Study on Emission Control of Berthing Vessels-Based on Non-Cooperative Game Theory

Abstract

To accomplish IMO’s emission reduction targets, the Chinese government has established emission control areas and implemented strict sulfur limitation policies. Faced with a downturn in the shipping industry and the challenge of an insufficient supply of compliant fuel, Hong Kong and Shenzhen in China have implemented different low-sulfur fuel oil subsidy policies. It is particularly important to study non-cooperative games between two ports considering low-sulfur fuel oil subsidies. In this paper, first, non-cooperative game models considering low-sulfur fuel oil subsidies are constructed. Second, the mechanisms of various factors affecting port pricing, throughput and profit are analyzed. Then, a case study is conducted using AIS data of container ships in Shanghai and Ningbo-Zhoushan ports. The study reveals that in both sequential and simultaneous games, the gross tonnage of a ship has an impact on the optimal service price, throughput and profit of the port. The subsidy rate has a positive impact on the profitability of the port itself, to the detriment of competitor ports. In conclusion, a low-sulfur fuel oil subsidy policy has a significant positive impact on the step-by-step implementation of more stringent air pollution reduction policies in port waters.

1. Introduction

International shipping contributes significantly to the world economy, but it also brings serious air pollution problems to ports and nearby cities. The various air pollutants and greenhouse gases produced by vessels, such as $\mathrm{S}{\mathrm{O}}_2$, $\mathrm{N}{\mathrm{O}}_2$, ${\mathrm{P}\mathrm{M}}_{10}$, $\mathrm{C}\mathrm{O}$, $\mathrm{C}{\mathrm{O}}_2$ and ${\mathrm{C}\mathrm{H}}_4$, have a negative impact on the environment of port cities and the health of people living around ports. In California, cardiovascular disease and premature death caused by ship emissions have sparked protests from local residents. Although more pollutants are emitted when the vessel is sailing than when it is in port, the density of the population around the port is particularly high, and the population is very large, so the pollution from emissions around the port cannot be ignored [1,2,3].

The International Maritime Organization (IMO) has been concerned about ship emissions since the 1960s, and from then on, early targets for greenhouse gas (GHG) reductions were set, advocating that the total annual emissions of GHGs from maritime transport should be reduced by more than half by 2050 compared to 2008 [4,5]. In response to the IMO’s plan, the United States, the European Union, Singapore, China and the Mediterranean Sea have enacted a number of bills to reduce air pollutants emitted by ships [6]. At present, there are two internationally popular means of reducing emissions from ships: switching to shore power for ships in port and using low-sulfur fuel oil (LSFO) [7].

As early as 2014, the California Air Resources Boad (CARB) in the United States imposed a mandatory limit on the number of transfers of shore power for ships calling at California ports (including container ships, cargo ships and cruise ships). In China, the coverage of shore power in ports is very high, but many ships calling at ports are not equipped with shore power reception facilities, resulting in low utilization of shore power, and allowing ships to still emit air pollutants during berthing. In response, in July 2019, China’s Ministry of Transport and other environmental authorities issued a series of regulatory policies requiring ships calling at ports to switch to shore power. Subsequently, Hong Kong, Shenzhen and Guangzhou introduced a series of policies to give different subsidies to ships switching to shore power and to ports retrofitting the shore power facilities, aiming to effectively reduce air pollutant emissions from ships.

Switching to low-sulfur fuel oil can effectively reduce $\mathrm{P}\mathrm{M}$ and $\mathrm{S}{\mathrm{O}}_x$ emissions from vessels [8,9]. In May 2019, the IMO released a mandatory sulfur restriction requiring that starting from 1 January 2020, the sulfur content of marine fuel for vessels outside the specific Emission Control Area (ECA) does not exceed 0.5% m/m. In October 2019, the Ministry of Transport and the environmental protection departments of China released a regulation on sulfur restriction of fuel for vessels, which stipulates that from 1 January 2020, the sulfur content of marine fuel used by vessels on international voyages entering port waters in China shall not exceed 0.5% m/m [10]. In December 2019, the European Commission released “The European Green Deal”, which calls for zero emissions from vessels calling at ports by 2030. In December 2022, The Maritime Environment Protection Committee (MEPC) established the Mediterranean ECA, requiring that from 1 May 2025, vessels should use marine fuel with less than 0.1% m/m sulfur content in the ECA. A large number of studies have shown that the enactment of such regulatory policies is very effective and has resulted in significant environmental benefits for port cities and residents.

The application of transferring low-sulfur fuel oil to vessels calling at ports is extremely widespread, but the challenges encountered include an insufficient supply of low-sulfur fuel oil in some ports, as well as the extremely high cost of retrofitting ship equipment and the price of low-sulfur fuel oil. In this scenario, port authorities adopted various subsidy policies to increase the incentive for vessels calling at ports. In Hong Kong, if low-sulfur fuel is used, the Hong Kong government will waive 50% of port charges and lighthouse taxes. In 2019, the port of Shenzhen in China implemented a 75% to 100% fuel price differential subsidy for vessels that switch to low-sulfur fuel oil with a sulfur content of no more than 0.5% m/m. As of 2019, for seagoing vessels entering the Pearl River Delta (PRD) ECA and consistently using low-sulfur oil with sulfur content <0.1% m/m, shipping enterprises can apply for low-sulfur fuel oil subsidies according to the difference in price between low-sulfur oil with sulfur content <0.1% m/m and low sulfur oil with sulfur content <0.5% m/m. In this scenario, study is essential when stricter fuel regulation policies are planned, or when compliant fuel supplies are inadequate.

