Evaluating Hack-a-Shaq

Years ago, a giant center used to destroy teams on his own, putting up monster statistics with high accuracy from the field. He was an unmovable object with superhuman strength, and he seemed like the ultimate offensive weapon. He did have one weakness, however, his kryptonite -- free-throw shooting. With several seasons below 50% from the line, teams would intentionally foul him because it was much better than the alternative. People argued that he was so poor from the line, in fact, that he was detrimental to the team whenever the strategy was employed.

At the time, the strategy wasn't called Hack-a-Shaq because it was used against Wilt Chamberlain. The league changed some rules regarding fouls called in the last two minutes and the strategy laid dormant for decades until eccentric basketball mind Don Nelson used it against Dennis Rodman and then Shaquille O'Neal. Unfortunately, it's still an active weapon, most notably against Dwight Howard. There are ongoing debates about the merits of keeping this a part of basketball -- it's ugly and slows down the game versus if a player doesn't want this used against him, he should work on his shooting -- but what's missed most often is how effective it is as a strategy. People discuss how sending a 55% foul-shooter to the line results in a "worse" offense, but don't compare it to how efficient the team normally is and ignore the possibility of offensive rebounds.

Breaking down Hack-a-Shaq, there are four components for evaluating its potency:
1) The foul-shooter's percentage from the line
2) The chance of an offensive rebound off a miss
3) The offensive rating after the offensive rebound
4) The offensive rating of the team if you don't intentionally foul

Combining 1) through 3), you can compare the offensive rating of the intentional foul to 4). If the resultant offensive rating for a Hack-a-Shaq isn't lower than the team's offensive rating without fouling them, then it's probably not a worthwhile strategy, unless you have other motivations like stopping the clock.

The formula for a Hack-a-Shaq offensive rating is pretty simple:
100*(2*FT%/100+(100-FT%)/100*FTORB*FTORTG/100)

where
FT%: a player's free throws percentage (in the form of 60% and not 0.60)
FTORB: the team's chance of grabbing an offensive rebound after a free throw miss (in the form of 0.20, not 20%)
FTORTG: the team's offensive rating after a FTORB (per 100 possessions, i.e. 110 points)

The 100 at the beginning is to translate the number into points per 100 possessions. The first part of the equation inside the parenthesis are the expected points you get sending the player to the line, and the second part is the expected points from an offensive rebound given the chance of an available rebound (a miss) and the chance of the offensive team grabbing it. So with the formula explained, here's a simplified version (with the extra 100's):

Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG

The FT% is the easier variable to find. You can simply use the player's season average, the average from his last three seasons, or his career average.

The variable FTORB, however, is trickier to find. It's not an official stat tracked by any site (least not that I could find) and it can't be deducted from conventional stats. There have been a handful of studies of offensive rebounding based on the shot type, like this early one from 82games.com or a recent one from hoopdon. The former study found a 13.9% OREB after free throw misses, while the latter using NBAWOWY's site found an average of 12%. The past couple seasons have seen an average of all offensive rebounding around roughly 26%, so there's clearly a substantial drop when rebounding after a missed free throw. (82game's higher figure, by the way, stems from the slightly higher offensive rebounding numbers in the mid-00's and because it included team rebounds.)

But the analysis doesn't end here. When, say, Dwight Howard is at the line, we cannot assume a 12% average because of two complicating factors: one being that Howard, the team's best offensive rebounder, is now far from the rim, and secondly when he's at the line players expect a missed free throw and fight harder for an offensive rebound. Reading play-by-play data, it's possible to calculate offensive rebounding values for when the infamous free throw chuckers miss. What's surprising is that despite having Houston's team offensive rebounder at the line and playing a significant portion with smallball lineups, the Rockets have rebounded at an above average rate when he misses a free throw.

Player.............	OREB% FT miss	FT%	FT misses
Dwight Howard	0.171	53.3	129
Andre Drummond	0.091	40.7	55
DeAndre Jordan	0.172	41.3	64

With some approximations of rebounding percentages, the next step is the expected points after the rebound off the free throw miss. This issue has been studied before, and one can expect a significantly higher effective field-goal percentage off a miss than most other actions (like a made field goal on the other end.) For instance, the Rockets have an eFG% of 56.4 after a free throw offensive rebound compared to their season average of 51.9%, while the Clippers have averages of 56.8% and 52.8%, respectively. Until I get more data specifically for offensive rating after misses, I'm going to assume a conservative increase in efficiency after these misses -- 5 points per 100 possessions, compared to nearly 10 points that I've found from NBAWOWY, depending on the team/personnel.

