Strategic Facility Location with Clients that Minimize Total Waiting Time

TL;DR

This paper studies a non-cooperative two-sided facility location game on a graph, where facilities choose locations and clients allocate purchasing power to minimize total waiting time. The client stage is modeled as an atomic splittable congestion game, ensuring existence, uniqueness, and polynomial-time computability of client equilibria, enabling facilities to anticipate client responses. However, subgame perfect equilibria for the full two-stage game do not always exist and deciding SPE existence is NP-hard; the authors provide a 3-approximate SPE and an efficient method to compute it via a Uniform-2-FLG proxy and an FPTAS, while proving related hardness results for the proxy model. The results delineate both tractable and intractable aspects of equilibrium behavior in two-sided facility location, with implications for predicting strategic placement and client behavior in networks.

Abstract

We study a non-cooperative two-sided facility location game in which facilities and clients behave strategically. This is in contrast to many other facility location games in which clients simply visit their closest facility. Facility agents select a location on a graph to open a facility to attract as much purchasing power as possible, while client agents choose which facilities to patronize by strategically distributing their purchasing power in order to minimize their total waiting time. Here, the waiting time of a facility depends on its received total purchasing power. We show that our client stage is an atomic splittable congestion game, which implies existence, uniqueness and efficient computation of a client equilibrium. Therefore, facility agents can efficiently predict client behavior and make strategic decisions accordingly. Despite that, we prove that subgame perfect equilibria do not exist in all instances of this game and that their existence is NP-hard to decide. On the positive side, we provide a simple and efficient algorithm to compute 3-approximate subgame perfect equilibria.

Figures (3)

• Figure 1: Two instances of the Min-$2$-FLG with their client equilibria visualized for a given facility placement profile. The clients on each node split their weight (above the nodes) among the facilities in their shopping ranges to minimize their cost. The facilities are marked by dots inside the nodes and receive the loads below in the respective unique client equilibria. The respective cost $L_1(\mathbf{s},\sigma)= \sigma(\mathbf{s},v_1)_1 \ell_1(\mathbf{s},\sigma) + \sigma(\mathbf{s},v_1)_2 \ell_2(\mathbf{s},\sigma)$ of client $v_1$ (and also $v_2$ on the right) is given below each instance.
• Figure 2: Best responses (dashed) to game states for the instance $G^*$ of the Min-$2$-FLG without SPE. The utilities of the facilities before and after the move are given on the right.
• Figure 3: An instance of the $2$-FLG for which an SPE in the Uniform-$2$-FLG is a $2$-approximate SPE in the Min-$2$-FLG. The improving facility improves by moving from $v_b$ to $v_a$.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Example 1
• Theorem 5