Up to now, there has been a great deal of research on the control of emissions from ships, and these studies mainly focus on the establishment of ship emission inventories for ports [11,12,13,14]. At present, a bottom-up approach to calculating emissions from vessels has become very popular [15,16,17,18]. Taking the port of Oslo as an example, López-Aparicio et al. used a classic bottom-up approach to estimate the emissions of ${\mathrm{P}\mathrm{M}}_{10}$, $\mathrm{S}{\mathrm{O}}_2$, $\mathrm{C}{\mathrm{O}}_2$, $\mathrm{C}{\mathrm{H}}_4$, and $\mathrm{N}{\mathrm{O}}_2$ from vessels and explored the impact of emission reduction measures on air pollutant emissions [19]. Zhang et al. calculated the vessel-source emissions and found that air pollutants emitted from unidentified vessels accounted for about 49% of the total emissions from Pearl River [20]. Tichavska et al. applied the classic Ship Traffic Emission Assessment Model (STEAM) to calculate ${\mathrm{P}\mathrm{M}}_{2.5}$, $\mathrm{C}\mathrm{O}$ and $\mathrm{C}{\mathrm{O}}_2$ emissions from vessels in the ports of St. Petersburg and Hong Kong based on data collected by automatic identification system (AIS) and finally suggested the need to revise the existing emission reduction policies [21]. It is worth noting that there are some differences in the ship exhaust emissions estimated using different models. Kwon et al. applied two methods to calculate emissions from vessels entering and leaving Incheon port, using a fuel consumption method and an emission calculation method based on basic ship activities, and found that the emissions estimated by the fuel consumption method were usually lower than those based on ship activities [22]. Ekmekcioğlu et al. collected AIS data for vessels arriving at Kocaeli port from 2017 to 2018, calculated the emissions produced by these vessels and discovered that the emission factor of $\mathrm{S}{\mathrm{O}}_2$ should be corrected [23].

More in-depth studies have applied a Weather Research and Forecasting (WRF) model to analyze the impact of vessel-source emissions on air quality around ports. Monteiro et al. assessed the impact of ship emissions on air quality in Portugal and simulated marine emission scenarios using the WRF model [24]. Chen et al. developed a vessel-source emissions inventory of Qingdao port and applied the WRF model to investigate the impact of emissions on the environment near Qingdao port [25].

To develop appropriate emission reduction policies, the factors influencing ship emissions need to be clarified. These factors include switching to low-sulfur fuel oil [26], the implementation of subsidies for shipping companies [27,28], monitoring by maritime authorities [29,30,31,32] and regional cooperation [33]. Dragovic et al. analyzed vessel-source emissions from two Nordic cruise ports and found that factors such as berth availability and berth accessibility affect the level of ship emissions [34]. Progiou et al. estimated emissions from the port of Piraeus and found that the increase in $\mathrm{N}{\mathrm{O}}_x$ emissions from the passenger port was mainly due to an increase in the number of ship calls, and the decrease in $\mathrm{S}{\mathrm{O}}_2$ emission levels was mainly attributed to a significant reduction in the sulfur content of marine fuels [35]. The choice of switching to low-sulfur marine fuel, retrofitting scrubbers or using LNG can all be effective in reducing $\mathrm{P}\mathrm{M}$ and $\mathrm{S}{\mathrm{O}}_2$ emissions from vessels [36]. However, as many shipping companies lack sufficient funds for retrofitting, more shipping companies are finally choosing to use low-sulfur marine fuel to comply with the emission regulations. The implementation of the low-sulfur fuel oil subsidy policy will help to alleviate air pollution in port waters, reduce the economic pressure on shipping companies and mitigate the shortage of compliant fuel supply in the port in the short term. Against the backdrop of a sluggish shipping economy, the implementation of this policy will help to attract more ships to call at the port and increase the incentive of shipping companies to reduce emissions. The subsidy policy is generally a temporary transitional policy, which helps to promote the soft landing of the IMO’s sulfur limitation policy in China.

Through the research, two issues were identified: (1) Most literature currently focuses on port emission inventories, the impact of ship emissions on air quality and the game strategies of multiple stakeholders in the port and shipping supply chain. There are relatively few studies analyzing the impact of low-sulfur fuel subsidy policies on game strategies between ports. (2) In particular, in much of the literature, the parameter values used for case studies differ significantly from the actual situation. In response to the above issues, this work investigates non-cooperative games between two ports considering low-sulfur fuel oil subsidies.

The work is structured as follows: After an introduction, Section 2 constructs a simultaneous game model and a sequential game model for duopoly ports considering low-sulfur fuel subsidies. Section 3 analyzes the factors influencing the optimal service pricefor ships and compares the results of simultaneous and sequential games. Section 4 conducts a case analysis and provides conclusions and policy recommendations.

2. Materials and Methods

2.1. Non-Cooperative Game Models for Air Pollution Control Considering Low-Sulfur Fuel Subsidies

Assumption 1. For the simultaneous game, the participants are ports 1 and 2, and the two ports make price decisions independently and unaware of the competitor’s price at the time of the decision. The payment is calculated by a profit function [37,38,39].

Assumption 2. For the sequential game, the participants are port 1 and port 2. Port 1 is a leader, and port 2 is a follower. Port 1 makes the price decision first, and port 2 makes the decision based on the observed price of port 1. Port 1 sets its service price knowing that port 2 will take the follow-up strategy. The payment is a profit function.

Assuming that when the port does not have sufficient compliant fuel, and ships choose to switch to cleaner fuel, the port authority will provide subsidies to shipping companies based on a certain percentage of the fuel price difference. Suppose that there are two ports in the region. According to the ECA’s regulation policy issued by the Ministry of Transport in 2018, vessels calling at a port should switch to low-sulfur fuel [10,13,14]. The parameters, decision variables and functions in the model are listed in

Table 1. Parameters, variables and functions in the non-cooperative game models.

