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Strategic Dispatching Equilibrium under Competition in Multi-Regional Ride-Hailing Markets
Ran Chen, Nikolas Geroliminis
TL;DR
The paper asks how competing ride-hailing companies should allocate fleets across regions when fleet sizes differ. It formulates a static non-cooperative game, proves restricted equilibrium existence, and studies a two-company, two-region case. The smaller company concentrates in one region until its fleet share exceeds a threshold, while more balanced fleet sizes are associated with higher total industry profit and served demand under the Logit specification.
Problem
The paper examines how fleet-size heterogeneity affects regional dispatching strategies and equilibrium outcomes in competing multi-regional ride-hailing markets.
Method
The paper formulates fleet allocation as a static non-cooperative game and analyzes equilibrium existence through asymptotic quasi-concavity on a restricted feasible set.
Results
The smaller company concentrates its fleet in one region until its fleet share exceeds a threshold, while more balanced fleet sizes are associated with higher total industry profit and served demand under Logit demand.
Takeaways & Limitations
Relative fleet size strongly shapes equilibrium dispatching, and balanced fleet configurations are associated with better aggregate industry outcomes in the studied numerical setting.
Abstract
from arXiv · showhide
This paper studies competition between ride-hailing companies in a multi-regional market through a static non-cooperative game, in which each company allocates its fleet across regions to maximize profit. Under the assumptions of the proposed framework, we establish an equilibrium existence result on a restricted feasible set by analyzing the asymptotic quasi-concavity of the payoff functions. A numerical study of a two-company, two-region market shows that relative fleet size strongly affects equilibrium dispatching strategies: the smaller company tends to concentrate its fleet in one region and serves both regions only after its fleet share (in percentage) exceeds a threshold. In the numerical setting considered here and under the Logit demand specification, more balanced fleet sizes are associated with higher total industry profit and higher total served demand than strongly asymmetric fleet configurations.
I. INTRODUCTION
The paper examines how competition and fleet-size heterogeneity shape regional dispatching in ride-hailing markets. It combines a stylized non-cooperative game with equilibrium analysis and numerical study.
- Ride-hailing fleet management is complicated by spatial demand imbalances, local traffic conditions, and competition across regions.
- The paper studies how relative fleet sizes affect aggregate dispatching strategies and equilibrium outcomes in competing multi-regional ride-hailing markets.
- The model treats each company’s regional fleet allocation as a non-cooperative profit-maximization decision.
- Equilibrium existence is analyzed on a restricted feasible set using asymptotic quasi-concavity of payoff functions.
- The numerical study uses a two-company, two-region example to examine how market structure shapes dispatching behavior.
II. STATIC DISPATCHING GAME
The static dispatching game models regional fleet allocation through aggregate flow and demand relationships. It combines steady-state assumptions, Logit customer choice, and competitor responses to characterize payoffs.
- Each company allocates a total fleet across multiple regions, with regional empty and occupied fleets linked through aggregate conservation relationships.
- The model assumes stationary regional conditions, productive fleet decomposition, cost-sensitive demand, and accessible competitor information.
- Customer demand is modeled using linear generalized costs, exponential utility, and a Logit choice specification.
- Fares are treated as fixed because pricing decisions are assumed to change more slowly than dispatching decisions.
- Waiting time is represented as a convex decreasing function of empty fleet, while demand depends on service conditions and competitors’ generalized costs.
- An increase in one company’s allocated fleet reduces a competitor’s occupied fleet as improved service draws away demand.
- The collective fleet representation groups variables by region, producing regional block structure in the Jacobian because cross-region derivatives vanish.
- The Jacobian mapping from allocated fleets to empty fleets is well-defined on an operationally restricted domain when the regional blocks are invertible.
III. EXISTENCE OF NASH EQUILIBRIUM
The paper proves existence of a Restricted Nash Equilibrium on a finite restricted feasible set by establishing payoff continuity, compactness, and asymptotic quasi-concavity.
- III. EXISTENCE OF NASH EQUILIBRIUM: The proof handles non-global concavity by showing that each payoff becomes asymptotically quasi-concave beyond a fleet threshold.The payoff generally lacks a closed form, so the analysis uses asymptotic curvature properties instead.
- III. EXISTENCE OF NASH EQUILIBRIUM: A Restricted Nash Equilibrium exists on the restricted game G when the finite threshold fleet defines an admissible feasible set.The result applies to fleet allocations in X_ac and follows from the stated equilibrium conditions.
- III. EXISTENCE OF NASH EQUILIBRIUM: The restricted feasible set is non-empty, convex, and compact because it is defined by finitely many linear inequalities.These properties support application of standard equilibrium-existence conditions.
- III. EXISTENCE OF NASH EQUILIBRIUM: Continuity of demand and fleet-occupancy mappings, together with the asymptotic quasi-concavity result, completes the existence argument.The proof also relies on the existence of the relevant Jacobian mapping for sufficiently large fleet levels.
