A Blotto Game Approach to Ride-hailing Markets with Electric Vehicles

TL;DR

The paper addresses strategic EV fleet allocation for a ride-hailing entrant competing with an incumbent across multiple regions, accounting for regional demand, charging prices, and customer abandonments. It models the interaction as a Blotto-like contest with a modified Tullock payoff $u_i(x^i,x^{-i})=\sum_j x^i_j(\frac{\beta^m_j}{x^i_j+x^{-i}_j+\epsilon_j}-\beta^c_j)$, proving that a unique Nash equilibrium exists and deriving a general interior-NE characterization via KKT conditions and a nonlinear scalar $t_\lambda^*$; for two regions, boundary equilibria are also tractable. The interior-NE form yields explicit region-level allocations driven by $\kappa_j^*$ and $t_\lambda^*$, while numerical cases show charging prices can substantially shift fleet splits and profits, and that boundary NE can emerge in small-region settings. Overall, the framework provides a tractable, equilibrium-based tool for planning market entry and charging-price strategies in EV-based ride-hailing markets, with potential extensions to more regions, more players, and rebalancing costs.

Abstract

When a centrally operated ride-hailing company considers to enter a market already served by another company, it has to make a strategic decision about how to distribute its fleet among different regions in the area. This decision will be influenced by the market share the company can secure and the costs associated with charging the vehicles in each region, all while competing with the company already operating in the area. In this paper, we propose a Colonel Blotto-like game to model this decision-making. For the class of games that we study, we first prove the existence and uniqueness of a Nash Equilibrium. Subsequently, we provide its general characterization and present an algorithm for computing the ones in the feasible set's interior. Additionally, for a simplified scenario involving two regions, which would correspond to a city area with a downtown and a suburban region, we also provide a method to check for the equilibria on the feasible set's boundary. Finally, through a numerical case study, we illustrate the impact of charging prices on the position of the Nash equilibrium.

Figures (5)

• Figure 1: Representation of the Blotto game setup with two companies and six 'battlefields' $\mathcal{J}=\{J_1,J_2,J_3,J_4,J_5,J_6\}$.
• Figure 2: Illustrative example plot of the function \ref{['eq:14']} for $j\in\{1,2,3\}.$
• Figure 3: Four-region setup - Fleet distribution and earned profit of the companies in the attained NE for different values of parameter $\alpha$.
• Figure 4: Two-region setup - Attained NE of the companies and secured profits in different regions for different values of parameter $\alpha$. The vertical black line represents the experimentally obtained critical value $\alpha_{\text{crit}}$.
• Figure 5: Two-region setup - Attained NE of the companies and total profits of the companies for different values of $\mathbf{X}_b$

Theorems & Definitions (13)

• Definition 1: Nash equilibrium
• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• Lemma 2
• proof
• Lemma 3
• proof