Parameters	Explanation
$a$	Monthly throughput of regional ports
$\mu$	The impact factor of service price on the throughput
$v$	The impact factor of competitors’ service price on the throughput
$l_i$	Service cost per TEU for port $i$
$f_i$	The average fuel consumption of auxiliary engines per hour per ship calling at port $i$ during berthing period
$r_i$	Handling efficiency of a crane per hour in port $i$
$h_i$	Subsidy rate of low-sulfur fuel at port $i$
$c_A$	Price of MGO with a sulfur content of 0.1% m/m
$c_B$	Price of VLSFO with a sulfur content of 0.5% m/m
$c_0$	The price difference between the two types of low-sulfur fuel oil
Variables and functions	Explanation
$p_i$	Service price per TEU at port $i$
$q_i$	Monthly throughput of port $i$
$Q_i$	Total fuel consumption of vessels calling at port $i$ during berthing
$W_i$	Subsidies from ports for switching to ultra-low marine fuel when the supply of compliant fuel at port $i$ is insufficient
$J_i$	Profit function of port $i$

.

According to the Bertrand model [37,38,39], the demand function of port i is shown as follows:

$$q_i=a-u{p}_i+v{p}_j\left(i\ne j,i,j=\mathrm{1,2}\right)$$

(1)

where, $a>0$, $p_i$ and $p_j$ represent the service prices of ports $i$ and $j$, which are competitors to each other. The symbol $u$ denotes the impact factor of the port’s service price on its own throughput, and $v$ denotes the impact factor of the service price $p_j$ of the competitor’s port on the throughput $q_i$, and also represents the substitutability factor of the services provided by the two ports [38]. As the impact on itself is much greater compared to the competitor, so $0<v<u<1$.

The auxiliary engines are not switched off when the ship is at berth, in order to provide the required electrical and thermal energy for various mechanical equipment on board. Let $Q_i$ indicate the total fuel consumption of ships berthed at port $i$, which can be shown as follows:

$$Q_i=f_i\frac{q_i}{r_i}$$

(2)

where $f_i$ denotes the average fuel consumption of auxiliary engines per hour per ship for container ships calling at port $i$ (ton/h), and $\frac{q_i}{r_i}$ represents the total berthing time for all the ships regardless of congestion (h).

In fact, according to Dong and Lee’s research [40],

$$f_i=SFOC·LF·P_i·{10}^{-6}$$

(3)

where $SFOC$ denotes the average fuel consumption rate of auxiliary engines (g/kW·h), $LF$ represents the load factor of auxiliary engines (%), and $P_i$ is the average power of the auxiliary engines of container ships calling at port $i$ (kW).

The subsidy given by port $i$ to the ships switching to ultra-low marine fuel is:

$$W_i=h_i\left(c_A-c_B\right)Q_i$$

(4)

If $c_0=c_A-c_B$, it follows that:

$$W_i={c_{0}h}_i{f}_i\frac{q_i}{r_i}$$

(5)

The profit function of port $i$ is:

$$J_i=(p_i-l_i)q_i-W_i$$

(6)

After some simplification, it turns out that:

$$J_i=\left(p_i-l_i\right)\left(a-u{p}_i+v{p}_j\right)-\frac{h_i{c}_0{f}_i\left(a-u{p}_i+v{p}_j\right)}{r_i},\left(j\ne i,i,j=\mathrm{1,2}\right)$$

(7)

2.2. A Simultaneous Game Model for Two Ports

The two ports simultaneously make price decisions that maximize their own profits, and the price response functions of the two ports to their competitors can be obtained as:

$$p_1=\frac{v{p}_2+a}{2\mu }+\frac{c_0{h_{1}f}_1}{2r_1}+\frac{l_1}{2}$$

(8)

$$p_2=\frac{v{p}_1+a}{2\mu }+\frac{c_0{h_{2}f}_2}{2r_2}+\frac{l_2}{2}$$

(9)

It is assumed that $p_1^b$ and $p_2^b$ represent the optimal prices of the two ports in the simultaneous game. Similar to Cui and Notteboom [41], it is proven that:

$$p_1^b=\frac{2a{r}_1{r}_2\mu +a{r}_1{r}_{2}v+2c_0{h}_1{f}_1{r}_2{\mu }^2+c_0{h}_2{f}_2{r}_1\mu v+l_2{r}_1{r}_2\mu v+2l_1{r}_1{r}_2{\mu }^2}{r_1{r}_2(4{\mu }^2-v^2)}$$

(10)

$$p_2^b=\frac{2a{r}_1{r}_2\mu +a{r}_1{r}_{2}v+2c_0{h}_2{f}_2{r}_1{\mu }^2+c_0{h}_1{f}_1{r}_2\mu v+l_1{r}_1{r}_2\mu v+2l_2{r}_1{r}_2{\mu }^2}{r_1{r}_2\left(4{\mu }^2-v^2\right)}$$

(11)

Let $q_1^b$ and $q_2^b$ denote the optimal throughputs of the two ports in the simultaneous game, and insert the optimal prices into the demand functions. It turns out that:

$$q_1^b=\frac{\mu [-2c_0{h}_1{f}_1{r}_2{\mu }^2+c_0{h}_2{f}_2{r}_1\mu v+a{r}_1{r}_2(2\mu +v)+c_0{h}_1{f}_1{r}_2{v}^2-2l_1{r}_1{r}_2{\mu }^2+l_1{r}_1{r}_2{v}^2+l_2{r}_1{r}_2\mu v]}{r_1{r}_2\left(4{\mu }^2-v^2\right)}$$

(12)

$$q_2^b=\frac{\mu [-2c_0{h}_2{f}_2{r}_1{\mu }^2+c_0{h}_1{f}_1{r}_2\mu v+a{r}_1{r}_2\left(2\mu +v\right)+c_0{h}_2{f}_2{r}_1{v}^2-2l_2{r}_1{r}_2{\mu }^2+l_2{r}_1{r}_2{v}^2+l_1{r}_1{r}_2\mu v]}{r_1{r}_2\left(4{\mu }^2-v^2\right)}$$

(13)

Substituting Equations (10)–(13) into Equation (7), the profit functions of the two ports can be easily calculated.

