Platooning, driverless cars and ride hailing services have all been suggested as ways to reduce congestion. In this post I want to examine the use of coordination via ride hailing services as a way to reduce congestion. Assume that large numbers of riders decide to rely on ride hailing services. Because the services use Google Maps or Waze for route selection, it would be possible to coordinate their choices to reduce congestion.

To think thorough the implications of this, its useful to revisit an example of Arthur Pigou. There is a measure 1 of travelers all of whom wish to leave the same origin (${s}$) for the same destination (${t}$). There are two possible paths from ${s}$ to ${t}$. The top’ one has a travel time of 1 unit independent of the measure of travelers who use it. The bottom’ one has a travel time that grows linearly with the measure of travelers who employ it. Thus, if fraction ${x}$ of travelers take the bottom path, each incurs a travel time of ${x}$ units.

A central planner, say, Uber, interested in minimizing total travel time will route half of all travelers through the top and the remainder through the bottom. Total travel time will be ${0.5 \times 1 + 0.5 \times 0.5 = 0.75}$. The only Nash equilibrium of the path selection game is for all travelers to choose the bottom path yielding a total travel time of ${1}$. Thus, if the only choice is to delegate my route selection to Uber or make it myself, there is no equilibrium where all travelers delegate to Uber.

Now suppose, there are two competing ride hailing services. Assume fraction ${\alpha}$ of travelers are signed up with Uber and fraction ${1-\alpha}$ are signed up with Lyft. To avoid annoying corner cases, ${\alpha \in [1/3, 2/3]}$. Each firm routes its users so as to minimize the total travel time that their users incur. Uber will choose fraction ${\lambda_1}$ of its subscribers to use the top path and the remaining fraction will use the bottom path. Lyft will choose a fraction ${\lambda_2}$ of its subscribers to use the top path and the remaining fraction will use the bottom path.

A straight forward calculation reveals that the only Nash equilibrium of the Uber vs. Lyft game is ${\lambda_1 = 1 - \frac{1}{3 \alpha}}$ and ${\lambda_2 = 1 - \frac{1}{3(1-\alpha)}}$. An interesting case is when ${\alpha = 2/3}$, i.e., Uber has a dominant market share. In this case ${\lambda_2 = 0}$, i.e., Lyft sends none of its users through the top path. Uber on the hand will send half its users via the top and the remainder by the bottom path. Assuming Uber randomly assigns its users to top and bottom with equal probability, the average travel time for a Uber user will be

$\displaystyle 0.5 \times 1 + 0.5 \times [0.5 \times (2/3) + 1/3] = 5/6.$

The travel time for a Lyft user will be

$\displaystyle [0.5 \times (2/3) + 1/3] = 2/3.$

Total travel time will be ${7/9}$, less than in the Nash equilibrium outcome. However, Lyft would offer travelers a lower travel time than Uber. This is because, Uber which has the bulk of travelers, must use the top path to reduce total travel times. If this were the case, travelers would switch from Uber to Lyft. This conclusion ignores prices, which at present are not part of the model.

Suppose we include prices and assume that travelers now evaluate a ride hailing service based on delivered price, that is price plus travel time. Thus, we are assuming that all travelers value time at \$1 a unit of time. The volume of customers served by Uber and Lyft is no longer fixed and they will focus on minimizing average travel time per customer. A plausible guess is that there will be an equal price equilibrium where travelers divide evenly between the two services, i.e., ${\alpha = 0.5}$. Each service will route ${1/3}$ of its customers through the top and the remainder through the bottom. Average travel time per customer will be ${5/3}$. However, total travel time on the bottom will be ${2/3}$, giving every customer an incentive to opt out and drive their own car on the bottom path.

What this simple minded analysis highlights is that the benefits of coordination may be hard to achieve if travelers can opt out and drive themselves. To minimize congestion, the ride hailing services must limit traffic on the bottom path. This is the one that is congestible. However, doing so makes its attractive in terms of travel time encouraging travelers to opt out.