IV. NUMERICAL RESULTS
The numerical study models two ride-hailing companies competing across two regions under a total fleet cap and reports normalized equilibrium quantities.
- IV. NUMERICAL RESULTS: The numerical market contains two companies and two regions, with total supply N_tot constrained by an upper bound representing a regulatory cap.The cap is intended to represent policy aimed at mitigating congestion and market saturation.
- IV. NUMERICAL RESULTS: Region A represents a congested city center, and the simulation parameters for both regions are summarized in Table I.The parameter values are described as inspired by real-world operations.
- IV. NUMERICAL RESULTS: Equilibrium dispatching and occupancy quantities are normalized as percentages to compare outcomes across different total supply levels.The normalized quantities are expressed relative to each company’s fleet.
A. Variation of Equilibrium Strategy
Equilibrium dispatching depends strongly on relative fleet size: the smaller company concentrates in one region before expanding service to both regions, with the threshold varying by total supply.
- A. Variation of Equilibrium Strategy: For N_tot = 500, Company 1 allocates all vehicles to Region B when its fleet fraction α_1 ≤ 7.6%, then serves both regions once α_1 exceeds 7.6%.Beyond 10%, it assigns a larger share to Region A than to Region B.
- A. Variation of Equilibrium Strategy: As the smaller company’s fleet share increases, newly added vehicles shift proportionally toward Region A; when α_1 approaches 100%, Company 2 exhibits the analogous concentration pattern.The roles reverse as Company 2 becomes the smaller competitor.
- A. Variation of Equilibrium Strategy: The threshold for serving both regions rises from 13.7% at N_tot = 300 to 22.9% at N_tot = 200.Lower total supply keeps the smaller company concentrated in one region over a wider fleet-share range, with sharper transitions at low supply.
B. Equilibrium Financial and Social Outcomes
Equilibrium outcomes vary with fleet-share balance: total industry profit and total realized demand peak near equal fleet shares, while regional demand remains higher in Region B.
- Outcome measures: Each company’s equilibrium profit is examined separately before total industry profit, reflecting the firms’ individual payoff objectives.The analysis evaluates service outcomes using total realized regional demand.
- Regional demand: Across all Ntot scenarios, equilibrium demand is higher in Region B than in Region A.The paper attributes this pattern to Region A being harder to serve efficiently because of stronger congestion and public-transport competition.
- Industry outcomes: Total industry profit and total realized demand are highest near α1 = 50% across all three supply levels.Under the adopted Logit demand specification, more balanced fleet sizes are associated with higher total profit and higher total served demand.
V. CONCLUSION
The paper models regional fleet dispatching as a static non-cooperative game and establishes equilibrium existence on a restricted feasible set. Its numerical study links fleet-size balance to dispatching patterns and aggregate market outcomes, while identifying several extensions beyond the current scope.
- V. CONCLUSION: The model establishes equilibrium existence on a restricted feasible set despite generally non-concave payoffs.The proof uses asymptotic quasi-concavity beyond an operational fleet threshold.
- V. CONCLUSION: In the two-company, two-region numerical study, the smaller company concentrates in one region until its fleet-share percentage exceeds a threshold.The larger company tends to serve both regions, and the threshold increases as total supply decreases.
- V. CONCLUSION: Under the adopted Logit demand model, more balanced fleet sizes are associated with higher total industry profit and higher total served demand.This conclusion is limited to the numerical setting considered in the paper.
- V. CONCLUSION: Future work could address uniqueness, joint pricing and dispatching, and dynamic control under time-varying or uncertain demand.These directions relax assumptions or extend the static model’s decision variables and demand setting.
A. Proof of Invertible Jacobian JN(V)
The proof connects invertibility of the regional Jacobian blocks to existence of the fleet-to-variable Jacobian. The Logit demand structure and a positive test vector then establish invertibility, a well-defined inverse, and positivity of JV(N).
- A. Proof of Invertible Jacobian JN(V): Invertibility of the regional blocks is equivalent to existence of JV(N), and the Jacobian exists for N ≥ Ninv.The supplied passages state that the proof proceeds through regional Jacobian blocks.
- A. Proof of Invertible Jacobian JN(V): The proof applies a Z-matrix criterion: a real square matrix with non-positive off-diagonal entries is invertible when some positive vector maps to a positive vector.The selected vector has components set to reciprocals of total occupied time.
- A. Proof of Invertible Jacobian JN(V): The Logit demand model implies a positive regional Jacobian-vector product, yielding invertibility of JN(V).The passage explicitly connects this result to the positive test vector and the invertibility argument.
- A. Proof of Invertible Jacobian JN(V): By the Inverse Function Theorem, JV(N) = JN(V)^−1 is well-defined, and its entries are positive because JN(V) belongs to the Z-matrix class.The conclusion concerns the Jacobian relationship used in the equilibrium analysis.