2.3. A Sequential Game Model for Two Ports

The leader, port 1, makes the price decision first, and the follower, port 2, sets its price according to port 1’s decision, such that port 2’s profit is maximized, i.e.:

$$\underset{p_2}{\mathit{max}}{J}_2\left(p_1,p_2\right)=\left(p_2-l_2\right)\left(a-{\mu p}_2+v{p}_1\right)-\frac{{c_{0}h}_2{f}_2\left(a-{\mu p}_2+v{p}_1\right)}{r_2}$$

(14)

The price response function for port 2 is shown as follows:

$$p_2=\frac{a{r_2+l_2{r}_2\mu +p_{1}r}_{2}v+c_0{h}_2{f}_2\mu }{2r_2\mu }$$

(15)

Port 1 predicts the follower behavior of port 2 and then makes a price decision that maximizes port 1’s profit; thus:

$$\underset{p_1}{\mathit{max}}{J}_1\left(p_1,p_2\right)=(p_1-l_1)\left(a-{\mu p}_1+v{p}_2\right)-\frac{{c_{0}h}_1{f}_1\left(a-{\mu p}_1+v{p}_2\right)}{r_1}$$

(16)

where $p_2$ represents the price response function of port 2. Let $p_1^o$ and $p_2^o$ denote the optimal prices of ports 1 and 2 in the sequential game. Then, it follows that:

$$p_1^o=\frac{c_0{h}_1{f}_1{r}_2(2{\mu }^2-v^2)+c_0{h}_2{f}_2{r}_1\mu v+a{r}_1{r}_2(2\mu +v)+r_1{r}_2(2l_1{\mu }^2-l_1{v}^2+l_2\mu v)}{2r_1{r}_2\left(2{\mu }^2-v^2\right)}$$

(17)

$$p_2^o=\frac{\mu \left(4{\mu }^2-v^2\right)(c_0{h}_2{f}_2{r}_1+l_2{r}_1{r}_2)+v\left(2{\mu }^2-v^2\right)(c_0{h}_1{f}_1{r}_2+r_1{r}_2{l}_1)+a{r}_1{r}_2\left(4{\mu }^2+2\mu v-v^2\right)}{4r_1{r}_2\mu \left(2{\mu }^2-v^2\right)}$$

(18)

Let $q_1^o$ and $q_2^o$ represent the optimal throughputs of ports 1 and 2 in the sequential game, and insert the optimal prices into the demand functions. The optimal throughputs of two ports can be obtained as:

$$q_1^o=\frac{c_0{h}_1{f}_1{r}_2{v}^2+l_1{r}_1{r}_2{v}^2-2l_1{r}_1{r}_2{\mu }^2+a{r}_1{r}_{2}v+c_0{h}_2{f}_2{r}_1\mu v+l_2{r}_1{r}_2\mu v-2c_0{h}_1{f}_1{r}_2{\mu }^2+2a{r}_1{r}_2\mu }{4r_1{r}_2\mu }$$

(19)

$$q_2^o=\frac{\mu \left(3v^2-4{\mu }^2\right)(c_0{h}_2{f}_2{r}_1+l_2{r}_1{r}_2)+v(2{\mu }^2-v^2)(c_0{h}_1{f}_1{r}_2+l_1{r}_1{r}_2)+a{r}_1{r}_2\left(4{\mu }^2+2\mu v-v^2\right)}{4r_1{r}_2\left(2{\mu }^2-v^2\right)}$$

(20)

Substituting Equations (17)–(20) into Equation (7), the profit functions of the two ports can be easily calculated.

3. Results

This section analyzes the various factors affecting the optimal service price and profit of two ports and compares the results of the simultaneous game model with those of the sequential game model.

3.1. Analysis of the Factors Affecting the Optimal Service Prices of Two Ports

Proposition 1.

For both simultaneous and sequential games, the optimal prices of two ports $p_1^b$, $p_2^b$, $p_1^o$ and $p_2^o$ are positively correlated with $a$, $v$, $c_0$, $h_1$, $h_2$, $f_1$, $f_2$ and negatively correlated with $\mu$, $r_1$, $r_2$.

Proof: 

The optimal prices in the simultaneous game model are shown as Equations (10) and (11):

$$p_1^b=\frac{2a{r}_1{r}_2\mu +a{r}_1{r}_{2}v+2c_0{h}_1{f}_1{r}_2{\mu }^2+c_0{h}_2{f}_2{r}_1\mu v+l_2{r}_1{r}_2\mu v+2l_1{r}_1{r}_2{\mu }^2}{r_1{r}_2(4{\mu }^2-v^2)}$$

$$p_2^b=\frac{2a{r}_1{r}_2\mu +a{r}_1{r}_{2}v+2c_0{h}_2{f}_2{r}_1{\mu }^2+c_0{h}_1{f}_1{r}_2\mu v+l_1{r}_1{r}_2\mu v+2l_2{r}_1{r}_2{\mu }^2}{r_1{r}_2\left(4{\mu }^2-v^2\right)}$$

The optimal service prices of the two ports in the sequential competition are shown as follows Equations (17) and (18):

$$p_1^o=\frac{c_0{h}_1{f}_1{r}_2(2{\mu }^2-v^2)+c_0{h}_2{f}_2{r}_1\mu v+a{r}_1{r}_2(2\mu +v)+r_1{r}_2(2l_1{\mu }^2-l_1{v}^2+l_2\mu v)}{2r_1{r}_2\left(2{\mu }^2-v^2\right)}$$

$$p_2^o=\frac{\mu \left(4{\mu }^2-v^2\right)(c_0{h}_2{f}_2{r}_1+l_2{r}_1{r}_2)+v\left(2{\mu }^2-v^2\right)(c_0{h}_1{f}_1{r}_2+r_1{r}_2{l}_1)+a{r}_1{r}_2\left(4{\mu }^2+2\mu v-v^2\right)}{4r_1{r}_2\mu \left(2{\mu }^2-v^2\right)}$$

Using MATLAB software to find the first-order derivatives separately, it is easy to find that the conclusion holds. The study discovers that reducing the difference in fuel prices, decreasing the rate of low-sulfur fuel subsidies and improving the handling efficiency of cranes at ports can keep the optimal service prices lower. □

3.2. Comparative Analysis of Simultaneous and Sequential Game Models

Proposition 2.

