Metric Hedonic Games on the Line

TL;DR

This work introduces metric hedonic games on a line where each agent has a fixed value and coalition costs are derived from intra-coalition distances using $cost_{AVG}$, $cost_{MAX}$, or $cost_{CUT-\lambda}$, with a cap on the number of coalitions. It establishes the existence of equilibria for all variants under swap and jump stability, analyzes sorted versus unsorted equilibria, and evaluates efficiency via PoA and PoS, noting often unbounded PoA but 1-PoS for several Swap and nice Cutoff cases. The study leverages potential-function methods, reductions to modified fractional hedonic games, and monotone-cost frameworks to derive equilibrium existence, stability properties, and PoS/PoA characterizations, including a grand-coalition stability result in the UIS setting and IRC phenomena for certain Jump variants. Overall, the results reveal a rich landscape of stability and efficiency in simple line-embedded distance-based hedonic games, with implications for distributed coalition formation and clustering under metric constraints.

Abstract

Hedonic games are fundamental models for investigating the formation of coalitions among a set of strategic agents, where every agent has a certain utility for every possible coalition of agents it can be part of. To avoid the intractability of defining exponentially many utilities for all possible coalitions, many variants with succinct representations of the agents' utility functions have been devised and analyzed, e.g., modified fractional hedonic games by Monaco et al. [JAAMAS 2020]. We extend this by studying a novel succinct variant that is related to modified fractional hedonic games. In our model, each agent has a fixed type-value and an agent's cost for some given coalition is based on the differences between its value and those of the other members of its coalition. This allows to model natural situations like athletes forming training groups with similar performance levels or voters that partition themselves along a political spectrum. In particular, we investigate natural variants where an agent's cost is defined by distance thresholds, or by the maximum or average value difference to the other agents in its coalition. For these settings, we study the existence of stable coalition structures, their properties, and their quality in terms of the price of anarchy and the price of stability. Further, we investigate the impact of limiting the maximum number of coalitions. Despite the simple setting with metric distances on a line, we uncover a rich landscape of models, partially with counter-intuitive behavior. Also, our focus on both swap stability and jump stability allows us to study the influence of fixing the number and the size of the coalitions. Overall, we find that stable coalition structures always exist but that their properties and quality can vary widely.

Figures (1)

• Figure 1: Coalition structure $\mathcal{C}=\{C_1,C_2,C_3\}$ with $n=10$ agents and $k=3$ coalitions. Bold black numbers represent agents with this value. Green areas are the intervals with radius $\lambda=1$ (green) and $\lambda=2$ (light green) around the value $7$. The blue lines show the maximum distance of the circled agent to both sides for coalitions $C_1$ and $C_3$.

Theorems & Definitions (50)

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