**As you have come to know by now, there is perhaps no subject I am better versed in than the bobblehead.**

**I have argued ad nauseam about how Major League Baseball (MLB) teams are not revenue maximising when it comes to scheduling which home games will feature the cherished novelty.**

**And while I have suggested that teams are losing out on as much as $5 million on this inefficiency in 2019 only, I now have evidence to suggest that teams are leaving even**

__more__money on the table by offering the wrong__number__of bobbleheads on the days they choose to offer them.Scarcity is the concept that there are finite resources yet possibly infinite desires. It is the decisions and actions made by individuals in the face of scarcity makes up the very root of the study of economics. In the supply and demand framework, an item's scarcity impacts it's supply and thereby it's price.

Alternatively, scarcity bias in behavioural economics suggests that individuals value a good more if they perceive it to be scarce. Mixed with human's inane sense of loss aversion, or Fear Of Missing Out (FOMO), scarcity can be used to induce demand. Think of the times you have heard the phrase 'Act Now' or 'Limited Time Only' or the McDonald's McRib.

Major League Baseball teams have been known to use scarcity on their promotional giveaways to induce demand for tickets. You will see conditions attached to their promotional giveaway schedules such as "first 20,000 fans," or "while supplies last," used to signal their rareness of the item. While teams may deploy various strategies, one thing that becomes clear is the 20,000 bobbleheads promised by the Los Angeles Dodgers would not satiate the same share of fans as it would if promised by, say, the Miami Marlins: 20,000 bobbleheads would imply on average 40% of Dodgers fans get one, while nearly every Marlins fan could have two.

I calculate the probability of a consumer receiving a bobblehead as the number of bobbleheads promised divided by the capacity of the stadium. If a team does not provide a number of bobbleheads, I assume they plan to give everyone a bobblehead, or that the probability is 100%. Below is a histogram of these probabilities.

I calculate the probability of a consumer receiving a bobblehead as the number of bobbleheads promised divided by the capacity of the stadium. If a team does not provide a number of bobbleheads, I assume they plan to give everyone a bobblehead, or that the probability is 100%. Below is a histogram of these probabilities.

What can be observed from this histogram is that there is a slightly skewed distribution of the probability of receiving a bobblehead, with a giant spike at 90-100% (note that no team signaled what could be interpreted as 'false' scarcity - a limit of bobbleheads that is greater than it's capacity - although it could be a very interesting strategy). Ignoring the 100% values, the average bobblehead day sees a 40% probability a fans receives one. Including the 100%, the average is closer to 50%.

To determine what sort of effect the scarcity generates in ticket sales, I create a model to predict the paid attendance at MLB games during these years. Similar to other studies by yours truly, the unit of observation is the attendance at a single game and I include controls for the following:

- Home team fixed effects,
- Away team fixed effects,
- Day of the week fixed effects,
- Month fixed effects,
- Year fixed effects,
- Day/evening game indicator,
- Divisional rival indicator,
- Interleague indicator,
- Opening day (team's first home game of the season) indicator, and
- Predicted Season Wins.
^{1}

Then, if all is done correctly, the only thing I do

The leftover attendance that is not explained by the list of controls from above is known as the residual. If the residual is positive then the actual attendance is higher than what the model predicts. Conversely, if the residual is negative, the model predicts a higher attendance than is actually observed.

If we plot the residuals, we begin to see a pattern emerge for bobblehead days that is correlated with the probability of consumers receiving one. Computers are much better at recognising these types of patterns, so just in case you are not able to see it, I have drawn in a non-parametric line of best fit.

**not**control for is idiosyncratic shocks and ... the bobblehead scarcity. (Yes, I understand that there are likely more things I do not control for than things that I do, but this is a free blog).The leftover attendance that is not explained by the list of controls from above is known as the residual. If the residual is positive then the actual attendance is higher than what the model predicts. Conversely, if the residual is negative, the model predicts a higher attendance than is actually observed.

If we plot the residuals, we begin to see a pattern emerge for bobblehead days that is correlated with the probability of consumers receiving one. Computers are much better at recognising these types of patterns, so just in case you are not able to see it, I have drawn in a non-parametric line of best fit.

