Characterization of Group-Strategyproof Mechanisms for Facility Location in Strictly Convex Space (1808.06320v3)

Abstract: We characterize the class of group-strategyproof mechanisms for the single facility location game in any unconstrained strictly convex space. A mechanism is \emph{group-strategyproof}, if no group of agents can misreport so that all its members are \emph{strictly} better off. A strictly convex space is a normed vector space where $|x+y|<2$ holds for any pair of different unit vectors $x \neq y$, e.g., any $L_p$ space with $p\in (1,\infty)$. We show that any deterministic, unanimous, group-strategyproof mechanism must be dictatorial, and that any randomized, unanimous, translation-invariant, group-strategyproof mechanism must be \emph{2-dictatorial}. Here a randomized mechanism is 2-dictatorial if the lottery output of the mechanism must be distributed on the line segment between two dictators' inputs. A mechanism is translation-invariant if the output of the mechanism follows the same translation of the input. Our characterization directly implies that any (randomized) translation-invariant approximation algorithm satisfying the group-strategyproofness property has a lower bound of $2$-approximation for maximum cost (whenever $n \geq 3$), and $n/2 - 1$ for social cost. We also find an algorithm that $2$-approximates the maximum cost and $n/2$-approximates the social cost, proving the bounds to be (almost) tight.

• The paper characterizes deterministic and randomized group-strategyproof mechanisms for facility location in strictly convex spaces.
• Deterministic group-strategyproof mechanisms are shown to be dictatorial, meaning one agent's input determines the facility location.
• The problem inherently faces a minimum approximation bound of 2 for maximum cost and n/2 - 1 for social costs, limiting mechanism performance.

Analysis of Group-Strategyproof Facility Location Mechanisms in Strictly Convex Spaces

The paper "Characterization of Group-Strategyproof Mechanisms for Facility Location in Strictly Convex Space" presents a comprehensive examination of group-strategyproof mechanisms within the context of a single facility location game, specifically focusing on strictly convex spaces. This research significantly contributes to the field of algorithmic mechanism design by providing detailed characterizations and insights into both deterministic and randomized mechanisms.

Facility location games involve a set of agents each reporting their preferred location, and the mechanism then decides on a facility location to minimize personal costs, which are typically measured as distances. The concept of strategyproofness—where no agent can benefit from misreporting their location—is extended in this paper to group-strategyproofness, ensuring no group of agents can collaborate to improve their collective outcome.

Key Findings

1. Characterization of Deterministic Mechanisms: The paper establishes that any deterministic, unanimous, group-strategyproof mechanism must be dictatorial in a strictly convex space. This implies that for a valid mechanism, there must exist a dictator agent whose reported location determines the facility's location, disregarding the preferences of other agents.
2. Randomized Mechanisms and 2-Dictatorship: For randomized mechanisms, the authors derive that any mechanism that is unanimous, translation-invariant, and group-strategyproof has to be 2-dictatorial when assessed under similar conditions. This means the distribution of the facility location is influenced by the inputs from exactly two agents (two dictators), providing a more flexible yet controlled form of strategyproofness compared to the deterministic case.
3. Approximation Bounds: The research concludes that the inherent structure of the problem allows for a minimum approximation bound of 2 for the maximum cost, implying that no mechanism can perform significantly better in minimizing the maximum individual costs. This extends to a bound of $n/2 - 1$ for social costs, up to a tight approximation.

Implications and Future Work

The characterization of group-strategyproof mechanisms in this paper is highly instructive, highlighting fundamental limitations in mechanism design within multidimensional settings. The finding that any deterministic mechanism must be dictatorial points to the limited scope for designing fair and inclusive solutions, thereby opening avenues for exploring hybrid or partially randomized strategies that might relax some constraints.

Furthermore, the results obtained for strictly convex spaces prompt interest in extending this paper to other domain geometries, such as Euclidean spaces where alternative metrics might reveal different error bounds or potentials for better approximations. The challenge remains to develop mechanisms that can achieve closer approximations without compromising group-strategyproofness, particularly in more complex multi-facility or networked environments.

In conclusion, this paper advances the understanding of facility location problems by rigorously addressing the constraints imposed by group-strategyproofness in strictly convex spaces, offering a foundation for both theoretical exploration and practical mechanism design in multi-agent systems.