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Facility Location Games with Entrance Fees: Approximation Ratios Beyond Single-Peakedness


Core Concepts
The authors introduce a novel facility location game model where each facility charges an entrance fee, leading to agent preferences that may no longer be single-peaked. They design strategyproof mechanisms with favorable approximation ratios for the total cost and maximum cost objectives, and provide nearly-tight impossibility results.
Abstract
The authors introduce a novel facility location game model where each facility charges an entrance fee, in addition to the travel fee. This causes agent preferences to no longer be single-peaked, introducing additional challenges compared to the classical model. The key insights and results are: The authors show that the optimal location for each agent can be found efficiently, and establish important structural properties of these optimal locations. For the one-facility game with the total cost objective, the authors design a deterministic strategyproof mechanism (MEDIAN) that achieves an approximation ratio of 3 - 4/(re+1), where re is the max-min ratio of the entrance fee function. They also provide a matching lower bound, showing this ratio is tight. For the one-facility game with the maximum cost objective, the authors design deterministic and randomized strategyproof mechanisms with approximation ratios parameterized by re. The authors extend their results to the two-facility game, providing upper and lower bounds for both objectives. The authors show that the optimization problems can be solved in polynomial time, despite the additional complexity introduced by the entrance fee function. Overall, the authors demonstrate that classical mechanisms can be extended to preference domains beyond single-peakedness while maintaining strategyproofness, and provide a comprehensive analysis of the impact of entrance fees on the approximation ratios of facility location games.
Stats
The total cost of the optimal solution is at least the sum of the optimal costs of all agents. The total cost of the randomized mechanism is at most (3 - 2/n) times the optimal total cost.
Quotes
"The entrance fee of the facility is decided by the building cost and thus also decided by the location where the facility is built." "The arbitrariness of the entrance fee function introduces new challenges in designing strategyproof mechanisms. Agent preferences may no longer adhere to single-peakedness, and standard mechanisms for the classical model cannot be directly extended to our setting while preserving strategyproofness."

Deeper Inquiries

How can the results be extended to settings with heterogeneous facilities, where agents have different acceptable sets of facilities

The results obtained in the paper can be extended to settings with heterogeneous facilities by considering different acceptable sets of facilities for each agent. In this scenario, each agent would have a unique set of facilities that they find acceptable based on their preferences or requirements. The mechanism design would need to take into account these heterogeneous preferences and optimize the facility locations to minimize the total or maximum cost based on the diverse acceptable sets of facilities. To extend the results to settings with heterogeneous facilities, the mechanism design would need to incorporate the individual preferences of each agent and ensure that the facilities are located in a way that serves the diverse needs of the agents. This extension would require a more complex optimization process to account for the varied acceptable sets of facilities and the different cost considerations for each agent.

What are the implications of allowing the entrance fee function to be a more complex function of the facility location, beyond the linear relationship considered in this work

Allowing the entrance fee function to be a more complex function of the facility location beyond the linear relationship considered in the paper would have significant implications for the mechanism design and optimization process. By introducing a more complex entrance fee function, such as a non-linear or piecewise function, the cost calculation for each agent would become more intricate and potentially nonlinear. The implications of this complexity would include: Increased Computational Complexity: The optimization process would become more computationally intensive as the cost calculations involve non-linear functions and potentially more variables. Enhanced Realism: A more complex entrance fee function would better reflect real-world scenarios where entrance fees are determined by various factors in a non-linear manner, leading to more realistic modeling. Challenging Optimization: Designing strategyproof mechanisms and achieving favorable approximation ratios would become more challenging due to the non-linear relationships between the facility locations and entrance fees. Improved Customization: A more complex entrance fee function would allow for a more customized and nuanced approach to cost calculation, catering to specific scenarios and preferences. Overall, allowing for a more complex entrance fee function would add realism to the model but also introduce additional challenges in terms of optimization and mechanism design.

Can the techniques developed in this paper be applied to other mechanism design problems beyond facility location games, where agent preferences exhibit a departure from single-peakedness

The techniques developed in this paper for facility location games with entrance fees can be applied to other mechanism design problems beyond single-peaked preferences. The departure from single-peakedness in agent preferences introduces challenges in designing strategyproof mechanisms, which are common in various mechanism design problems. Some applications of the techniques to other mechanism design problems include: Auction Design: In auction settings where bidders have complex preferences over items and prices, the techniques can be used to design mechanisms that elicit truthful bidding and optimize outcomes. Resource Allocation: For resource allocation problems where agents have diverse utility functions and constraints, the techniques can be adapted to design mechanisms that allocate resources efficiently while ensuring truthfulness. Matching Markets: In matching markets where agents have complex preferences over potential matches, the techniques can be utilized to design mechanisms that result in stable and efficient matches. By adapting the methodology and principles from facility location games with entrance fees, researchers can address a wide range of mechanism design problems with non-trivial preferences, leading to more robust and effective solutions.
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