Facility Location Games Beyond Single-Peakedness: the Entrance Fee Model

TL;DR

This work introduces a general entrance-fee extension to one- and two-facility facility location games on the real line, where each facility at location $\ell$ charges an entrance fee $e(\ell)$ and an agent's cost is $\min_{\ell_j}( |x_i-\ell_j| + e(\ell_j) )$, potentially breaking single-peakedness. The authors develop structure-based mechanisms that are strategyproof and analyze their approximation ratios for utilitarian ($TC$) and egalitarian ($MC$) objectives, parameterized by the max-min ratio $r_e$ of the entrance-fee function. They provide polynomial-time optimization via dynamic programming, and present both deterministic and randomized mechanisms with explicit bounds for one and two facilities, along with nearly-tight lower bounds that depend on $r_e$. The results extend approximate mechanism design without money to a broader, more realistic setting where location-dependent fees influence agent choices, offering new insights into truthfulness, efficiency, and the design of practical facility configurations.

Abstract

The facility location game has been studied extensively in mechanism design. In the classical model, each agent's cost is solely determined by her distance to the nearest facility. In this paper, we introduce a novel model where each facility charges an entrance fee. Thus, the cost of each agent is determined by both the distance to the facility and the entrance fee of the facility. In our model, the entrance fee function is allowed to be an arbitrary function, causing agents' preferences may no longer be single-peaked anymore: This departure from the classical model introduces additional challenges. We systematically delve into the intricacies of the model, designing strategyproof mechanisms with favorable approximation ratios. Additionally, we complement these ratios with nearly-tight impossibility results. Specifically, for one-facility and two-facility games, we provide upper and lower bounds for the approximation ratios given by deterministic and randomized mechanisms with respect to utilitarian and egalitarian objectives.

Figures (11)

• Figure 1: The concept of the entrance fee function. The three black dots are agents. The vertical axis stands for the value of the entrance fee. The curve depicts an entrance fee function $e: \mathbb{R}\rightarrow \mathbb{R}_{\ge 0}\!\cup\! \{+\infty\}$. If the facility is located at $\ell$, the entrance fee is $e(\ell)$ and the cost of agent $i$ is $\mu|\ell-x_i|+e(\ell)$, where $\mu$ is the cost per unit-of-distance.
• Figure 2: A tight example for the approximation ratio of mechanism $m_{med}(\cdot,\cdot)$. The black dots are agents. The upper horizontal line represents the entrance fee function $e$. The height of the vertical line segment represents the entrance fee of the location.
• Figure 3: Definitions of $e(\cdot),\mathbf{x}_1,\mathbf{x}_2 \textrm{ and } \mathbf{x}_3$. Red triangles are facilities, and the dashed line denotes the deviation of the agent.
• Figure 4: A tight example for the approximation ratio in Theorem \ref{['thm_1f_tc_r_ub']}. The black dots are agents.
• Figure 5: Example for randomized lower bound $2$ for the total cost. The black dots are agents. The red letters denote the probabilities to locate the facility by the mechanism $f(e,\cdot)$. The dashed lines represent the deviations of the agents.

Theorems & Definitions (56)

• Remark 1
• Lemma 1: Monotonicity
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5