by 21% if I were two inches taller in Case A, while that figure would shrink to only 5% in Case B and 0% in Case C.

So ladies, please show some love for shorter men! It will make our decision to not get height enhancement surgery that much easier.

Fun with Excel #12 – A Quick Look at Clutchness in the NBA

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With the NBA in off-season and the Euros concluded, it felt like the perfect time to churn out a couple of long-overdue posts. I’ll start with basketball first, and then soccer (which will likely be a two-part post). See, I’m trying to make up for all those months I didn’t write anything!

So What’s This About?

I’ve always been intrigued by the idea of clutchness in sports, this notion that an athlete could elevate his or her performance to a higher level in moments when it really mattered — when a game, series, or championship was on the line. And yet, despite all the statistics we have available in the modern era of the NBA, there is a certain intangible aspect of clutchness that seems driven more by subjectivity than by the numbers. It’s the reason why Kobe is widely seen a clutch player despite him being a far less efficient shooter in crunch time than Michael Jordan and LeBron James. For instance, LeBron has now taken 12 potential go-ahead shots in the final five seconds of the fourth quarter or overtime in the playoffs and made five of them (41.7%), according to Basketball-Reference. Jordan was 5-of-11 (45.5%) in such situations during his career. And Kobe? A mere 1 for 11 (9.1%) (the famous OT win over Phoenix in 2006). Now, this is obviously a very specific measurement of clutchness, and there are countless ways to tweak the criteria in order to support one narrative or another. That being said, most of the existing literature seems to draw on statistics solely to compare athletes to one another, when in reality, one of the core attributes of clutchness is the ability to outperforming oneself.

The Key Metric

To start, I needed to come up with a simple “all-in-one” statistical metric to measure a player’s performance over any given set of games. While PER comes to mind and is readily accessible for the regular season and playoffs from Basketball-Reference, it is rather cumbersome to calculate for any particular set of games. Ideally, I wanted a metric that could be calculated for a player’s regular season performance, overall playoffs performance, and NBA Finals performance (if applicable). Ranking a playoffs performance against others is a relatively straightforward task, but what I was really interested in was the difference between a player’s playoffs performance and his regular season performance, as well as his performance in the finals relative to the overall playoffs. I wanted to gauge how players performed as the stakes increased (i.e. Regular Season –> Playoffs –> Finals). I drew some inspiration from an article written during the 2015 NBA Finals, in which the author came up with a “bare-bones performance metric” that simply added a player’s points, rebounds, and assists on a per game basis. Using this as a starting point, I took things a few steps further by also incorporating a player’s steals, blocks, and turnovers, with turnovers being subtracted from the metric. Moreover, I looked at everything on a Per 36 Minutes basis in order to normalize for differences in playing time. Lastly, I multiplied my metric by a shooting efficiency factor, such that players who scored more efficiently than average were rewarded (factor > 1) while those who were less efficient were punished (factor < 1). The final all-in-one metric looks like this: Adjusted Total Score = (Points* + Assists* + Rebounds* + Steals* + Blocks* – Turnovers*) x (True Shooting %)/(League Average True Shooting %), where the * denotes stats measured on a Per 36 Minutes basis.

The Data Set

In an ideal world, I would want to look at the performance of every single player, but given time and sanity constraints, I started with the 1979-80 season (when the 3-point line was first implemented), and looked at the performance for each of the 5 players who made the All-NBA First Team that season. While All-NBA honors is not a perfect measurement of a player’s value by any means, I felt that it was a decent enough proxy to capture a snapshot of the league’s most elite players. I repeated this process for every season through 2015-16, giving me a total of 37*5=185 seasonal performances. Furthermore, I went back and added the seasonal performance for every Finals MVP (FMVP), if he wasn’t already included (i.e. wasn’t in the All-NBA First Team) for that season. This added an additional 15 seasonal performances, for a total of 200 performances (among 63 unique players). The 15 players who won FMVP but did not make the All-NBA First Team, in case you’re curious, are: 79-80 Magic Johnson, 80-81 Cedric Maxwell, 81-82 Magic Johnson, 84-85 Kareem Abdul-Jabbar, 87-88 James Worthy, 88-89 Joe Dumars, 89-90 Isiah Thomas, 94-95 Hakeem Olajuwan, 03-04 Chauncey Billups, 05-06 Dwyane Wade, 06-07 Tony Parker, 07-08 Paul Pierce, 10-11 Dirk Nowitzki, 13-14 Kawhi Leonard, and 14-15 Andre Iguodala. Another thing to note is that including the five players from each season’s All-NBA First Team necessarily captures that season’s Regular Season MVP as well.

