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Equilibria in Two-Stage Facility Location with Atomic Clients

Simon Krogmann, Pascal Lenzner, Alexander Skopalik, Marc Uetz, Marnix C. Vos

TL;DR

This work analyzes a two-stage facility location game with atomic (unsplittable) clients, where facilities place first and clients respond by patronizing a single facility chosen from their neighborhood, possibly randomized. It introduces a novel potential-based approach with hierarchical client classification and rounding to establish the existence of pure subgame perfect equilibria for unit-weight clients, while showing weighted instances can lack SPE and are NP-hard to approximate within $\phi$; nonetheless, a $k$-approximate SPE always exists. The paper also proves a tight price of anarchy of $2$, indicating strong social welfare at equilibrium when equilibria exist, and provides practical constructions such as minimum neighborhood sets and $\pi$-favoring equilibria to ensure convergence. These results deepen understanding of two-sided competitive facility location and suggest robust techniques for multi-stage strategic interactions with atomic agents.

Abstract

We consider competitive facility location as a two-stage multi-agent system with two types of clients. For a given host graph with weighted clients on the vertices, first facility agents strategically select vertices for opening their facilities. Then, the clients strategically select which of the opened facilities in their neighborhood to patronize. Facilities want to attract as much client weight as possible, clients want to minimize congestion on the chosen facility. All recently studied versions of this model assume that clients can split their weight strategically. We consider clients with unsplittable weights but allow mixed strategies. So clients may randomize over which facility to patronize. Besides modeling a natural client behavior, this subtle change yields drastic changes, e.g., for a given facility placement, qualitatively different client equilibria are possible. As our main result, we show that pure subgame perfect equilibria always exist if all client weights are identical. For this, we use a novel potential function argument, employing a hierarchical classification of the clients and sophisticated rounding in each step. In contrast, for non-identical clients, we show that deciding the existence of even approximately stable states is computationally intractable. On the positive side, we give a tight bound of $2$ on the price of anarchy which implies high social welfare of equilibria, if they exist.

Equilibria in Two-Stage Facility Location with Atomic Clients

TL;DR

This work analyzes a two-stage facility location game with atomic (unsplittable) clients, where facilities place first and clients respond by patronizing a single facility chosen from their neighborhood, possibly randomized. It introduces a novel potential-based approach with hierarchical client classification and rounding to establish the existence of pure subgame perfect equilibria for unit-weight clients, while showing weighted instances can lack SPE and are NP-hard to approximate within ; nonetheless, a -approximate SPE always exists. The paper also proves a tight price of anarchy of , indicating strong social welfare at equilibrium when equilibria exist, and provides practical constructions such as minimum neighborhood sets and -favoring equilibria to ensure convergence. These results deepen understanding of two-sided competitive facility location and suggest robust techniques for multi-stage strategic interactions with atomic agents.

Abstract

We consider competitive facility location as a two-stage multi-agent system with two types of clients. For a given host graph with weighted clients on the vertices, first facility agents strategically select vertices for opening their facilities. Then, the clients strategically select which of the opened facilities in their neighborhood to patronize. Facilities want to attract as much client weight as possible, clients want to minimize congestion on the chosen facility. All recently studied versions of this model assume that clients can split their weight strategically. We consider clients with unsplittable weights but allow mixed strategies. So clients may randomize over which facility to patronize. Besides modeling a natural client behavior, this subtle change yields drastic changes, e.g., for a given facility placement, qualitatively different client equilibria are possible. As our main result, we show that pure subgame perfect equilibria always exist if all client weights are identical. For this, we use a novel potential function argument, employing a hierarchical classification of the clients and sophisticated rounding in each step. In contrast, for non-identical clients, we show that deciding the existence of even approximately stable states is computationally intractable. On the positive side, we give a tight bound of on the price of anarchy which implies high social welfare of equilibria, if they exist.
Paper Structure (20 sections, 17 theorems, 19 equations, 9 figures, 4 tables, 2 algorithms)

This paper contains 20 sections, 17 theorems, 19 equations, 9 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and Preliminaries
  3. The Basics.
  4. The Sequential Game.
  5. Feasible Client Strategies.
  6. Objectives.
  7. Equilibria.
  8. Social Welfare.
  9. Related Work
  10. Our Contribution
  11. Client Equilibria
  12. Subgame Perfect Equilibria
  13. Rounded Client Profiles
  14. Favoring Client Equilibria
  15. Constructing a Subgame Perfect Equilibrium
  16. ...and 5 more sections

Key Result

Corollary 1

For an instance of the 2-FLG and the FPP $\mathbf{s}$, the class set is unique. Moreover, we have $\ell(C_i) < \ell(C_{i+1})$ for all $i\ge 1$.

Figures (9)

  • Figure 1: Instance with four atomic clients ($=$ vertices) with weights $3,1,0,0$. Depicted are the three client equilibria for one given FPP where the two facilities ($=$ colored bullets) each select a $0$-weight node as location. The (mixed) client strategies of the clients with weights $3$ and $1$ are shown as pie charts within the client nodes, e.g., on the right, both clients select each facility with probability $\frac{1}{2}$.
  • Figure 2: Instance with unit weight clients ($=$ vertices) and FPP $\mathbf{s}$ for three facilities ($=$ colored bullets). Left: client equilibrium with non-atomic clients. The facilities receive equal load, e.g., red facility $f$ receives the full demand of $u$, and $\frac{2}{3}$ of the demand of $v$. This is also a mixed client profile for atomic clients, but not an equilibrium as client $v$ can improve by increasing her probability of patronizing $f$. Right: A possible client equilibrium for atomic clients.
  • Figure 3: In this unweighted instance, the class set contains the shown two classes for the given facility placement profile.
  • Figure 4: Right: An intermediate state of the algorithm to compute a rounded client equilibrium in \ref{['thm:rounded-eq-existence']} for the instance on the left. The dashed edges denote current client assignments to facilities. The red paths mark two (out of many) augmenting paths for \ref{['step-path']}.
  • Figure 5: The following rounded full client equilibrium produces the given best response cycle for the facilities in this instance: Clients prefer locations in clockwise order so that the facility in the preferred location receives load $2$ and the other facility load $1$.
  • ...and 4 more figures

Theorems & Definitions (33)

  • proof
  • Definition 1: Minimum Neighborhood Set (MNS)
  • Definition 2: Class Set
  • Corollary 1: Uniqueness of Class Set
  • Corollary 2: Clients in Lowest Class of Neighborhood
  • Corollary 3: Class Set Computation
  • Definition 3: Rounded Client Profile
  • Lemma 1
  • proof
  • Theorem 1: Existence of Rounded Profiles
  • ...and 23 more