Game Theoretic Analysis of Driver Behaviour in Choosing Between Two Harbor Crossings in Hong Kong
The congestion problem of three vehicular harbor crossings in Hong Kong, the Cross-Harbour Tunnel, the Eastern Harbour Crossing, and the Western Harbour Crossing becomes a pressing problem. According to the Hong Kong News on September 16, 2022, the government proposed to implement a “congestion charge” on these three cross-harbor tunnels during rush hours to encourage people to use public transportation. [1] Before adjusting the price, it is important to understand private vehicle drivers’ behaviors in choosing which tunnels to take.
In the lecture, we discussed the network traffic problem. The same mechanism can be applied to the investigation of game theory in drivers’ choices between the Cross-Harbour Tunnel and the Western Harbour Crossing, specifically the route from Mong Kok Station to Central Station. (The Eastern Harbour Crossing is not discussed in this blog because choosing this crossing takes a much longer time compared with choosing the other two tunnels). The red line indicated in the map below is the Cross-Harbour Tunnel. [2]
The red line indicated in the map below is the Western Harbour Crossing. [2]
In this game, the players are identified as 8000 private vehicle drivers. There are two strategies: taking the Cross-Harbour Tunnel (CHT) and taking Western Harbour Crossing (WHC). Payoffs contain two components: traveling time and toll. According to Google Map, the traveling distance between Mong Kok Station and Central Station is 7.9 km if the driver chooses the Cross-Harbour Tunnel, and the distance for Western Harbour Crossing is 7.3 km. The speed limit is 70 km/hr. Assume there are no other cars on the road and a car driver can drive at a speed of 70 km/hr, the fastest traveling time for each tunnel is calculated as below:
In a congested situation, there are 3900 cars per hour in CHT during rush hour in the morning, and there are 4100 cars per hour in WHC. [3] According to Google Map, it takes 40 minutes to travel from Mong Kok Station to Central Station if the driver chooses CHT during rush hour, and it takes 26 minutes if the driver chooses WHC. The number of cars, in a given hour, that chooses each tunnel under congested conditions is calculated as below:
The data above can be summarized in the table:
| Shortest traveling time in Uncongested Condition | Number of Cars in Congested Condition | Traveling Time in Congested Condition | |
| CHT | 6.77143 min | 2600 | 40 |
| WHC | 6.25714 min | 1777 | 26 |
Table 1. The number of cars and Traveling Time Data
The graph below is plotted with the number of cars as the x-axis and the traveling time as the y-axis. The number of cars and traveling time in the congested condition for the CHT are plotted as point C, and that for the WHC as point A. Points H and G respectively represent the shortest traveling time in uncongested conditions in the CHT and the WHC.
Therefore, the linear equations of traveling time to the number of cars in these two tunnels are found. y is the traveling time in minutes and x is the number of cars in the tunnel.
The toll can be converted into time as part of payoffs. According to the Transport Department of Government of Hong Kong Annual Transport Digest, the toll of CHT For private cars is $20, and the toll of WHC is $75. [4] The median hourly wage in Hong Kong is $75.7. [5] The time equivalent to the toll can be calculated.
The toll and traveling time are combined into payoffs as
Let the number of car drivers that choose CHT to be xc, and the number of car drivers that choose WHC to be xw. Since there are a total of 8000 car drivers as players in this game, we have
To find the Nash Equilibrium in which there are no benefits for the private car drivers to change from one route to the other, the expected payoffs are equalized.
xc is solved to be 5523.68719 ≈ 5524, and xw is solved to be 2476.31281 ≈ 2476. Therefore, when there are 5524 car drivers choose CHT and 2476 car drivers choose WHC, there is a Nash Equilibrium. The result is consistent with reality: more cars are choosing CHT than choosing WHC.
There are several realistic factors are not taken into account in this case. For example, toll may not be able to be directly converted to time because different people have various opinions on money and time. Also, some people may value the driving experience so they prefer the route with fewer cars even though it takes longer to arrive at the destination.
References:
- “Congestion Charges May Impose on Cross-Harbor Tunnels During.” Hong Kong News, https://hksar.org/congestion-charges-may-impose-on-cross-harbor-tunnels-during-peak-hours.
- “About Geoinfo Map.” 地理資訊地圖．香港特別行政區政府提供的香港地圖, https://www.map.gov.hk/gm/.
- Ye, Haoyuan. “‘Congestion Charge’ for Three Tunnels.” Ta Kung Pao, 6 Nov. 2022.
- “Section 4road Tunnels and Control Areas.” Road Tunnels and Control Areas – Toll Rates as at 31 December 2019 | Annual Transport Digest 2020, Transport Department of Government of Hong Kong, https://www.td.gov.hk/mini_site/atd/2020/en/section4_t_2.html.
- Wages and Labour Earnings, Census and Statistics Department of the Government of Hong Kong Special Administrative Region, https://www.censtatd.gov.hk/en/scode210.html.