What this non-parametric pattern reveals is that there appears to be a local maximum at around 40%, a local minimum at around 80%, and an uptick at the 100% mark, although there is lots of variation in between.

What is also illustrates is that perhaps a parametric approximation is appropriate - this is beneficial sa it allows us to calculate the actual value of the local minimum and maximum, as well as test what the maximum value is over the range of bobblehead probabilities [0,1]. I therefore fit a cubic function in order to test this hypothesis. This is accomplished by adding four more controls to the model described above:

- Bobblehead Day indicator (the cubic's constant term)
- Probability of a Bobblehead,
- Probability of a Bobblehead-Squared, and
- Probability of a Bobblehead-Cubed.

I graph cubic function returned by the regression below. The function tells us what percent increase in fans we should expect at a bobblehead game on the Y-axis given the probability of receiving a bobblehead on the X-axis.

The results suggest the most fans attend when the probability of receiving a bobblehead is 38.7%. This is also known as the revenue

The strategy of offering 100% is not quite as lucrative, although it is very close, bringing in about 10.5% more fans on average. Conversely, a bobblehead probability of 80.9% is revenue

__maximising__strategy and it will bring in about 13.6% more fans, all else equal. This means that 65% of bobblehead days give out too many bobbleheads.The strategy of offering 100% is not quite as lucrative, although it is very close, bringing in about 10.5% more fans on average. Conversely, a bobblehead probability of 80.9% is revenue

__minimising__, and it is not predicted to bring in any additional fans.
But how much does choosing the bobblehead probability actually matter? To answer this, we could calculate the change in attendance if every team chose the revenue maximising value of bobbleheads.

Similarly to how I reassign bobblehead

Then I take the difference of revenue-maximising predicted and observed attendance and multiple it by the team's average ticket cost from Team Marketing Report. The end result is the forgone revenue due to an inefficient bobblehead strategy. Below is a graphical depiction with the forgone revenue from bobblehead day reassignment added for comparison (I estimated the latter back in March - see here).

Similarly to how I reassign bobblehead

**days**, I reassign the**number**of bobbleheads given out on the bobblehead days to 38.7%, and predict attendance. Because the predicted attendance will be higher than the observed attendance, I cap the predicted attendance at the stadium's capacity to avoid overestimating the impact.Then I take the difference of revenue-maximising predicted and observed attendance and multiple it by the team's average ticket cost from Team Marketing Report. The end result is the forgone revenue due to an inefficient bobblehead strategy. Below is a graphical depiction with the forgone revenue from bobblehead day reassignment added for comparison (I estimated the latter back in March - see here).

The sum total of revenue gained from changing the bobblehead day schedule is approximately $16M from 2016 to 2019. But the sum total from handing out the correct number of bobbleheads is ...

$25.6 million!

**A sobering thought ...**

If my model is correct, most teams do not revenue maxmise when deciding the number of bobbleheads to give out on promo days. About 65% of days promise too many bobbleheads. Therefore, from 2016 to 2019, more than 4 million extra bobbleheads were handed out than would

have otherwise been suggested.

At 18cm (7 inches) in height, 4 million bobbleheads is enough to stretch from San Franscisco to San Diego!

(or, for my readers in Canada, from Winnipeg to the Paris of the Prairies, Saskatoon)

But in other not-so-fun terms, a bobblehead weighs about 700g (~1.5lbs) and a garbage truck holds 12.5 tonnes (~14 US tons) meaning MLB produced 227 extra garbage trucks worth of bobbleheads and lost $25.6 million at the same time.

While a bobblehead day can be a fun way for teams to attract fans, it has also been shown that not getting the timing and quantity correct may be quite costly and not just in forgone revenue...

[1] Predicted Season Wins is the number of wins a team can expect to end the season with given their play in prior games and the probability of winning in future games. It is calculated as the actual number of wins prior to the observation plus the expected future number of wins from the observation to the end of the season (using the moneyline odds for information on the probability of winning each future game).

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