With these numbers, one can now calculate Hack-a-Shaq offensive ratings. For example, using the averages for Howard and his team this season:

Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG

Hack-a-Shaq ORTG = 2*53.3+(100-53.3)/100*0.171*(109.3+5)

Hack-a-Shaq ORTG = 115.7

Compared to Houston's rating of 109.3 for this season, it is not wise to put him on the line intentionally because you are effectively giving them a very high offensive efficiency (like the Nash-Suns at their very best) and one that's much higher than what you'd expect without fouling him. I'm sure people also want a Shaq-specific example, so during the 2001 regular season in LA he shot 51.3% (his worst season during the title stretch) with an offensive rating of 108.4. Using estimates of a 15% FTORB and a conservative FTORTG 5 points above their regular season average, this translates to a Hack-a-Shaq ORTG of 110.9 points per 100 possesions -- so yes, even during one of his poorest shooting seasons in LA, it was still an unwise strategy.

Or you can solve for what FT% Howard would need for this to be a break-even strategy (meaning, a percentage lower than this means it's an effective strategy.)

Hack-a-Shaq ORTG = 2*FT%+(100-FT%)/100*FTORB*FTORTG

109.3 = 2*FT%+100/100*FTORB*FTORTG-FT%/100*FTORB*FTORTG

109.3 - 100/100*FTORB*FTORTG = 2*FT%-FT%/100*FTORB*FTORTG

109.3 - FTORB*FTORTG = FT%*(2-1/100*FTORB*FTORTG)

FT% = (109.3 - FTORB*FTORTG) / (2-1/100*FTORB*FTORTG)

FT% = 49.7

(Even with an OREB% of 10, the break-even point is still only around 51.9%.)

Fifty-percent is a reasonably good break-even line for big men on good teams in most scenarios. Note that there's a high degree of elasticity with regards to free-throw percentage (i.e. FT% greatly changes the Hack-a-Shaq efficiency.) Free-throw OREB% has a lower elasticity, while ORTG after a missed free-throw is greatly inelastic. To visualize this, I prepared a series of charts showing how Hack-a-Shaq efficiency changes with each respective variable. Since there are three independent variables and one dependent variable (the result), it's a bit like four-dimensional graphing, so there's a chart for five different OREB rates, where the x-axis is FT% and the y-axis for the Hack-a-Shaq efficiency, and then one line each for a team's ORTG after a missed free throw.

What's surprising is that Dwight Howard gets all the attention for that Hack-a-Shaq strategy when he's comfortably past the break-even line, but both Drummond and DeAndre are two of the worst foul shooters ever and are far below the efficiency threshold. Even if their team rebounds their misses at a high rate, the result is an offense that would perform worse than the 2012 Bobcats, who won only 7 games. The Clippers' offense is a high-powered one featuring the best point guard in the guard, one of the best scoring big men, and several outside shooters, but you can stall their offense by sending DeAndre Jordan to the line. He's only had one season above 50% and needs to improve so he's not a liability whenever the team is in the penalty.

The strategy is ugly and an effrontery to the beauty of the game, but it's also less effective in most situations than most people realize. With the relevant data, it's also fairly simple to calculate. (This will probably spawn another study looking at how a free throw miss offensive rebound percentage changes based on who's at the line.) For most of Shaq's career it was an inappropriate strategy because it gave his team a higher offensive rating than they had otherwise, and this is true of Howard too as his two worst seasons had free-throw percentages of 49, which is straddling the break-even line. When both guys were near 60%, it was an especially stupid strategy.

But you can still have fun with it.

Lakers' Season Start: Whose Head to Place on a Pole

Only a couple games into the season, people nonetheless are drawing conclusion, and obviously there are some scary and strange conclusions out there. Take a random sample of two to three consecutive games during a season, and you can find some really strange statistics like Andre Miller leading the league in points, Jordan in blocks, Manute Bol the three-point sniper, or Jamaal Tinsley the shotblocking king. Last season, people were excited to see DeMar DeRozan's development as an outside shooter, but by the end of the season he had slumped to 26% from the three-point line. Are we really going to live in a world where Brandon Jennings is the assist leader, Durant in rebounds, Hawes in blocks, and Chandler at 100% from the field? Of course not.

The Lakers are being trashed right now for not winning anything, even a preseason game, while the hiccups are becoming serious from a Nash injury to Bryant's dead foot. With a new offense and an expectation of greatness from the mere presence of the Canadian overlord, the fact that they aren't scoring 140 points a game has Lakers fans readying with pitchforks to storm the castle. My prediction of the Lakers' success this year looks dumber by the minute, and I should have included more of a penalty for team fit and coherence along with Dwight's injury. But in actuality, looking at the stupidly small three game sample, their offense has been fine; it's their defense.

