Testing Hypothesis 1 to 3

There are four models in part A. We first focus on whether CumSelf_ikt is having a positive relation to W eeklySalesikt and if CumOtherikt has a negative relation. This is directly investigated by model A₁. We add variables that belong to the basic performance information people can get when buying tickets into consideration in Model A₂. P eriod_k, T imek, Regionk, and Y eark are added. The interaction effects of performance infor-mation to CumSelf_ikt are further added in Model A₃ to see more detailed interactions.

We further add Show_k, P erf ormanceId_k, and W eek_kt as control variables to Model A₃ as Model A^D₃ to control the variation among performances. The effect of CumSelf_ikt and CumOther_ikt to W eeklySales_ikt might not be linear. Model A₄ is then proposed to examine the relationship. We take Af ternoon_k and Central_k as factors for T ime_k and Region_k. The reference table of variable names and math notations is in Table 4.11.

Model A₁:

W eeklySales_ikt = β₀+ β₁^SCumSelf_ikt+ β₂^SCumOther_ikt+ 

Model A₂:

W eeklySalesikt = β0+ β₁^SCumSelfikt+ β₂^SCumOtherikt+ β₁^PP eriodk

Variable Name Math Notation

In Model A₃ and A^D₃, we added the interaction effects and are denoted by β_j^XI, where X ∈ {S, P, T, R}.

We can find that the three models all have a significant correlation between cumulated sales to sales speed and a negative correlation between other price band’s cumulated sales to sales speed (Table 4.12). Hypothesis 1 and 2 are therefore proven. As for Hypothesis 3, the results of Model A2 and A3 both show that the length of sales period negatively affects sales speed and verified Hypothesis 3; however, when control variables Show_k, P erf ormanceId_k, and W eek_kt are added, P eriod_k becomes positive. W eek_kt are mostly negative with the level of W eekk1, which indicates that the sales speed of Week 1 is normally the fastest.

It is also considered that if the effect of Hypothesis 1 and 2 are not only linearly correlated. We add two variables: CumSelf_ikt² and CumOther_ikt² to Model A₃ as Model A₄.

Model A₄:

W eeklySales_ikt = β₀ + β₁^SCumSelf_ikt+ β₂^SCumOther_ikt+ β₁^PP eriod_k+ β₂^PW eek_kt +β₁^TM orning_k+ β₂^TEvening_k+ β₁^RN orthern_k+ β₂^RSouthern_k+ β₁^YY ear_k

+β₁^DShow_k+ β₂^DP erf ormanceId_k+ β₁^{P I}P eriod_k× CumSelf_ikt +β₁^{T I}M orning_k× CumSelf_ikt+ β₂^{T I}Evening_k× CumSelf_ikt +β₁^RIN orthern_k× CumSelf_ikt+ β₂^RISouthern_k× CumSelf_ikt

+β₃^SCumSelf_ikt² + β₄^SCumOther_ikt² + 

The previous variables maintain the same effects in Model A₄ (Table 4.12). If we fix all other variables and the equation that only consider the effect of CumSelfikt and

Regression Model A1 Model A2 Model A3 Model A^D₃

Adjusted R² 12.49% 18.51% 20.78% 39.60%

***p < 0.001, ** p < 0.01, * p < 0.05

Table 4.12: The Results of Part A

CumOther_ikt will be

W eeklySales_ikt= 0.469CumSelf_ikt− 0.157CumSelf_ikt²

+0.247CumOther_ikt− 0.369CumOther_ikt² .

The effects of CumSelf_ikt² and CumOther_ikt² are both interactive effects of that of CumSelf_ikt and CumOther_ikt.

We can obtain the first derivatives of y of CumSelf_ikt and CumOther_ikt.

∂W eeklySales_ikt

∂CumSelf_ikt = 0.469 − 0.314CumSelf_ikt.

∂W eeklySales_ikt

∂CumOther_ikt = 0.247 − 0.738CumOther_ikt.

Both CumSelf_ikt and CumOther are concave functions. Negative CumSelf_ikt² explains that the effect of CumSelf_ikt continues to increase positively with decreasing margin (Figure 4.3a). As for the relationship of CumOther_ikt to W eeklySales_ikt is that it first decreases positively and then when cumulated sales of other price bands is high, customer tend to have less incentive to purchase the price band’s ticket (Figure 4.3b). The situation is due to the interaction of two effects. One is that when people find that other price bands are selling good, they consider the show worthy to watch and purchase their most preferable price band of the show. The second one is that when people find that other price bands are selling good, they turn to buy the price bands with higher sales. From Graph 4.3b, we can conclude that the first effect is stronger during the start of the sales period and the second effect is stronger in the end of the sales period.

We verify the correlation coefficients between variables (Figure 4.4). We find that no two variables are highly correlated.

(a) CumSelf_ikt (b) CumOther_ikt

Figure 4.3: Change of W eeklySales_ikt.

Figure 4.4: Correlation Coefficients of Variables

4.4.2 Testing Hypothesis 4

When it comes to part B, we investigate if the effects of a price band to another are stronger when their prices are closer to each other. Price bands need to separately examined. We categorize other price bands as two types: Near and Far. For P1k, the price bands that are near is P_2k and the rest are far. For P_2k, the near price bands are P_1k and P_3k. The CumSold_ikt for Near and Far are CumN ear_ikt and CumF ar_ikt separately.

