Facility Assignment with Fair Cost Sharing: Equilibrium and Mechanism Design

TL;DR

The paper studies a one-dimensional Facility Assignment with Fair Cost Sharing (FAG-FCS), where each facility incurs a building cost shared among users and each agent bears a personal connection cost. It analyzes two game-theoretic settings: (i) agents directly choose facilities and a dynamic-programming approach computes a Pure Nash Equilibrium in time $O(n^2 m^2)$, achieving a $\ln n$-approximation to the social optimum; (ii) mechanisms that assign based on reported positions, showing a strong lower bound that any strategyproof anonymous mechanism cannot bound the social-cost approximation, while providing a universal class of unanimous, anonymous, strategyproof mechanisms via rank-based constructions. Key contributions include the DP-based equilibrium computation, a potential-function framework for FAG-FCS, a complete $m=n=2$ mechanism characterization, and the general rank-mechanism design for all $n,m$. The results illuminate fundamental trade-offs between truthfulness and welfare in FAG-FCS and motivate extensions to metric FAG-FCS and broader parameter regimes. The open questions address the complexity of PNE in Metric FAG-FCS and extending the $m=n=2$ characterization to arbitrary $(m,n)$.

Abstract

In the one-dimensional facility assignment problem, m facilities and n agents are positioned along the real line. Each agent will be assigned to a single facility to receive service. Each facility incurs a building cost, which is shared equally among the agents utilizing it. Additionally, each agent independently bears a connection cost to access a facility. Thus, an agent's cost is the sum of the connection cost and her portion of the building cost. The social cost is the total cost of all agents. Notably, the optimal assignment that minimizes the social cost can be found in polynomial time. In this paper, we study the problem from two game-theoretical settings regarding the strategy space of agents and the rule the assignment. In both settings, agents act strategically to minimize their individual costs. In our first setting, the strategy space of agents is the set of facilities, granting agents the freedom to select any facility. Consequently, the self-formed assignment can exhibit instability, as agents may deviate to other facilities. We focus on the computation of an equilibrium assignment, where no agent has an incentive to unilaterally change her choice. We show that we can compute a pure Nash equilibrium in polynomial time. In our second setting, agents report their positions to a mechanism for assignment to facilities. The strategy space of agents becomes the set of all positions. Our interest lies in strategyproof mechanisms. It is essential to note that the preference induced by the agents' cost function is more complex as it depends on how other agents are assigned. We establish a strong lower bound against all strategyproof and anonymous mechanisms: none can achieve a bounded social cost approximation ratio. Nonetheless, we identify a class of non-trivial strategyproof mechanisms for any n and m that is unanimous and anonymous.

Theorems & Definitions (33)

• definition thmcounterdefinition: pure Nash equilibrium
• theorem thmcountertheorem
• definition thmcounterdefinition: congestion game
• theorem thmcountertheorem
• lemma thmcounterlemma: $\clubsuit$
• lemma thmcounterlemma
• proposition thmcounterproposition: $\clubsuit$
• definition thmcounterdefinition: strategyproofness
• definition thmcounterdefinition: anonymity
• theorem thmcountertheorem