	Los Angeles Lakers	League average 2013	League average 2011
Offensive efficiency	104.4	102.3	107.3
Defensive efficiency	114.4	102.3	107.3
Pace	93.2	94.0	92.1
TS%	56.8	52.0	54.1
Effective field-goal %	53.9	48.0	49.8
FT/FTA	0.267	0.214	0.229
FT %	63.7	74.5	76.3
Turnover %	18.7	14.5	13.4
Offensive rebound %	31.9	26.5	26.4

I included the data from the 2011 season because lockouts typically have reduced, messier offenses. The Lakers' offense is ranked 11th (according to basketball-reference), but looking closer the problem is turnovers. Their shooting percentages are stellar, and although their free-throw percentage is low (because of Dwight Howard) they're getting to the line a lot (also because of Dwight Howard), and shooting 63.7% is still better than an average possession shooting from the field. They're also offensively rebounding very well (again, Dwight Howard), pulling their numbers further up. The turnovers, however, are stunting the offense, which should be near the top four or three in the league. With Nash at the helm and Bryant still a major force, someone who can take a large number of shots without losing the ball, this shouldn't happen. An easy explanation is that they're trying to incorporate new players onto the team and they're using a new offensive system, and that's reasonable enough. Strangely though, the Heat's defense is rated as the worst, an indication we shouldn't put faith into early season numbers, but we can analyze how the Lakers are losing.

The defense is a complete wreck. They're second from last in the league giving up an effective field-goal percentage of 52.0 while never forcing turnovers (second from the bottom in that respect too). They're also letting the opponent get to the line at a high rate. In summary, their defense is failing in multiple respects, and it's hard to blame that on their schedule: Dallas was without Nowitzki, Portland's relying on rookies, and the Clippers are the only formidable team.

So who's to blame? Mike Brown is the obvious scapegoat for fans, but in terms of players Kobe Bryant's name is coming up too often.

	Steve Nash	Kobe Bryant	Pau Gasol	Dwight Howard
Points/36 mins	6.5	28.8	14.8	21.3
Assists/36 mins	5.8	1.2	3.3	2.9
Rebound %	7.3	6.8	18.5	17.8
Turnover %	20.0	18.9	9.7	15.9
TS%	37.5	71.3	48.8	61.4
PER	7.4	26.1	19.6	24.5
Win Shares/48 mins	-0.056	0.160	0.121	0.162
+/-	-3	+1	+1	-18

No one saw this coming, but as much as people want Pau Gasol executed for his shooting percentages he's leading the foursome in turnover % by a landslide, and he's even outrebounding Howard. Dwight's basic stats seem alright until you get to his plus/min: -18 in three games, and considering their bench this is impressive. Nash has been complete trash and he'll be out a few games, where he's not even hitting his jumpers, but at least the Lakers played decently when he was on the court. Kobe Bryant, however, has a sterling record: an amazing 71.3 TS% and a plus/minus of +1, which on an 0-3 team is not easy to tally when you play big minutes. His turnovers, however, are historically high, and it would be silly to assume both he and Nash will continue to turn the ball over like they have been. Even if just those two control the turnovers, and Nash shoots near his career averages, the offense should be spectacular. Controlling turnovers will also help the defense by limiting opponent's transition plays.

Some people will try to blame Kobe's defense since he's been playing hurt, and it's hard to find evidence to support or deny the conclusion, especially so early in the season. Trying to go deeper into the numbers, when Kobe Bryant is off the court the opponents average 130 points per 100 possessions (offensive efficiency), which compared to the Lakers' average of 114 and the league is an absurdly high number. During one stretch the Blazers killed the Lakers on the offensive boards, which isn't Kobe Bryant's job to prevent, but even when that's excluded from the numbers it's 120 points per 100 possessions. I will stress that one should not hold these plus/minus numbers with any confidence because it's only been three games, but the evidence available suggests Kobe isn't the problem on offense.

There's another player pushing through an injury: Dwight Howard, who's replacing Andrew Bynum, not exactly a Defensive Player of the Year candidate. Healing from his back injury, his recoveries are slow and opposing guards are zipping to the basket unimpeded. Last year the Lakers were middle of the pack in defense with the Magic barely ahead, but the Howard drama zapped both the team and the center of the effort necessary to dominate; the season before the Magic were ranked third despite really only possessing one good defensive player: Dwight. Given that the Lakers are decent without Howard, if Defensive Player of the Year Howard shows up at a game and the Lakers control their turnovers, this team should easily be one of the best in the league. He's only 26 and should regain his mojo eventually, but we don't know when that will be, and Los Angeles will have to wait patiently.