The price difference in the leader port is positively proportional to the price difference in the follower port.

Proof: 

Based on the calculated optimal service prices of the two models, the difference between the optimal service prices $∆p_1$ and $∆p_2$ for the sequential competition and simultaneous competition are calculated as follows:

$$∆p_1=-\frac{v^2\left[c_0{h}_1{f}_1{r}_2\right(2{\mu }^2-v^2)-\mu v(c_0{h}_2{f}_2{r}_1+l_2{r}_1{r}_2)+l_1{r}_1{r}_2(2{\mu }^2-v^2)-a{r}_1{r}_2(2\mu +v\left)\right]}{2r_1{r}_2\left(2\mu +v\right)\left(2\mu -v\right)\left(2{\mu }^2-v^2\right)}$$

(21)

$$∆p_2=-\frac{v^3\left[c_0{h}_1{f}_1{r}_2\left(2{\mu }^2-v^2\right)-\mu v\left(c_0{h}_2{f}_2{r}_1+l_2{r}_1{r}_2\right)+l_1{r}_1{r}_2\left(2{\mu }^2-v^2\right)-a{r}_1{r}_2\left(2\mu +v\right)\right]}{4r_1{r}_2\mu \left(2\mu +v\right)\left(2\mu -v\right)\left(2{\mu }^2-v^2\right)}$$

(22)

Suppose $∆p_i$ denote the difference between $p_i^o$ and $p_i^b$ for port $i$, it. It follows that:

$$\frac{∆p_1}{∆p_2}=\frac{p_1^o-p_1^b}{p_2^o-p_2^b}=\frac{2\mu }{v}>0$$

(23)

Thus, the above formula indicates that the price difference in the leader port is positively proportional to the price difference in the follower port. □

Proposition 3.

The difference in optimal throughput of the leader port between sequential and simultaneous games, $∆q_1$, is negatively proportional to the difference in optimal throughput of the follower port between sequential and simultaneous games, $∆q_2$.

Proof: 

Let $∆q_i$ represent the difference between $q_i^o$ and $q_i^b$ for port $i$. Using MATLAB software, it is shown that:

$$∆q_1=\frac{v^2\left[c_0{h}_1{f}_1{r}_2(2{\mu }^2-v^2)-\mu v\left(c_0{h}_2{f}_2{r}_1+l_2{r}_1{r}_2\right)+l_1{r}_1{r}_2\left(2{\mu }^2-v^2\right)-2a{r}_1{r}_2\mu -a{r}_1{r}_{2}v\right]}{4r_1{r}_2\mu \left(4{\mu }^2-v^2\right)}$$

(24)

$$∆q_2=-\frac{v^3\left[c_0{h}_1{f}_1{r}_2(2{\mu }^2-v^2)-\mu v\left(c_0{h}_2{f}_2{r}_1+l_2{r}_1{r}_2\right)+l_1{r}_1{r}_2\left(2{\mu }^2-v^2\right)-2a{r}_1{r}_2\mu -a{r}_1{r}_{2}v\right]}{4r_1{r}_2\left(4{\mu }^2-v^2\right)\left(2{\mu }^2-v^2\right)}$$

(25)

It is proven that:

$$\frac{∆q_1}{∆q_2}=-\frac{\left(2{\mu }^2-v^2\right)}{\mu v}$$

(26)

As $0<v<u<1$, it can be concluded that $∆q_1$ is negatively proportional to $∆q_2$. □

Let $∆J_1$ denotes the difference between $J_1^o$ and $J_1^b$ for port 1. $J_1^o$ and $J_1^b$ represent the optimal profit of port 1 obtained from the sequential game and the simultaneous game. Then, it follows that:

$$∆J_1=\frac{v^4{\left[(2{\mu }^2-v^2)(c_0{h}_1{f}_1{r}_2+l_1{r}_1{r}_2)-c_0{h}_2{f}_2{r}_1\mu v-a{r}_1{r}_2(2\mu +v){-l}_2{r}_1{r}_2\mu v\right]}^2}{8r_1^2{r}_2^2\mu \left(2{\mu }^2-v^2\right){\left(2\mu -v\right)}^2{\left(2\mu +v\right)}^2}>0$$

(27)

It can be easily deduced that if the optimal service price in the sequential game in port 1 is higher than that in the simultaneous game, the profit in the sequential game in port 1 is greater than that in the simultaneous game.

There are a number of factors that affect the difference in optimal profit between the sequential and the simultaneous games for port 2, and it is theoretically impossible to determine whether the optimal profit of port 2 increase during the sequential competition, so this study conducts a numerical analysis in the case study section.

4. Discussion and Case Analysis

AIS data is provided by HiFleet (www.HiFleet.com, accessed on 6 May 2021) and provides shipping enterprises with accurate services such as ship location enquiries, historical trajectories and port information analysis. Based on these data, vessels are classified into three categories according to their gross tonnage. The values of parameters such as the average handling efficiency of container bridge cranes were analyzed by consulting a large number of statistical yearbooks and other reference materials. Finally, taking Ningbo-Zhoushan port and Shanghai port as examples, a case study is conducted.

4.1. Data Source

For simplicity, it is assumed that port 1 is Shanghai port, and port 2 is Ningbo- Zhoushan port. Then, the AIS data of container ships calling at two ports were analyzed to obtain the average power and fuel consumption of auxiliary engines.