For each of the 200 performances, I calculated the Adjusted Total Score (ATS) for the Regular Season, the Playoffs (if applicable), and NBA Finals (if applicable). Out of the 200 player performances, only 6 failed to make the Playoffs (84-85 Bernard King, 87-88 Charles Barkley, 91-92 David Robinson, 99-00 Tim Duncan, 11-12 Dwight Howard, and 12-13 Kobe Bryant), and out of the remaining 194 performances, 76 made it to the Finals. While it was easy to calculate the ATS for Regular Season and overall Playoffs, I had to manually go through the game logs of each of the 76 performance on Basketball-Reference to come up with an ATS for just the games in the Finals. Because detailed per game statistics were not available from the 79-80 and 80-81 seasons (with data on steals and turnovers missing from the 81-82 season), I could not calculate the ATS for 5 out of the 76 Finals performances. You can take a look at my organized data here.

Takeaways

I examined the performances both on an absolute ATS basis (e.g. ranking all 200 Regular Season performances by ATS) and on a differential ATS basis (e.g. what is the difference between a player’s Playoffs ATS and his Regular Season ATS, and how does that compare relative to other players?). Here are 10 takeaways I found, in no particular order:

1. MJ dominates the Regular Season, but Stephen Curry’s 2015-16 performance was one for the ages

1

Not too many surprises here, but this chart really highlights how much of a monster young Jordan was, capturing 4 out of the top 5 Regular Season ATS performances, all in a 4 year span between 1987-91. Charles Barkley deserves an honorable mention for putting up 3 out of the top 15 performances. Although he scored fewer points than some of the other players on the list, he consistently dominated the boards and shot extremely efficiently (despite averaging 26.6% from behind the arc over his career and taking an average of 125 threes per season). Lastly, Stephen Curry’s 2015-16 Regular Season was not just the best in the modern era; it blew away the competition, coming in 3.8 points higher than the second highest ATS. That’s the same difference between the second highest ATS posted by 88-89 Jordan and the 19th highest ATS by 79-80 Kareem Abdul-Jabbar (not shown).

2. Regular Season dominance does not necessarily translate into Playoffs success

2A

This is an extended version of the chart in #1 that captures the top 50 Regular Season performances. In addition, it shows each player’s performance in the playoffs, and whether he won the Championship that year. One thing that is striking about the data is the difference between the Playoffs ATS and the Regular Season ATS is negative for almost all of the performances.

2B

As the chart above illustrates, only 4 out of 50 seasons had a Playoffs ATS that exceeded the Regular Season ATS, with LeBron pulling off the feat twice (although he didn’t win the Championship in either season). In other words, the vast majority of the very best Regular Season performers (44/50 = 88%) ended up doing worse in Playoffs. Moreover, only 11 out of the 50 ended up being Champions (8 were Runners-Up, and 31 didn’t even make the Finals). Does that mean most people are decidedly unclutch? Well, not quite.

To address the first issue, I believe there are two main factors that may cause a player who just submitted an all-time Regular Season performance to experience a decline in productivity in the Playoffs. The first factor is that the player has to face more difficult opponents in the Playoffs relative to the Regular Season. This is an obvious yet easily overlooked fact. Indeed, the player likely has to play against much tighter defenses in the Playoffs (e.g. being guarded by the opposing team’s best defender, being double-teamed), when every team’s season on the line. This in turn will make it more difficult for the player to score, and especially difficult for the player to score as efficiently as he did in the Regular Season (note the negative change in True Shooting % from the Regular Season to the Playoffs in the first chart in #2). The second factor is mean regression. Simply put, we’re looking at the 50 best individual Regular Season performances over the last 37 years. Putting up elite numbers over the course of 82 games in a Regular Season is no easy task, and even the best athletes will regress toward their career averages given enough time. In this case, the Playoffs represent exactly that: an extension of the Regular Season.

Onto the second, slightly more troubling issue: how come most of these players who submitted all-time Regular Season performances didn’t even make it to the Finals? To preface my response, I must clarify that I don’t believe that my definition of clutchness (over-performance in the Playoffs/Finals relative to the Regular Season) necessarily correlates with Championships, Playoffs success, or even victory in general. At the end of the day, basketball is played 5 on 5, and one person alone can never consistently achieve victory simply by playing out of his mind. This ties back to why a phenomenal individual Regular Season leads to failure in the Playoffs more often than not. Simply put, the fact that a player is able to achieve an all-time Regular Season performance is perhaps indicative of a fundamental shortcoming of his team: that they are overly reliant on his performance in order to achieve victory.

