*4.3. Transition Matrix According to Relatedness and Complexity Framework*

Table 3 shows the transition matrix based on the expert interviews. Here are the basic assumed conditions of the Markov chain process: the number of system states in the forecast period remains constant; the system state transition probability matrix will not change over time; and the state transition is only affected by the previous state, namely the non-aftereffect property. A Markov chain is a Markov process with a discrete time and discrete state. In a Markov chain, the transition of system state needs a probability matrix. The probability of a state at any later time point can be forecasted by the probability of states at initial time points through the state transition probability matrix. Thus, this study reaches the transition matrix in Table 3 through interviews with native economic experts and the senior executives of large-scale gambling companies to derive the transition probability matrix of the three scenarios in Table 3, namely, rapid transition from the gambling industry to the non-gambling industry, stable transition from the gambling industry to the non-gambling industry and slow transition from the gambling industry to the non-gambling industry.


**Table 3.** Transition probability matrix based on expert interviews.

As shown by the probability results in Table 3, when the gambling industry stably transfers to the non-gambling industry, in the following year, the probability that the gambling industry transfers to the non-gambling industry is 8% and the probability that it remains there is 92%; the probability that the non-gambling industry transfers to the gambling industry is 0.5% and the probability that it remains there is 99.5%.

#### *4.4. Forecast by the Markov Chain*

This study applies the ring price in 2015 to calculate a total added value (based on the producer's price and one million Macao dollars per unit) of 16 industries in Macao and the proper scales and growth rates of Macao's gambling industry and non-gambling industry were forecast until 2021 through the forecast model of the Markov chain. The situation at the initial time points was that the gambling industry's total output value was much larger than the non-gambling industry's total output value. Table 4 shows the relevant results under the assumption that the gambling industry transfers to the non-gambling industry through a state transition probability matrix.

**Table 4.** The scales and growth rates for Macao's gambling industry and non-gambling industry forecasted by the Markov chain in three scenarios (unit: million Macao dollars).


The appropriate scales for Macao's gambling industry and non-gambling industry in 2021 were forecast by the Markov chain. Under scenario 1 referred to Figure 3, the industrial added value of the gambling industry is predicted to be 159.050 billion Macao dollars and the industrial added value of the non-gambling industry is predicted to be 340.567 billion Macao dollars. The growth rates for gambling and non-gambling industry are −0.5% and 10.9% respectively. The economic diversification entropy index is predicted to be 2.01.

Under scenario 2 referred to Figure 4, the gambling industry moves smoothly to the non-gambling industry. In 2021, the industrial added value of Macao's gambling industry is predicted to be 191.441 billion Macao dollars, the industrial added value of the non-gambling industry is predicted to be 308.176 billion Macao dollars, the growth rates for gambling and non-gambling industry are 3.1% and 9.6% respectively and the economic diversification entropy index is predicted to be 1.84.

**Figure 3.** The scale for Macao's gambling industry forecast by the Markov chain—scenario 1.

**Figure 4.** The scale of Macao's gambling industry forecast by the Markov chain—scenario 2.

Under scenario 3 referred to Figure 5, the gambling industry slowly moves to the non-gambling industry. In 2021, the industrial added value of the Macao's gambling industry is predicted to be 215.902 billion Macao dollars and the industry added value of the non-gambling industry is predicted to be 283.715 billion Macao dollars. The growth rates for gambling and non-gambling industry are 5.6% and 8.1% respectively. The economic diversification entropy index is predicted to be 1.70. Under the three scenarios, the overall added value of Macao is continuously increasing.

**Figure 5.** The appropriate scale of Macao's gambling industry forecasted by the Markov chain—scenario 3.

#### **5. Discussion**

The entropy index of economic diversification in Macao, which was calculated in Section 4.1, was 1.99 in 2009. According to Measuring Economic Diversification in Hawaii [17] published by the Department of Business, Economic Development and Tourism of Hawaii in 2011, the entropy index of economic diversification of Hawaii, which positions tourism as its pillar industry, was 2.61 in 2009, while the value of the same index in the state of Nevada, home to Las Vegas, which positions gambling as its pillar industry, was 2.64. Both of these relevant index values are higher than Macao's, which means their level of economic diversification is higher than Macao's. From the empirical calculation of the entropy index of economic diversification and the comparison to Hawaii and Nevada, which have mainstay industries of gambling and tourism, respectively, Macao is revealed to have a relatively high industrial concentration and insufficient industrial diversification.

The calculation of efficiency by bootstrapping-DEA based on VRS is more accurate than efficiency calculated by the traditional BCC-VRS model, because the traditional DEA method has an advantage in parameter estimation. However, the traditional DEA method may bias the sample evaluation so that the statistic test is ignored. A bootstrapping-DEA model can make up for the traditional DEA method's deficiency by simulating the process of data generation through repeated sampling to fix the bias in the sample evaluation results. This study shows that the bootstrapping-DEA efficiency computed by repeated sampling calculation is more accurate than the efficiency computed by the traditional BCC-VRS model. Bootstrapping-DEA takes statistical tests into consideration so that bias resulting from sample evaluation is avoided.

This result is consistent with actual observations. During the gambling industry depression that began in 2015, the central government of China had adjusted the number of visitors to Macao, which resulted in the decline of gambling tax revenue. When gambling tax revenue declined, the level of Macao's economic diversification improved to some degree. Therefore, it is necessary to transfer the gambling industry's added value to the non-gambling industry.

### **6. Conclusions**

#### *6.1. Summary and Implications*

The principle of determining an appropriate scale for Macao's gambling industry is based on the premise of achieving appropriate diversification in Macao's economy. The gambling industry is the dominant industry in Macao. The goal of achieving appropriate industrial diversification in

Macao is associated with issues about scale and efficiency in the gambling industry. Therefore, we need to calculate the efficiency of the gambling industry by a bootstrapping-DEA model based on VRS. In terms of the gambling industry's efficiency, in 2016, both the traditional DEA-BCC efficiency and the bootstrapping-DEA efficiency of the gambling industry were less than 1, which reveals insufficient efficiency in the operation of the gambling industry. In 2013 and 2014, both the DEA-BCC efficiency and the bootstrapping-DEA efficiency were 1, which means Macao's gambling industry was efficient in these two years. However, the DEA-BCC efficiency and bootstrapping-DEA efficiency showed big differences from 2008 to 2011.

Through the Markov chain forecast, the quantitative gambling industry scale for the goal of appropriate economic diversification can be given. The appropriate scales for Macao gambling industry and non-gambling industry in 2021 were forecast by a Markov chain. Under the three scenarios, the overall added value of Macao was predicted to continuously increase. Under scenario 1, although the entropy index of economic diversification is high and the growth rate for non-gambling industry is above 10%, the growth rate for gambling industry is negative. Therefore, rapid transition causes an issue of sustainable growth for the gambling industry. Under scenario 3, when the growth rate of gambling is around 5.5%, the growth rate of non-gambling is around 8.2%. This growth rate is lower than the growth rates in 2012 and 2013. The slow transition does not provide a satisfying economic diversification. Under scenario 2, the gambling industry is smoothly transferred to the non-gambling industry. In 2021, in case the growth rate of gambling is around 3%, an economic diversification entropy index of 1.84 will be more suitable for the actual situation that the growth rate for non-gambling industry is around 10%.