4.1.1. AIS Data of Container Ships in Shanghai Port and Ningbo-Zhoushan Port

This study obtained AIS data for container ships calling at Shanghai and Ningbo-Zhoushan ports in January 2018 and collected the data about callsign, gross tonnage (GT), net tonnage (NT), deadweight tonnage (DWT), the main engine power, building year, destination and maritime mobile service identify (MMSI). A sample of the AIS data for these container ships is given in

Table 2. Sample values of AIS data for container ships calling at two ports.

Callsign	IMO	MMSI	Gross Tonnage	Net Tonnage	DWT	Berth	Building Year	Distance (nm)
BLBX	8,901,755	416,260,000	17,123	7336	23,692	Waigaoqiao	1990	121.30
BIBP7	9,159,878	413,378,250	16,705	9118	24,336	Wusong	1997	164.18
D5IR9	9,189,500	636,016,980	66,526	29,460	67,712	Yangshan	2000	139.74
3EB09	9,320,403	371,860,000	50,963	30,224	59,587	Changxing	2006	125.34
9LU2532	9,258,210	667,001,729	1510	705	2212	Baoshan	2001	147.41

.

According to the statistics, in January, 603 container ships made approximately 851 ship calls. Among the container ships calling at Shanghai port, those with a gross tonnage of 50,000 GT or more made about 315 calls, accounting for approximately 37%. Container ships of 10,000 to 49,999 GT also made 315 ship calls. This was followed by container ships of less than or equal to 9999 GT, with 221 ship calls, accounting for about 26%. According to a report from the US Environmental Protection Agency (EPA), on average, a container ship is equipped with 3.6 auxiliary engines, and the average auxiliary engine power compared to the propulsion power of a container ship is 0.22 [42], from which the average auxiliary engine power of container ships can be estimated, as shown in

Table 3. Data of ship calls at Shanghai port.

	GT	$\le 9999$	$[\mathrm{10,000},\mathrm{49,999}]$	$\ge \mathrm{50,000}$
Container Ships
Number of ship calls		221	315	315
Percentage		26%	37%	37%
The average power of auxiliary engines $P_1$ (kW)		1728.73	4744.68	12,392.47
$f_1$ (ton/h)		0.19	0.52	1.37

below.

Removing the abnormal data and missing data, there are approximately 292 ship calls from 207 container ships in January. Among the container ships calling at Ningbo-Zhoushan port, container ships with a gross tonnage of over 50,000 are the most, with a total of 215 ship calls, accounting for 74%. This was followed by container ships of 10,000–49,999 GT, with 64 ship calls, accounting for 22%. Container ships of less than or equal to 9999 GT made 13 ship calls to this port, approximately 4%. According to a report from the US EPA, the average auxiliary engine power compared to propulsion power for container ships is 0.22 [42], from which the average auxiliary engine power for container ships can be estimated, as shown in

Table 4. Data of ship calls at Ningbo-Zhoushan port.

	GT	$\le 9999$	$[\mathrm{10,000},\mathrm{49,999}]$	$\ge \mathrm{50,000}$
Container Ships
Number of ship calls		13	64	215
Percentage		4%	22%	74%
The average power of auxiliary engines $P_2$ (kW)		1725.98	5022.15	11,684.20
$f_2$ (ton/h)		0.19	0.55	1.29

below.

4.1.2. Price of Marine Fuel

Taking very low-sulfur fuel oil (VLSFO) and marine gas oil (MGO) as examples, the emission control area implementation plan requires vessels calling at the port to use fuel oil with a sulfur content of no more than 0.5% m/m [10,13,14,30]. If VLSFO is not available, vessels calling at the port are required to refill with MGO with 0.1% m/m sulfur content. The vessels can apply for an economic subsidy from the port authority where the vessels are calling, depending on the type of fuel and the fuel price difference. The port authority subsidizes the shipping company where the container ships switched to MGO at a percentage of the difference in fuel price. By using Clarkson to obtain the price of bunker fuel, it is shown that from 2019 to early 2023, the average bunker prices of VLSFO and MGO in Singapore were 587 USD/ton and 728 USD/ton. The bunker prices of VLSFO and MGO are taken as $c_A=728$ USD/ton and $c_B=587$ USD/ton, so the subsidy cost of Shanghai port and Ningbo-Zhoushan port is:

$$W_i=141·h_i{f}_i\frac{q_i}{r_i}$$

(28)

where $h_i$ is the subsidy rate, $h_i\in \left(\mathrm{0,1}\right]$, $f_i$ is the average hourly fuel consumption per ship during berthing for auxiliary engines of the container ships, and $r_i$, $q_i$ are the handling efficiency of each bridge crane per hour and the monthly throughput of containers.

4.1.3. Parameter Values

In 2018, the container throughput of Shanghai port was ${4.201\times 10}^7$ TEUs, and the container throughput of Ningbo-Zhoushan port was ${2.635\times 10}^7$ TEUs, so the average regional monthly demand ${a=5.70\times 10}^6$ TEUs. Referring to Zheng and Negenborn [38], the price impact factor $\mu =1.50\times {10}^5$ TEUs/$, and the competitor’s price impact factor $v=1.00\times {10}^5 \mathrm{T}\mathrm{E}\mathrm{U}\mathrm{s}/\mathrm{\$}$. The service cost and fuel subsidy rate per TEU at Shanghai port and Ningbo-Zhoushan port are taken as $l_1=74$ USD/TEU, $l_2=88$ USD/TEU, $h_1=100\%,h_2=75\%$, i.e., the fuel difference is fully subsidized. The larger the gross tonnage, the greater the fuel consumption rate becomes. According to Sohu.com, the daily handling capacity of each crane in Yangshan port exceeded 1000 TEUs in 2021, so it is assumed that the average handling efficiency is about 42 TEUs for each crane per hour. The handling efficiency of cranes in Shanghai and Ningbo-Zhoushan ports are taken as $r_1=42$ TEUs/h, and $r_2=30$ TEUs/h.