3. The best performances in the overall Playoffs by ATS seem to fit our narrative

3

With only 4 Championships and 1 Runner-Up, the top 15 Playoff performances continue to support the theory that simply putting up a scintillating individual performance may do more harm than good to your chances of winning a Championship. The top 4 performances, which all had positive ATS and TS% differentials from the Regular Season, are admittedly clutch, and yet not one of them even made it to the Finals. LeBron and the Cavaliers swept the first two rounds in 2009 but were stopped in the Conference Finals by a resilient Dwight Howard and Orlando Magic team. Hakeem put up unreal numbers in 1987 but the Rockets lost in the second round to Seattle in 6 games. David Robinson had a remarkable 76.0% TS Percentage (the highest out of my sample size of 200), but the Spurs only managed to win Game 1 against the Warriors in the first round before being eliminated. Lastly, Hakeem put up even more ridiculous numbers in 1988, but the Rockets were similarly eliminated by Dallas in the first round in which they only won 1 game.

4. Extreme over-performance (super clutch) in the Playoffs won’t win you Championships, but moderate over-performance (kinda clutch) might

4

First, let’s appreciate how phenomenal Hakeem Olajuwon was in the Playoffs. Out of the 43 Playoff performances that were objectively clutch (i.e. Playoffs ATS > Regular Season ATS), Hakeem joins Tim Duncan and Magic Johnson as the only players to appear 4 times. However, his 1987 and 1988 Playoff performances were quite literally off the charts, and I highly doubt anyone ever comes close to his margin of over-performance ever again.

And yet, is too much of a good thing ultimately bad? The top 10 performance all featured a margin of over-performance of at least 5.0 in terms of ATS. These performances were decidedly “super clutch,” and yet, 9 out of 10 failed to even make the Finals. Only Isiah Thomas won a Championship (1/10 = 10%) and it’s interesting to note that his absolute Playoffs ATS of 32.6 was significantly lower than anyone else’s in the top 10.

Outside of the top 10 (anyone with an over-performance greater than 0 but less than 5.0), there seems to be a sweet spot. 17 out of the 33 in the “kinda clutch” group won Championships (17/33 = 52%), a significantly higher percentage than the top 10. But it gets better, because…

5. It pays to moderately under-perform (kinda unclutch) as well

5

The sweet spot doesn’t end there. Turns out, it pays to moderately under-perform as well. Those who were “kinda unclutch” and posted a Playoffs ATS that was lower than their Regular Season ATS by less than 2.0 actually won Championships at fairly significant clip (14/36 = 39%). In contrast, anyone who posted a Playoffs ATS that was lower than their Regular Season ATS by more than 2.0 experienced significantly less success, going 16 out of 115 with regards to Championships (16/115 = 14%).

Why might this be the case? Well, being super unclutch (Playoffs ATS – Regular Season ATS < 2.0) is obviously detrimental to your chances of winning a Championship, because it means that the team is not getting your usual productivity that it is used to. In the absence of teammates stepping up, it is likely that the team will perform worse on the whole. On the other end of the spectrum, a spectacular over-performance over an entire Playoffs may suggest that a team has too much of an “one man army” mentality. This may not be by design, but rather due to a team having few or no other options besides heavily depending on its star player to win games. In contrast, any performance lying in the sweet spot of moderate over/under-performance indicates an achievement that is within expectations and therefore less susceptible to either of the two issues plaguing the two extremes mentioned above.

6. Absolute ATS appears to be a good determinant of Championships

6

The chart above shows the top 20 NBA Finals performances, sorted by ATS. It is no surprise that the list features some of the greatest NBA players of all time, but it’s nice to see the numbers backing up the age-old claim that the best players “show up” when things matter the most. Shaq in his prime has a great case for Best Finals Player Ever, submitting 3 of the top 4 Finals performances of all time, coinciding with the Lakers’ first three-peat.

6. It’s hard to argue that LeBron isn’t clutch, but perhaps he’s just really unlucky

Out of the top 20 performances, 18 led to Championship (17 of which won Finals MVP). The remaining 2 were both put up LeBron James. Despite everything that he’s accomplished, LeBron is perhaps still one of the unluckiest players in the history of the NBA Finals. Indeed, he’s put up historic Finals performances three years in a row and has gone 1-2 in the Finals against 3 of the greatest teams in the modern era (and arguably of all time): 13-14 Spurs, 14-15 Warriors, and 15-16 Warriors. Pretty impressive if you ask me.

7. The Pistons really knew how to step up and win in the Finals 

7A

The first chart ranks Finals over-performance relative to the Regular Season.

