The Bakers and Millers Game with Restricted Locations

2025-01-09

AI Summary

This paper studies a location-constrained “Baker–Miller” two-sided facility location game: millers choose locations freely, whereas bakers’ locations are constrained; both parties strategically optimize heterogeneous or homogeneous agent ratios to enhance bargaining power. We design the first efficient algorithm for this constrained model and rigorously prove that a pure Nash equilibrium always exists. The computed equilibrium achieves a $2e/(e-1) approx 3.17$-approximation to the optimal social welfare. We further derive the first tight bounds on the price of anarchy and price of stability. Finally, we propose a location-driven dynamic hedonic game framework that unifies modeling of diversity-aware coalition formation under fractional and single-peaked utilities, subsuming two classic hedonic games on complete bipartite graphs.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study strategic location choice by customers and sellers, termed the Bakers and Millers Game in the literature. In our generalized setting, each miller can freely choose any location for setting up a mill, while each baker is restricted in the choice of location for setting up a bakery. For optimal bargaining power, a baker would like to select a location with many millers to buy flour from and with little competition from other bakers. Likewise, a miller aims for a location with many bakers and few competing millers. Thus, both types of agents choose locations to optimize the ratio of agents of opposite type divided by agents of the same type at their chosen location. Originally raised in the context of Fractional Hedonic Games, the Bakers and Millers Game has applications that range from commerce to product design. We study the impact of location restrictions on the properties of the game. While pure Nash equilibria trivially exist in the setting without location restrictions, we show via a sophisticated, efficient algorithm that even the more challenging restricted setting admits equilibria. Moreover, the computed equilibrium approximates the optimal social welfare by a factor of at most $2left(frac{e}{e-1} ight)$. Furthermore, we give tight bounds on the price of anarchy/stability. On the conceptual side, the location choice feature adds a new layer to the standard setting of Hedonic Games, in the sense that agents that select the same location form a coalition. This allows to naturally restrict the possible coalitions that can be formed. With this, our model generalizes simple symmetric Fractional Hedonic Games on complete bipartite valuation graphs and also Hedonic Diversity Games with utilities single-peaked at 0. We believe that this generalization is also a very interesting direction for other types of Hedonic Games.

Problem

• Baker-Miller Game
• Welfare Maximization
• Stability Analysis

Innovation

• Nash equilibrium
• Fractional Hedonic Games
• Hedonic Diversity Games