Table 5. Parameter values.

Parameters	Explanation	Value
$a$	Monthly throughput of regional ports	$5.70\times {10}^6$ TEUs
$\mu$	The impact factor of service price on the throughput	$1.50\times {10}^5$ TEUs/$
$v$	The impact factor of competitors’ service price on throughput	$1.00\times {10}^5$ TEUs/$
$r_1$	Handling efficiency in Shanghai port	$42$ TEUs/h
$r_2$	Handling efficiency in Ningbo-Zhoushan port	$30$ TEUs/h
$l_1$	Service cost per TEU in Shanghai port	$74$ USD/TEU
$l_2$	Service cost per TEU in Ningbo-Zhoushan port	$88$ USD/TEU
$h_1$	Fuel subsidy rate at Shanghai port	$1$00%
$h_2$	Fuel subsidy rate at Ningbo-Zhoushan port	75%
$c_A$	Price of MGO	$728$ USD/ton
$c_B$	Price of VLSFO	$587$ USD/ton
$c_0$	The price difference between the two types of low-sulfur fuel oil	141 USD/ton

summarizes the values of the parameters used in the case study.

4.2. A Case Study of Non-Cooperative Games Considering Low-Sulfur Fuel Oil Subsidies

In this section, firstly, a case analysis of simultaneous and sequential competition models is conducted to examine the impact of the gross tonnage of container ships on the optimal service price, throughput and profitability of the ports. Then, the results of the empirical analysis of the two competitive models are compared. Secondly, the impact of the subsidy rates on the profit of two ports is investigated to further refine the mechanism of the impact of the low-sulfur fuel oil subsidy policy on the optimal price, throughput and profit of the ports.

4.2.1. Discussion of the Case Analysis

To analyze the effect of gross tonnage of container ships on the optimal service price ($p_i^b,p_i^o$), optimal throughput ($q_i^b,q_i^o$) and profit ($J_i^b,J_i^o$), the container ships are first classified into three categories according to gross tonnage. Then, the optimal service price, optimal throughput and profit of Shanghai port and Ningbo-Zhoushan port under the two competition models are calculated. Finally, the obtained numerical results are compared.

1.

Simultaneous competition between two ports.

The results of the case study are shown in

Table 6. Results of simultaneous competition between two ports.

	GT	Unit	$\le 9999$	$[\mathrm{10,000},\mathrm{49,999}]$	$\ge \mathrm{50,000}$
Parameters
$p_1^b$		USD	87.11	87.97	90.06
$p_2^b$		USD	92.37	93.29	95.30
$q_1^b$		${10}^6$ TEUs	1.87	1.83	1.72
$q_2^b$		${10}^6$ TEUs	0.56	0.50	0.41
$J_1^b$		${10}^6$ USD	23.33	22.42	19.72
$J_2^b$		${10}^6$ USD	2.06	1.69	1.13

. When the two ports play the game simultaneously, the increase in the gross tonnage of container ships makes the optimal service price of Shanghai port and Ningbo-Zhoushan port increase slightly, and it makes the optimal throughput and profit for both ports decrease. The larger the gross tonnage, and the greater the number of container ships, the more significant the effect of gross tonnage on optimal service price, throughput and profit becomes.

2.

Sequential competition between two ports

The results are shown in

Table 7. Results of sequential competition between two ports.

	GT	Unit	$\le 9999$	$[\mathrm{10,000},\mathrm{49,999}]$	$\ge \mathrm{50,000}$
Parameters
$p_1^o$		USD	88.89	89.72	91.70
$p_2^o$		USD	92.97	93.88	95.84
$q_1^o$		${10}^6$ TEUs	1.66	1.63	1.53
$q_2^o$		${10}^6$ TEUs	0.64	0.59	0.49
$J_1^o$		${10}^6$ USD	23.70	22.77	20.03
$J_2^o$		${10}^6$ USD	2.77	2.32	1.63

. When the two ports compete sequentially, the increase in the gross tonnage of container ships makes the optimal service price of Shanghai and Ningbo-Zhoushan ports increase slightly, and it makes the optimal throughput and profit of both ports decrease. The larger the gross tonnage of container ships, and the greater the number of ships, the more significant the influence becomes. The result is similar to the situation in which two ports compete simultaneously.

3.

Comparative analysis of two non-cooperative game models

The results of the comparative analysis are shown in

Table 8. Comparative analysis of two non-cooperative game models.

	GT	Unit		$\le 9999$	$[\mathrm{10,000},\mathrm{49,999}]$	$\ge \mathrm{50,000}$
Parameters
$∆p_1$			USD	1.78	1.75	1.64
$∆p_2$			USD	0.59	0.58	0.55
$∆p_1/∆p_2$			/	3	3	3
$∆q_1$			${10}^6$ TEUs	−0.21	−0.20	−0.19
$∆q_2$			${10}^6$ TEUs	0.09	0.09	0.08
$∆q_1/∆q_2$			/	−2.33	−2.33	−2.33
$∆J_1$			${10}^6$ USD	0.37	0.36	0.31
$∆J_2$			${10}^6$ USD	0.71	0.64	0.49
$∆J_1/∆J_2$			/	0.52	0.56	0.63

. The formula $∆p_i=p_i^o-p_i^b>0 (i=\mathrm{1,2})$ indicates that regardless of whether the gross tonnage of container ships takes a large or small value, the optimal service price of container ships increases in sequential competition compared to simultaneous competition, regardless of whether it is the port of Shanghai or Ningbo-Zhoushan. The formula $\frac{∆p_1}{∆p_2}=\frac{p_1^o-p_1^b}{p_2^o-p_2^b}=3>0$ holds, indicating that $∆p_1$ and $∆p_2$ are positively correlated. The fluctuation in the optimal service price of Shanghai port, which plays the role of leader, is much larger than the fluctuation in the optimal service price of Ningbo-Zhoushan port. $∆q_1=q_1^o-q_1^b<0$ and $∆q_2=q_2^o-q_2^b>0$ indicate that the optimal throughput of Shanghai port decreases and the optimal throughput of Ningbo-Zhoushan port increases in sequential competition compared to simultaneous competition, regardless of whether the gross tonnage is taken as large or small. The formula $\frac{∆q_1}{∆q_2}=\frac{q_1^o-q_1^b}{q_2^o-q_2^b}=-2.33$ holds, indicating that for container ships, the decrease in the optimal throughput of Shanghai port is greater than the increase in that of Ningbo-Zhoushan port.

