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.