Los Angeles Lakers 2012-13 Preview

I know this sounds crazy, but I feel the Lakers are underrated this year. They essentially traded Bynum for Nash and Dwight Howard. Dwight Howard, a two-time defensive player of the year and best center by a mile, carried weak supporting casts far into postseasons and past reasonable win totals. His best teammate previously was ... Jameer Nelson? Turkoglu? Ryan Anderson? Nash is a great-grandfather in NBA terms, but he had another impressive season and his shooting and passing skills won't erode with time. He's replacing Ramon Sessions and Steve Blake. They now have a core of Nash, Bryant, Pau Gasol, and Dwight Howard.

Some forecasts have the Lakers modestly accumulating win totals. Hollinger predicts only 53 wins, behind the Nuggets, Thunder and the Spurs, who he pegs to come out of the west. Basketball Prospectus estimated the Lakers with Howard and Nash would only win about 55 to 56 games next season; it's hardly a slight but it's a stratosphere from what experts were saying would happen when LeBron, Wade, and Bosh joined forces. Lakers fans, obviously, are confidently predicting 73-9 with not just one but two titles in a single year, but there are a few doubters out there.

Throwing together a cast of all-stars doesn't mean you can just add some static value of each player to predict the win total for the season. In other words, the team is not the sum of its parts. This is a case of diminishing returns. There is only one basketball on the court, and the Lakers gang can only shoot one at a time. If you add up the big four's shot attempts and compare it to what the Lakers averaged per game last year, those four players would comprise 72% of their shot attempts. Do the same with free throws and it's ... 104%. In fact, if you take a weighted average of the usage rates from last season and apply it to this season's Lakers, you get something around 21.8%. Note that because there are five players on the court at once the average should be 20 (100% divided by 5). For the math to make sense, you have to subtract shots from someone, and the question is will that be a center on track for the Hall of Frame in his prime or arguably the best shooter ever in Nash, or will that be Kobe Bryant, whose last season's shooting quantities were only rivaled by Jordan, Iverson, Wade, and himself?

The table below is showing every with a usage % (percentage of team plays used) of 35 and over with a minimum of 1500 minutes played in the season.

Player	Year	Age	Tm	MP	FGA	3PA	FTA	Ast	Tov	Pts	FG%	3P%	TS%	USG%
Kobe Bryant	2006	27	LAL	3277	27.2	6.5	10.2	4.5	3.1	35.4	45.0	34.7	55.9	38.7
Michael Jordan	1987	23	CHI	3281	27.8	0.8	11.9	4.6	3.3	37.1	48.2	18.2	56.2	38.3
Allen Iverson	2002	26	PHI	2622	27.8	4.5	9.8	5.5	4.0	31.4	39.8	29.1	48.9	37.8
Dwyane Wade	2009	27	MIA	3048	22.0	3.5	9.8	7.5	3.4	30.2	49.1	31.7	57.4	36.2
Jermaine O'Neal	2005	26	IND	1530	19.4	0.1	8.9	1.9	3.0	24.3	45.2	16.7	52.0	36.2
Michael Jordan	2002	38	WAS	2093	22.1	0.9	5.6	5.2	2.7	22.9	41.6	18.9	46.8	36.0
Allen Iverson	2001	25	PHI	2979	25.5	4.3	10.1	4.6	3.3	31.1	42.0	32.0	51.8	35.9
Allen Iverson	2006	30	PHI	3103	25.3	3.1	11.5	7.4	3.4	33.0	44.7	32.3	54.3	35.8
Kobe Bryant	2012	33	LAL	2232	23.0	4.9	7.8	4.6	3.5	27.9	43.0	30.3	52.7	35.7
Allen Iverson	2004	28	PHI	2040	23.4	4.1	9.5	6.8	4.4	26.4	38.7	28.6	47.8	35.3
Tracy McGrady	2003	23	ORL	2954	24.2	6.0	9.7	5.5	2.6	32.1	45.7	38.6	56.4	35.2
Dominique Wilkins	1988	28	ATL	2948	25.1	1.7	8.4	2.9	2.8	30.7	46.4	29.5	53.4	35.2
Jerry Stackhouse	2001	26	DET	3215	24.1	5.9	10.1	5.1	4.1	29.8	40.2	35.1	52.1	35.2
Kobe Bryant	2011	32	LAL	2779	20.0	4.3	7.1	4.7	3.0	25.3	45.1	32.3	54.8	35.1
Bernard King	1985	28	NYK	2063	23.7	0.2	10	3.7	3.7	32.9	53.0	10.0	58.5	35.1
George Gervin	1982	29	SAS	2817	25.2	0.5	8.1	2.4	2.7	32.3	50.0	27.8	56.2	35.0