$∆J_1=J_1^o-J_1^b>0$ and $∆J_2=J_2^o-J_2^b>0$ indicate that the optimal profit of both Shanghai and Ningbo-Zhoushan ports increases in sequential competition compared to simultaneous competition, regardless of whether the value of gross tonnage is taken as large or small. The formula $0<\frac{∆J_1}{∆J_2}=\frac{J_1^o-J_1^b}{J_2^o-J_2^b}<1$ reveals that the difference in optimal profit of the leader port between sequential and simultaneous games, $∆J_1$, is positively related to the difference in optimal profit of the follower port between sequential and simultaneous games,$∆J_2$. The difference in the optimal profit of Shanghai port is smaller than the difference in the optimal profit of Ningbo-Zhoushan port.

4.2.2. The Impact of the Subsidy Rates on the Profit of Two Ports

The impact of the subsidy rates on the profit of two ports is analyzed using container ships of 50,000 GT or more as an example. Figure 1 and Figure 2 show the interaction between $J_1^b$, $J_2^b$ and the low-sulfur fuel oil subsidy rates $h_1$, $h_2$. Figure 1 uses different colors to indicate the optimal profit for the port of Shanghai when competing simultaneously. The highest profits are indicated in red, up to $2.52\times {10}^{10}$ USD, followed by purple, blue, green and yellow, and the lowest profits are indicated in orange. Figure 2 uses different colors to indicate the optimal profit for Ningbo-Zhoushan Port when competing simultaneously. The highest profits are indicated in red, up to $2.18\times {10}^9$ USD, followed by purple, blue, green and yellow, and the lowest profits are indicated in orange. The figures discover that the profit gained by Shanghai port is much greater than the profit gained by Ningbo-Zhoushan port during the simultaneous game. The numerical results show that the profit of Shanghai port is positively correlated to $h_1$ and negatively correlated to $h_2$ when the game is played simultaneously. Similarly, for Ningbo-Zhoushan Port, the optimal profit $J_2^b$ is positively correlated with $h_2$ and negatively correlated with $h_1$. The implication is that the implementation of low-sulfur fuel oil subsidies in a port can attract more ships to call at this port, making it more profitable.

In Figure 3, the highest profit for Shanghai port, when competing sequentially with Ningbo-Zhoushan port, is shown in red, up to $2.56\times {10}^{10}$ USD, followed by yellow, green and blue. In Figure 4, the highest profit for Ningbo-Zhoushan port, when competing sequentially with Shanghai port, is shown in red, up to $2.85\times {10}^9$ USD, followed by yellow, green and blue. Figure 3 and Figure 4 reveal that the conclusions in the sequential game are similar to those in the simultaneous game. The low-sulfur fuel subsidy rate has a positive effect on the profit of the port itself, to the detriment of competitor ports. The profit gained by the leader port is higher than that gained by the follower port.

Figure 5 and Figure 6 illustrate that regardless of whether the two subsidy rates take large or small values, there is always $∆J_1>0$, i.e., $J_1^o>J_1^b$, which means that the optimal profit of the leader port in the sequential competition is higher than the profit obtained in the simultaneous competition. Figure 6 shows that regardless of the subsidy rates, $∆J_2>0$ always holds, i.e., $J_2^o>J_2^b$. Moreover, the profit difference $∆J_1$ for Shanghai port is positively related to $h_1$ and negatively related to $h_2$, and vice versa. The figures discover that the greater the port’s subsidy rates, the larger the difference in profit, implying that the sequential game is more favorable to this port in this scenario.

4.3. Conclusions and Policy Recommendations

This study investigates non-cooperative games between two ports considering low-sulfur fuel oil subsidies and conducts a case study of Shanghai port and Ningbo-Zhoushan port as an example. The main conclusions and policy recommendations are as follows:

(1)

In both sequential and simultaneous games, the gross tonnage of a ship has a significant impact on the optimal service price, throughput and profit of the port. When formulating subsidy policies for air pollution prevention in port waters, it is recommended that port authorities could consider implementing different subsidy criteria depending on the gross or net tonnage of the ship.

(2)

The low-sulfur fuel oil subsidy rate has a positive impact on the profitability of the port itself, to the detriment of competitor ports. When a port authority implements a low-sulfur fuel oil subsidy policy, it can attract more ships to call at that port, making it more profitable. The greater the subsidy rate of the port, the greater the difference in profits between the two games, which means that in this case, sequential competition is more favorable to both ports, making both ports more profitable.

(3)

The low-sulfur fuel oil subsidy policy can help alleviate the plight of port compliance fuel shortages and excessive pressure on shipping companies to reduce emissions and can lead to a soft landing for IMO’s emissions reduction policy, which plays a positive and important role in improving air quality in port waters. In designing an effective subsidy policy, government departments need to consider the impact of incentive policy on the service price, throughput and competitiveness of ports.

However, in reality, bunker prices, subsidy rates, exchange rates and other factors will change, which will have an impact on the service price, throughput and profit of ports, and maritime sectors should take these factors into account for decision making. It is proposed that future research could consider the following two aspects: (1) exploring the long-term effects of low-sulfur fuel oil subsidy policies and evaluate the corresponding economic and environmental impacts; and (2) further enriching the relevant theories of non-cooperative games by combining air pollution control policies in port waters.