Question

Choosing between public transportation and a private vehicle There is a single highway leading to a central business district (CBD). Commuters can travel on the highway using their own car, or use a public shuttle that travels on its own fast lane. We assume the following: - Commuters arrive according to a Poisson process with rate Λ>0. - Upon arrival, every commuter chooses between driving their own car or taking the shuttle. - The regular lane (for cars) is modelled as a FCFS M/M/1 queue with capacity μ. - A shuttle has n≥1 seats and departs when it is full. Every time a shuttle departs a new one arrives immediately. - The shuttle travel time is a constant d>0. - Commuters want to minimize the expected total commuting time to the CBD (=the sum of travel time and the waiting time if they choose the shuttle). 1 Nash equilibria 1.2. Compute all possible Nash equilibria in this game. Guidance. consider the following cases separately; (i) n=1, (ii) n>1,Λ<μ; (iii) n>1,Λ≥μ. Note that the possible equilibrium solutions in all cases depend on the value d !

Answer 1

Step-by-step explanation:

To compute the possible Nash equilibria in this game, we need to analyze the strategies of the commuters and find the situations where no commuter can benefit by unilaterally changing their strategy. We'll consider the three cases mentioned separately:

(i) **n=1:** In this case, the shuttle can only accommodate one passenger at a time. Each commuter must decide whether to take their own car or wait for the shuttle.

- If Λ < μ (arrival rate is less than the service rate of the car lane), then the car lane is always faster. Therefore, all commuters will choose to drive their own car, and there is no equilibrium involving the shuttle.

- If Λ ≥ μ (arrival rate is greater than or equal to the service rate of the car lane), then there is a possibility of using the shuttle. However, the shuttle can only serve one passenger at a time, and if multiple passengers arrive simultaneously, there will be a competition to board the shuttle. This can be a game with no pure Nash equilibrium since passengers may have different preferences. Mixed strategies could be possible, but the exact equilibrium depends on the specific details of passenger preferences and boarding rules.

(ii) **n>1, Λ < μ:** In this case, the shuttle can accommodate more than one passenger at a time, and the arrival rate is less than the car lane's capacity.

- The car lane is still faster for individual passengers, so all commuters would choose to drive their own cars, and there is no equilibrium involving the shuttle.

(iii) **n>1, Λ ≥ μ:** Here, the shuttle can accommodate more than one passenger at a time, and the arrival rate is greater than or equal to the car lane's capacity.

- In this case, the shuttle may be an attractive option for some commuters, especially when it's close to full capacity. However, if all commuters decide to use the shuttle, it will always depart immediately when full, and no one will benefit from switching to the car lane.

- An equilibrium might exist where some commuters choose the shuttle, and others choose to drive, depending on the specific values of Λ, μ, and d. However, finding the exact equilibrium requires detailed knowledge of passenger preferences and how they value their time.

In summary, the presence and nature of Nash equilibria in this game depend on the specific values of Λ, μ, n, and d, as well as the preferences and behavior of the commuters. The equilibrium strategies are not immediately apparent without more information about these factors.