Model B:

W eeklySales_ikt= β₀+ β₁^SCumSelf_ikt+ β₂^SCumN ear_ikt+ β₃^SCumF ar_ikt

+β₁^PP eriodk+ β₂^PW eekkt+ β^T₁M orningk+ β₂^TEveningk+ β₁^RN orthernk+ β₂^RSouthernk

+β₁^YY eark+ β₁^DShowk+ β₂^DP erf ormanceIdk+ β₁^{P I}P eriodk× CumSelfikt

+β₁^{T I}M orning_k× CumSelf_ikt+ β₂^{T I}Evening_k× CumSelf_ikt

+β₁^RIN orthern_k× CumSelf_ikt+ β₂^RISouthern_k× CumSelf_ikt+ 

From Table 4.13 we find that the correlation between CumSelf_ikt, CumN ear_ikt, and CumF arikt to W eeklySalesikt has the same pattern as that of Model A^D₃ and A4. From Figure 4.5, we can also find that Model B has the same pattern as A₄. CumSelf_ikt, CumN ear_ikt, and CumF ar_ikt are all concave. CumN ear_ikt and CumF ar_ikt both are with decreasing marginal increase first and then reach an U-turn and decrease with in-creasing margin. Interestingly, CumF ar_ikt has larger effects compare with CumN ear_ikt. Hypothesis 4 is not verified and can be further investigated in the future.

Regression Model B

Table 4.13: The Results of Part B

(a) CumSelfikt (b) CumN earikt

(c) CumF ar_ikt

Figure 4.5: Change of W eeklySales_ikt for Model B.

Chapter 5

Application: Dynamic Seat Allocation

After verifying our hypotheses, we come up with a scenario that can apply our findings.

When a show with several price bands has been selling tickets for some time, people can verify if a specific price band is selling well or not with its cumulated sales outcome.

If a price band is selling very well and is about to sold out and another price band is not selling as well and has a lot of tickets remain to be sold, we can do dynamic seat allocation. We consider a case which is possible to adjust a portion of ticket capacity of the price bands selling inferior to the price bands that is about to sell out. That is to say in the end the fewer tickets are likely to be unsold.

We use regression models to predict the future sales and adjust ticket capacity of var-ious price bands based on the prediction. As we find that cumulated sales of price bands affect customer utility, we compared the profit difference of using the correct regression and the one neglecting cumulated sales.

A Performance’ Number of Price Bands Four Five Six Number of Performances 54 158 112

Table 5.1: Number of Price Bands of Performances

As previously mentioned, each performance has various numbers of price bands. We select some performances of five price bands as example for in our data set the perfor-mances with five price bands have the largest quantity (Table 5.1).

5.1 Exploratory data analysis

Having the same procedure as Chapter 4, we conduct exploratory data analysis in prior to implementing the application. The data with five price bands contains 20,125 rows of data and is examined. We examine the sales of performances by adding up N umSold_ikt of the five price bands. Then, we separately analyze each price band’s N umSold_ikt and W eeklySales_ikt. Due to the resemblance of distribution, we listed out only the information of N umSold_ikt in Table 5.2. The distributions are also all exponential and are very similar. Figure A.2a is taken as an example. As for the cumulated sales of the five price bands the variation is also huge and Figure A.2b is taken as an example.

P eriod_k is also right skewed for the performances with five price bands. From Table 5.2, we can identify that variations of the variables are all high.

T ime_kand Region_k have resembling patterns to the first part (Figure A.2d and A.2e).

The performances played in the morning is still the least (26). In the evening, there are 58 performances played during this time, which is slightly lower than the ones played in the

Variable Mean Standard Deviation Maximum Minimum

N umSold_kt of Performance k 35.378 66.779 728 −164

N umSold_1kt of P_1k 3.409 8.664 110 −33

N umSold_2kt of P_2k 4.250 9.655 138 −8

N umSold_3kt of P_3k 3.669 8.500 100 −26

N umSold_4kt of P_4k 3.637 7.994 71 −6

N umSold_5kt of P_5k 2.964 7.795 123 −27

CumP_1kt× Capacity_1k 57.699 56.872 261 0

CumP_2kt× Capacity_2k 75.800 68.662 331 0

CumP_3kt× Capacity_3k 70.460 74.712 549 0

CumP_4kt× Capacity_4k 66.463 72.878 509 0

CumP_5kt× Capacity_5k 50.686 54.196 391 0

P eriod_k 161.224 92.433 359 33

Table 5.2: Exploratory Analysis of the Variables of the Second Part

afternoon (59). Northern Taiwan continues to be the location where most performances take place (111). Central Taiwan has 23 performances and Southern Taiwan only has 9.

Y ear_k of the performances of the five price bands is very different from that of the first part. Numbers of performances of five price bands progressively decrease from 42 in 2008, 34 in 2009, 13 in 2010 to 9 in 2011. In 2012, it suddenly escalates to 45 (Figure A.2f).