Kobe tops the list with his hallowed 2006 scoring season of 35 points and his single game of 81. Kobe last season was ranked 9th all-time. With his new all-star teammates, will we see Happy Olympics Kobe, content to take less shots, play more defense, and step up as a veteran when needed? As a side note, beyond Kobe, Jordan, and Iverson, who comprise 9 of the 16 seasons, there are a few surprising players, but there's certainly a pattern: it's all wing players and mostly shooting guards, and with the weak exception of Jerry Stackhouse they're all subpar three-point shooters. There is one player who sticks out like a bleeding thumb. Which one of these players is not like the others? I certainly would not expect him. Going further off topic, I put together a simple graph of TS% (field-goal percentage adjusted for free throws and three pointers) and usage % to show how efficiency varies among the seasons with the historically greatest "ballhogging".

Bringing it back to the Lakers: as a counterpoint, let's consider how these players bring their value. Howard can replace Bynum's shots, and he can also soak up more minutes (presumably) to leave little room for replacement level types like Jordan Hill. Nash doesn't shoot all that much anyway, and Ramon Sessions loved to shoot. Howard's value is primarily on the defensive end where he's strong enough to defend big centers but quick enough to snuff out pick and roll's while protecting the rim and the weak side, and offensively a lot of his value is the mere thread of his presence in the paint. Nash is a wizard at the helm of the offense, dribbling wherever he wants on the court because if you give him a sliver of daylight he'll throw up an accurate jump shot. He shoots a staggering 50+ percent on midrange jump shots, near the top of the leaderboard year after year. His passing and pick and roll skills will combine with Howard and Gasol to form an impossible to defend play. Can you stop Howard, a huge athletic beast, from rolling to the basket while preventing one of the best shooters and passers ever an open shot? The Lakers are adding high efficiency shooting and a defensive player of the year for their interior. For another aside, one study found that when you increase a player's usage by 1% his offensive efficiency increases by 1.25 points per 100 possessions and vice versa for a decrease in usage. For the Lakers, they should hope their reduction in shot attempts for their stars translates to better offensive efficiency.

The key to the season is how they cope with new roles; this is paramount for their all-star guards. Nash regularly has the ball in his hands, directing the offense, while Kobe does the same though instead of passing he sets up for midrange jump shots. Kobe does have the ability to catch and shoot more as the mere thought of leaving him open despite his shooting percentages from 3 is enough for teams to react defensively. If the floor stays open in the middle, Howard and Gasol will destroy opponents with their length and size like they have with Bynum before. Howard will control the boards and the paint while Pau will keep evolving into a jump shooter (who doesn't actually jump). It's not ideal, but it's the same problem as last year with a better center. With Artest and Jamison likely to receive major playing time, floor spacing is the key word. Hopefully, Meeks and Goudelock can earn more playing time with Artest as a power forward, kicking out the defensively inflammable and offensively useless Jamison for outside shooting. Defensively, as long as Jamison doesn't play too many minutes at the same level he has recently, the swap of Bynum for Howard will net huge dividends as long as Howard can stay on the court. I'd rather bet on Howard than Bynum, however.

Of course, the Lakers view the regular season differently than most teams. Mike Brown notwithstanding, they should rest their best players as much as they can, especially the veterans like the nearing 40 years-old Nash. Injuries are also a concern, and those issues could affect this team more than almost any other one because their bench is, simply put, deplorable. Antawn Jamison leads the cast of misfits, but he's an undersized power forward who is a weak outside shooter, sports a low-percentage conversion rates, and can't play defense against most professional players. Their best bench player is arguably Jodie Meeks because at least his outside shooting is valuable.

During the playoffs this will matter less, and as a result I'm seeing the Lakers representing the west in the finals, especially now that the Thunder are Beard-less. Nash and Howard seem to possess a magic ability to lift offenses and defenses, respectively, and with this odd mash of players it feels like a mad experiment in the lab. The Lakers retain their status as the league's trust fund baby, gaining Dwight because he wanted big city lights and the allure of the purple and gold, and lucking out on Nash simply because they were closest to his kids. Looking at the new Lakers' stars box score numbers doesn't tell the whole story; this team could be utterly transformed. For the rest of the league, this isn't fair.

Model win total prediction:
59-23