An economic approach: More referees in NBA games?

December 14, 2016 by J.H. Yeh

On December 12, 2016, the NBA announced that it will experiment a four and five-person crews for nine Long Island Nets games in late December. This is not the first time the NBA has experimented new change in the Development League. In the past, the NBA has experimented new technology and equipment on several Development League games. Moreover, the NBA has experimented four-person crews in the summer league earlier this year. In anticipation of potentially expanding the officials, the NBA has decided to revisit such trial and bring it to the Development League level. If change does come in the next couple years, this would be the first major shake-up in the officials since 1988–when the league added a third official to the two-man crew for better management and control of the games.

Calling the four and five-referee initiative as “a prime example” that will help the growth of the league, Bob Delaney, the NBA’s Vice President for Referee Operations and Director of Officials said that he is “confident in how [their] three-person system works and are constantly thinking of ways to improve our game.”

Personally, I think it’s a good change and an experiment worth studying. As the game of basketball progressed and players are increasingly athletic, powerful and faster (and also more creative), it has been more difficult for the referees to have better management of the game than in the past as officials in today’s NBA have to keep track of many little things such as traveling and three second violation. As the game and players evolve, these actions and events have been a lot harder to get a handle on. It’s a different era now and a change in the number of officials is a good way to ascertain the future direction of the role of the NBA officials.

And now here’s a question: How many officials is enough? We have three options: Three officials, four officials, and five officials. So how many officials is enough for an NBA game?

From an economic approach, there’s an important principle called the “Law of Diminishing Marginal Returns.” In short, in economics, this law states that diminishing returns is the gradual decrease in the marginal output in a production process. Note that in economics, the word marginal means a small change, or an incremental change. Holding other factors constant, or ceteris paribus, adding one more unit of factor of production will, at one point, yield lower incremental per-unit returns. For another word, each additional unit of the factor of production will become increasingly less effective in converting inputs into outputs.

Please note that the Law of Diminishing Returns does not expressly insinuates that adding one additional unit of factor will cause a decrease in total production–an economic condition called negative returns, meaning additional input will decrease the total output. Although this phenomenon is quite common and often related to the Law of Diminishing Returns. And we will save this for some other day.

In short, the Law of Diminishing Returns, with the output remain unchanged and all other factors constant, states that each extra unit of input will become less effective in producing the output. The graph below illustrates the relationship between 1) one extra unit of output and 2) the output each extra unit of input produces.


The first unit of input can give you substantial return of output, whereas the next and each subsequent unit of input gives you less and less return of output. Thus, explaining the negative parabola presented in the chart. Or simply put, as you move rightward along the graph, the slope of the curve becomes less steep, and eventually, you reach the peak (where slope equals zero). This perfectly illustrates the Law of Diminishing Returns. For the first 2/3 of the upward slope, each unit of input is yielding significant amount of return, and then the next 1/3 of the upward slope gives you very little return (diminishing return). And finally, at the top where you hit the peak, that additional unit gives you nothing in return. And then from there, each additional unit will hurt the overall production because each extra unit hurts (negatively affect) the total output (negative returns). Again, we will save this for some other day.

Now let’s put this in real life example. Imagine you’re the owner of a sports bar. There are 10 bar stools and you only have one bartender (worker). It’s obvious the bartender has to do all the work. Now you hire one new bartender and this additional bartender provides significant help to the old bartender by alleviating half of the work load. Each bartender will be responsible of five bar stools (or customers). The act of hiring a second bartender is major because it yields significant amount of return. Not only that one extra bartender is a huge help, he can also work more effectively. Now you decided to employ a third bartender, the hiring of this third bartender is not so significant because his presence does not help as much as the second bartender to alleviate the workload. Regardless, he still helps–but by a little bit. In here, “by a little bit” means diminishing marginal returns, meaning the third bartender is less effective. Now that each bartender will be responsible for 3.33 bar stools; now that’s a big difference! We can also look at the difference to see how each additional bartender affect the returns. The act of employing the second bartender helps you with five (10-5+5) bar stools. The act of hiring the third bartender helps you with just 1.67 (5-3.33=1.67) bar stools! Now this is a huge difference! The act of employing the third bartender hugely reduces the effectiveness of each bartender. Their responsibility is a lot smaller and each one of them is doing a lot less work.

This can go so on and so forth and the difference (diminishing marginal return) will just get increasingly smaller with each extra bartender. Below is a list. I purposely made this list to include up to 15 bartenders just to clearly demonstrate the economic notion of diminishing marginal returns. If you hire 15 bartenders then each one of them will be responsible for 0.67 bar stools and the 15th bartender will yield a diminishing return of 0.048 bar stool! Such an extremely small return! And definitely the 15th bartender adds very little value in helping the sport bar to operate effectively.




So far we have covered how each additional unit of input, now let’s talk about the cost. We know the Law of Diminishing Marginal Returns states the negative correlation between each additional unit and the return of such investment. So as the input increases, the return of each unit decreases. Moreover, the Law of Diminishing Marginal Returns also implies an increase in marginal cost, which can contribute to rise in cost.