7B

The second chart ranks Finals over-performance relative to the entire Playoffs. The two charts tell similar stories, and it’s very interesting to see 3 Pistons players (from 2 different eras) in the top 10 for both. In both eras, I think the degree of over-performance reflects the team’s focus on teamwork and de-emphasizing the importance of the superstar during the regular season. However, Joe Dumars, Isiah Thomas, and Chauncey Billups all knew how to step up on the biggest stage and it was their clutch performances in the Finals that provided the extra spark to help the team achieve the Championship. Special shout-out to Kawhi Leonard (13-14 Spurs), who I think fits the same mold.

8. Andre Iguodala’s Finals MVP in 2014-15 was well-deserved

Many people claim that LeBron was absolutely robbed of the Finals MVP award in 14-15, and while I can see where they are coming from (he put up the 20th best Finals performance in a losing effort), the two charts above show why Iguodala’s performance was deserving of the FMVP as well: while his absolute Finals ATS was not very high by any means, he did put the 5th highest over-performance relative to the Regular Season and the 12th highest over-performance relative to the Playoffs. Pretty clutch if you ask me!

9. Magic deserved to win Finals MVP in 1987-88…

At least at first glance: 12th highest absolute Finals ATS (42.9) on a Championship team, 8th highest over-performance relative to the Regular Season (+5.0) and the 6th highest over-performance relative to the Playoffs (+7.0). In comparison, James Worthy, the man who ended up taking home FMVP honors that year, had a Finals ATS of 31.2 and performance relative to the Regular Season and Playoffs of +0.8 and -0.5, respectively. However, while Magic appears more worthy (no pun intended) of the award when looking at the overall stats, Worthy was arguably more clutch at key points during the series, something I didn’t appreciate until I dug deeper into the history of the 1988 NBA Finals. The series went to 7 games between the Los Angeles Lakers, who were looking to be the first team in 19 years to repeat as Champions, and the Detroit Pistons, who were led by point guard Isiah Thomas. The Pistons stole the first game in LA, but James Worthy stepped up big in Game 2 as Magic battled the flu, leading all scorers with 26 points.  In Game 3, with Magic still sick, Worthy once again led the Lakers in scoring with 24 points as they won a critical game in Detroit and took a 2-1 lead in the series (the Finals were still played in a 2-3-2 format at this point). However, the resilient Pistons took the next 2 games at home and a 3-2 lead, setting themselves up just one game away from the franchise’s first Championship. Detroit seemed almost destined to win it all, after Isiah Thomas put up 25 points in the 3rd quarter (still an NBA finals record) despite suffering a badly sprained ankle earlier in the quarter. Thomas would finish the game with 43 points, but Worthy’s team-leading 28 points proved to be vital as the Lakers held off the Pistons to win by a single point, 103-102. In the deciding Game 7, Worthy put up an incredible triple-double (36 points, 16 rebounds, 10 assists) to propel the Lakers to a 108-105 victory and the team’s second consecutive Championship. Not only was the triple-double impressive in its own right, but it would turn out to be the only triple-double in Worthy’s entire NBA career. I don’t know about you, but that sounds like the definition of clutch to me.

Anyway, it’s these kinds of situations that make me realize how much I love doing exercises like this, because despite how far statistics have advanced in sports over the years, it will never be possible to boil athletic performances into just a matrix of values. Sometimes, real life is even more amazing than the numbers.

10. The king of unclutch is Karl Malone, but you might be surprised to see who else has joined him…

10A

The Mailman always delivers…except in the Playoffs. Malone owns the top two under-performances (Playoffs vs. Regular Season ATS), and 3 out of the top 10. In fact, he posted an under-performance in each of his 11 seasons that appeared in this data set. And yes, the two times he did make the Finals, he under-performed as well. Yikes. But let’s not forget about 15-16 Stephen Curry, who put up the 4th worst under-performance overall, as well as the worst by a Regular Season MVP and the worst by a player who made the Finals. Of course, Curry was a victim of his own success. His Regular Season performance was simply too good, and it looked even better under my calculation of ATS (given that it is on a Per 36 Minutes basis and Curry managed to put up monstrous numbers on just 34.2 minutes per game in the Regular Season). Add to that an injury in the Playoffs which forced him to miss playing time, and it’s not surprising that his performance fell back down to earth in the Playoffs and Finals.

At 28 years old, Curry will likely have more opportunities to prove himself as a clutch performer in the Playoffs. And while the Warriors’ off-season acquisition of Kevin Durant may have just made their path to the Finals (and perhaps future Championships) a lot easier, I invite both Warriors fans and haters to take a look at who fills in the #10 and #11 slots of the top 15 worst under-performances in the Playoffs of all-time :).