Contribution IV: Dynamic pricing in ridesharing

Pricing is arguably the most basic setting where the system designer needs to take the incentives of the agents into consideration. From sports events to airline tickets, pricing is the simplest revenue management technique and therefore lies at the heart of many works in economics, operations research, and theoretical computer science. The simplest online pricing setting is the so called prophet inequalities [85] where the different agents i arrive online and have values drawn

from distributions Fiknown to the designer. The designer wants to set prices in

an online manner aiming to maximize her revenue knowing that the agents are price-taker, i.e. they will only purchase the good if their value is above the price. This problem can be formulated as a Markov Decision Process where the goal is to find the desired stopping time but there are also simple threshold-based schemes with a single threshold that achieve constant approximation ratios.

Main question. The rise of online markets has significantly complicated the complexity of these online pricing decisions; one of the best examples to illustrate this is a ridesharing application such as Lyft or Uber. In traditional pricing, there is a straightforward relation between the price displayed to a user and the availability of the good in the future: if the price is higher, the user is less likely to purchase the good and therefore the good is more likely to be available for future users. In ridesharing, we tend to have users in many different locations and the good is reusable as it corresponds to a driver providing a ride to the customer and this driver can be useful for future customers as well. As a result, a lower price at a location means that the driver is less likely to stay there to serve future local requests, but may help the driver serve another possibly profitable request in the destination of the customer – this can propagate throughout the system (affecting its state). These complex network state externalities of any single pricing decision makes this setting significantly more complicated than traditional pricing. Tackling these complexities, we pose the following question:

Can we design effective pricing at the face of network state externalities?

Result. To study this question, in joint work with Siddhartha Banerjee and Daniel Freund [17], we focus on a queueing-theoretic modeling of the setting as prominent in the literature of shared vehicle systems. In our model, we have n discrete locations (nodes) that correspond to the discretizations that such ridesharing companies employ in all their decisions; we also assume that there are m drivers (units). To isolate the first-order effect that we wish to study, we assume that the number of drivers is fixed and that the drivers are not strategic. For any pairs of nodes (i, j), there is a demand of price-taker customers that want to get rides; we assume a continuous-time (Poisson) arrival model and fixed

value distributions Fi j for any pair of nodes. The designer needs to select prices,

possibly in a state-dependent way (depending on the configuration of drivers across locations), aiming to maximize some desired objective such as revenue or social welfare. Since such systems tend to operate in fast timescales, we ignore the initial mixing time and focus on the steady-state performance of the resulting processes. In queueing-theoretic terms, the prices create an alternative Markov Decision Process (MDP) whose arrival rates are thinned via removing part of the demand; the goal is to create the MDP that optimizes the desired objective.

In this model, we derive a general approximation framework. The approxi- mation ratio of our approach is 1+n/m: asymptotically optimal as ratio of drivers per location increases and very close to 1 in the real-system parameters (there are typically significantly more drivers than locations). Notably, our pricing policy is state-independent (it outputs only one price for each pair of locations) but the guarantee stands even against state-dependent policies. Our framework applies to a large class of objective functions including throughput, welfare, revenue (un- der a regularity distributional assumption common in the revenue management literature), Ramsey pricing (max. revenue subject to lower bound on welfare). It also extends to constrained pricing settings such as cases where the prices need to come from some discrete set and to various other rebalancing controls such as deciding which driver to match to a particular customer and allowing for empty-vehicle rebalancing. Finally, our results apply generally to optimization in closed queueing networks (where the number of units remains unaltered), even outside the ridesharing application. We elaborate on these results in Chapter 5.

Technical highlight. Our framework which we term Elevated Flow Relaxation is based on solving a convex relaxation of the problem and deriving the approxi-

mation ratio via a three-step argument. In revenue management, it is easier to express the objectives in terms of quantiles associated to prices (percentage of de- mand that has value higher than the price) instead of prices. If there was always a driver available to serve any request, then the objective would be concave and as a result we could apply convex optimization techniques to find the optimal price. However, the difficulty arises due to the network supply externalities: each pricing scheme induces a Markov chain that has some probability of driver unavailability in each node. Unfortunately, the resulting system is non-convex with respect to the quantiles (or the prices) and therefore not easily optimizable. To tackle this issue, we first drop the dependence on the unavailability prob- ability from the objective function. This makes our objective concave but now the solution of the program does not necessarily correspond to some quantiles derived by some pricing scheme (as it does not deal with unavailability). To address this, we add flow conservation constraints which is a necessary con- dition for the solution to be actually achievable as quantiles of some pricing policy. We finally need to connect the solution of the relaxation to the m-unit system objectives we are interested in. For that, we show three properties: a) this solution is no less than the optimal state-dependent solution, b) this solution can be achieved by an infinite-unit system, and c) the objective of the m-unit system differs to the one of the infinite-unit system by at most a factor of 1+ n/m.