Marginal (small change and incremental) cost is defined as the additional cost added to produce each extra unit of output. In here, and from an economic standpoint, cost is often not associated with money. Sure money is definitely involved since (in the above example) you have to pay money to the bartenders. However, cost can also be measured in different terms. Two of the important costs that people often omit are the opportunity cost and the external cost. These are hard to see because they are the hidden costs that often entail within an operation.

Opportunity cost is the cost of a potential gain from an alternative decision. It is the cost of forgoing to do something. For example, after a basketball game, I can choose to watch the film to better myself or I can improve my skills by staying at the court and hoisting a few extra shots. If I choose to watch the game tape, then my opportunity cost is potentially improving my physical skills through practicing. Often times, in life, there are more than two options. I can also have a third option where I can choose to go home and sleep so I can rest my body. Moreover, opportunity cost is not strictly limited to choosing one option. I can choose to watch game tape for 25 minutes and shooting some hoops for 50 minutes and spend the rest of the day sleeping. By mixing what to choose and how much to do, different combinations will give us different opportunity costs since everyone has different preference. In the sport bar example, the opportunity cost of not hiring a third bartender is hiring the third bartender. And vice versa, the opportunity cost of hiring a third bartender is to use that money you pay to the third bartender on somewhere else such as giving raise to the first two bartenders or invest that money on elsewhere like a bigger TV or better drink choices–or both. The concept of opportunity cost may sound strange and ambiguous to most newcomers (it took me two months to get accustomed to this concept) and we will explore this topic in depth next time.

The second cost is the external cost. In short, the external cost is the cost that the producer and the consumer impose the unwanted cost on other producers, consumers, or even on themselves. This is a cost that can’t really be measured with money. For another words, it is the invisible )and sometimes inconvenient) cost inevitably created by producing each additional unit of output. Again, in the sport bar example, if I hire 10 bartenders so each one can take care of one seat, theoretically this may sound nice, but the external cost is this may create a logjam and a crowded staff area or kitchen, which will hinder the efficiency and effectiveness of the bartenders. In addition, having 10 bartenders in a small sport bar (only has 10 bar stools) can also create communication problems since it’s not easy to manage such a large group for a small operation. And adding one more (the 11th) bartender is only going to make the situation worse, hence validating the theory of increasing marginal cost. Another example is air pollution. factories create goods and by doing so, they also emit unwanted smokes. And the more factories there are, the more the air (and water, radiation, noise, etc) will be polluted, which can affect both other producers and consumers, and the factories themselves. Since it’s difficult to measure such external cost, the local government typically impose a special tax (and license fee) on the factories to roughly estimate the production of such externalities.

After spending a great deal of time introducing the Law of Diminishing Returns, opportunity cost, and external cost, now it’s time to put everything together and apply to the referee situation.


The question is: Will the game be improved by having more referees? My prediction is no.

The benefit from having an extra or even two referees officiating the game is that one extra person, or two, can see the game from a different angle, hence increasing the likelihood that the appropriate call is made at the moment. Moreover, having one extra brain to pick also helps during the replay sessions when an unclear call is made. All these positives are pointing direction that increasing th number of referees can improve the overall quality of the game. However, along with the good, here comes the negative that may affect the game in a subtle but tangible ways.

Adding one or two referees leads to a huge diminishing in returns. There are 10 players on the court at any given time and back in 1988, adding a third referee allowed each referee to hold accountable for 3.3 players (obviously officiating in the NBA doesn’t work this way). The third referee yielded a diminishing returns of 1.67. If we expand that number to including the fourth referee, this will yield a diminishing returns of 0.83, very close to half of 1.67. Moreover, adding a fifth referee would give us a diminishing return of 0.50. And a sixth would yield 0.33.


Of course, this is a very simple (and perhaps baseless) look at the effect of adding more officials in an NBA game. I excluded many factors such as foul calls, turnovers, and time outs. The former two are very important because these are what coaches are having the most problems with. And it is not easy to put a measuring metric on that. I do agree that having an extra official can help see the game better by precisely catching the proper calls such as traveling, charging and other turnovers. Nonetheless, this seemingly pleasant benefit is outnumbered by the cost. Here, we are strictly talking about opportunity cost and external cost. These two costs outweighs the benefit and, with that said, having an extra referee on the court is more likely to lower the quality of officiating than to raise it. Again, these are just my prediction only.

Let’s start with the external cost. Adding one referee could cause a logjam in the referee rotation in the already crowded basketball court that also features 10 players and occasionally a couple coaches (who like to run up to the court). This could lead to inefficiency and could slow the referee rotation and increase the likelihood a player runs into one by accident. Moreover, having one more referee on the court can also lead to management problem both during the game and at the instant replay review. The game has been playing at a faster pace each year since 2005-06, and this trend is expected to increase in future seasons. Having more officials in games playing at increasingly faster pace may cause on court chemistry problems between the officials, which could cause conflicts that disrupt the flow and quality of the competition, since it is always easier to manage with a smaller team. And these problems could be further magnified during the playoffs, the last few minutes of a tight game, and overtime period, where the intensity level rises as the players compete harder. So far this is just four officials. Imagine how cumbersome and crowded it would be if there were five officials on the court?

