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Two-Stage Facility Location Games with Strategic Clients and Facilities

Simon Krogmann, Pascal Lenzner, Louise Molitor, Alexander Skopalik

TL;DR

This work introduces a two-sided facility location framework (the 2-FLG) on a directed, vertex-weighted host graph $H=(V,E,w)$ with $k$ facilities and $n$ clients, where facilities choose locations and clients distribute their spending across nearby facilities to minimize their maximum waiting time. It proves the existence of subgame perfect equilibria for the load-balancing variant, and provides a polynomial-time algorithm to compute client equilibria and resulting facility loads, enabling facilities to anticipate client behavior and check equilibrium efficiently. For arbitrary feasible client distributions, it derives a tight bound of $\mathrm{PoA}\le 2$ with a lower bound of $2-\frac{1}{k}$ on both the PoA and PoS, and proves that computing a socially optimal placement is NP-hard; the results rely on a combinatorial approach using minimum neighborhood sets/ratios and maximum-flow computations. The work thus delivers both theoretical insights and practical algorithms for two-sided competitive facility location, highlighting how anticipatory client behavior can stabilize equilibria and bound inefficiency in large, strategic systems.

Abstract

We consider non-cooperative facility location games where both facilities and clients act strategically and heavily influence each other. This contrasts established game-theoretic facility location models with non-strategic clients that simply select the closest opened facility. In our model, every facility location has a set of attracted clients and each client has a set of shopping locations and a weight that corresponds to her spending capacity. Facility agents selfishly select a location for opening their facility to maximize the attracted total spending capacity, whereas clients strategically decide how to distribute their spending capacity among the opened facilities in their shopping range. We focus on a natural client behavior similar to classical load balancing: our selfish clients aim for a distribution that minimizes their maximum waiting times for getting serviced, where a facility's waiting time corresponds to its total attracted client weight. We show that subgame perfect equilibria exist and give almost tight constant bounds on the Price of Anarchy and the Price of Stability, which even hold for a broader class of games with arbitrary client behavior. Since facilities and clients influence each other, it is crucial for the facilities to anticipate the selfish clients' behavior when selecting their location. For this, we provide an efficient algorithm that also implies an efficient check for equilibrium. Finally, we show that computing a socially optimal facility placement is NP-hard and that this result holds for all feasible client weight distributions.

Two-Stage Facility Location Games with Strategic Clients and Facilities

TL;DR

This work introduces a two-sided facility location framework (the 2-FLG) on a directed, vertex-weighted host graph with facilities and clients, where facilities choose locations and clients distribute their spending across nearby facilities to minimize their maximum waiting time. It proves the existence of subgame perfect equilibria for the load-balancing variant, and provides a polynomial-time algorithm to compute client equilibria and resulting facility loads, enabling facilities to anticipate client behavior and check equilibrium efficiently. For arbitrary feasible client distributions, it derives a tight bound of with a lower bound of on both the PoA and PoS, and proves that computing a socially optimal placement is NP-hard; the results rely on a combinatorial approach using minimum neighborhood sets/ratios and maximum-flow computations. The work thus delivers both theoretical insights and practical algorithms for two-sided competitive facility location, highlighting how anticipatory client behavior can stabilize equilibria and bound inefficiency in large, strategic systems.

Abstract

We consider non-cooperative facility location games where both facilities and clients act strategically and heavily influence each other. This contrasts established game-theoretic facility location models with non-strategic clients that simply select the closest opened facility. In our model, every facility location has a set of attracted clients and each client has a set of shopping locations and a weight that corresponds to her spending capacity. Facility agents selfishly select a location for opening their facility to maximize the attracted total spending capacity, whereas clients strategically decide how to distribute their spending capacity among the opened facilities in their shopping range. We focus on a natural client behavior similar to classical load balancing: our selfish clients aim for a distribution that minimizes their maximum waiting times for getting serviced, where a facility's waiting time corresponds to its total attracted client weight. We show that subgame perfect equilibria exist and give almost tight constant bounds on the Price of Anarchy and the Price of Stability, which even hold for a broader class of games with arbitrary client behavior. Since facilities and clients influence each other, it is crucial for the facilities to anticipate the selfish clients' behavior when selecting their location. For this, we provide an efficient algorithm that also implies an efficient check for equilibrium. Finally, we show that computing a socially optimal facility placement is NP-hard and that this result holds for all feasible client weight distributions.

Paper Structure

This paper contains 10 sections, 19 theorems, 30 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Further Related Work
  3. Model and Preliminaries
  4. Our Contribution
  5. Load Balancing Clients
  6. Facility Loads in Polynomial Time
  7. Existence of Subgame Perfect Equilibria
  8. Comparison with Utility Systems
  9. Arbitrary Client Behavior
  10. Conclusion and Future Work

Key Result

Theorem 1

For a facility placement profile $\mathbf{s}$, a client equilibrium $\sigma$ exists and every client equilibrium induces the same facility loads $(\ell_1(\mathbf{s},\sigma),\ldots,\ell_k(\mathbf{s},\sigma))$.

Figures (5)

  • Figure 1: Example of the load balancing $2$-FLG. The clients (vertices) split their weight (shown by numbers) among the facilities (colored dots) in their shopping range. The client distributions are shown by colored pie charts. Left: The blue facility receives a load of $2$ while all other facilities get a load of $\frac{4}{3}$. The left client with weight $2$ distributes weight $\frac{4}{3}$ to the yellow facility and $\frac{1}{3}$ to both the green and the red facility. The state is not in SPE as the red facility can improve her load to $\frac{3}{2}$ by co-locating with the blue facility. Right: A SPE for this instance, all facilities have a load of $\frac{3}{2}$.
  • Figure 2: An instance of the load balancing $2$-FLG with a facility placement profile marked by dots and $10$ clients with weight $1$ each. \ref{['alg:load-balancing-utilities']} successively finds and removes the minimum neighborhood sets $S_1=\{f_1\}$, $S_2=\{f_2, f_3\}$ and $S_3=\{f_4\}$.
  • Figure 3: Left: An instance of the load balancing $2$-FLG with the graph $H$ and the facility placement profile $\mathbf{s}$ marked by dots. Right: The maximum flow instance constructed by \ref{['alg:minimum-neighborhood-set']}.
  • Figure 4: The host graph $H$ of an instance $I$ of the $2$-FLG with arbitrary client behavior with a unique SPE.
  • Figure 5: An example of a corresponding host graph $H$ to the 3SAT instance ${(x\vee \neg y \vee z)} \wedge {(\neg x\vee y \vee z)}$.

Theorems & Definitions (38)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 28 more