Additionally, another external cost I suspect that may come with having more referees on the court is the whistles blown. I anticipate having more officials can lead to an increased likelihood that two or more officials may 1) blow the whistles at the same time, 2) make different calls (ex: blocking vs. offensive foul), and 3) blow too many whistles throughout the game. Attached below (courtesy to is the league per game averages from 1978-79 to 2016-17 seasons (as of Dec. 13, 2016). I included the ten years prior to the league implementing three-person crews in the 1988-89 season so we can have an adequate sample of ten years to see how the games have changed before adding a third official to the games. And the results are quite interesting. (click here for the complete stats.) The stats on this page are up to date as of December 14, 2016.


To my surprise, personal fouls and free throws attempted have gone down significantly in the past 28 years since 1988. Turnovers have gone down but this category is not entirely dependent on the officials. Certain turnovers such as traveling, offensive fouls have impact because they are fully dependent by the officials’ judgement. Still, I was fascinated to learn that all three categories saw a decrease in the past 28 years because the statistics do not align with my anticipation. I anticipated an increase in all three categories and I was wrong. The table below depicts the averages of each category in the 10 years prior, and after the implementation of the third NBA official. I also included the average for the past ten years to better reflect contemporary averages.


Free throw attempts, turnovers, and personal fouls all have dropped. This may allude to the fact that players are playing smart and making good decisions by avoiding mistakes. However, it’s rather interesting to see the drop in free throw attempts. It’s hard to tell if the players are playing smart by avoiding contact or playing smart by drawing fouls. Or this may reflect a change in playing style in modern basketball where teams are emphasizing spacing by hoisting three pointers. Interestingly, the minutes (per game) saw an increase from the 10 years before 1988 and have remained virtually the same in the past 28 years. I suppose this may have something to do with the rise of the overall quality of NBA players which resulted in more overtime games being played. Despite that, it’s somewhat amusing to see the in modern basketball games the minutes are longer while there are less free throw attempts, turnovers, and fouls.

Regardless, let’s hold the player decisions and game strategies constant. If we solely focus on the officials, it may be attributed to the external cost, where having a third official decreases each official’s accountability, which gives them less incentive to strive to make the best calls because they are less focused since they rely on the others and believe the presence of the other two officials will make up for his or her missed calls, unbeknownst to the third official that the other two officials are probably thinking about the same, too.

This phenomenon reflects human inclination which is quite natural and understandable. Assigning three officials to a basketball game gives each official less responsibility, which may make them less concentrated on each possession. This explains MLB umpires are burdening heavy responsibility because they are pressured to make the correct call each time since the MLB assigns only one umpire on each base. Moreover, in NBA, since there are multiple officials managing one game, if a major mistake occurs during a critical possession that can alter the outcome of the game, each official also have a sense of anonymity since each one of them carries less blame from the critics. As a result, due to the fact there are three officials working in a single game, each one of them feels less responsible when a mistake happened–and this can be a serious problem if the league does decide to add a fourth or even a fifth official top the game. In a fascinating way, this situation perfectly illustrates how diminishing marginal returns and external cost are interrelated, which validates what we discussed earlier–the Law of Diminishing Marginal Returns implies an increase in marginal cost (in this case, the external cost).

And finally, moving on to the opportunity cost. We always think opportunity cost as a positive gain that never happened. In this case, the opportunity cost of having one additional official is the positive outcomes of not having that having that one additional official, such as easier crew management (because smaller team), fewer opinion conflicts during replay sessions, and many of the potential scenarios we discussed earlier. On the other hand, the opportunity cost of not having that one extra official is less diversity in opinions during replay sessions and one less pair of eyes to catch the proper calls. Remember that the benefit you gain from choosing one option is the opportunity cost of not choosing such option.

Consequently, people usually don’t see the costs in this way. Our society has taught us to look at cost from a financial standpoint and we often neglect the underlying costs that are not that readily visible. Oftentimes we can perceive the opportunity cost. But, in my personal opinion, I think external cost is what really matters because it is something that hinders efficiency and you cannot really measure such cost with a price.

Revisit our initial question that was about 3,200 words earlier in this article: How many officials is enough? My answer would be three. Four is simply too many people and each official bears even less responsibility and may have less incentive to concentrate on each possession. This human nature can lead to inefficiency and, like what I mentioned earlier, the additional officials are more likely to lower the quality of officiating. In conclusion, I think the NBA is better off with just three officials because basketball games can benefit more from less regulation. It’s all about quality over quantity and three officials are just enough. Again, this is simply my prediction. If a change will be made, I have a feeling we are still a long way from seeing an NBA game being officiated by four or five referees. Still, I am excited and curious to see the impact the four and five officials can have on the upcoming nine Long Island Nets games (they chose the Nets because the league wants to see the effect of different officials managing games that involve one same team–they want to hold at least one factor constant).

I will write again in a few weeks after the conclusion of all nine games. I hope by then I will learn something I did not know previously.


–J.H. Yeh

(all graphs are created by me, and all sources of information are courtesy of and as of December 14, 2016)


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