On the closest balanced game

2026-01-16

AI Summary

This study addresses the issue of empty cores in cooperative games by proposing to orthogonally project an imbalanced game onto the set of balanced games using a weighted Euclidean distance, thereby identifying the nearest game with a non-empty core. To this end, we introduce a novel solution concept—the least-squares core—and develop an efficient algorithm that circumvents exponential complexity, enabling the handling of games with up to 20 players. Leveraging tools from convex optimization, polyhedral geometry, and combinatorics, we theoretically demonstrate that as the number of players grows, the core of the projected game almost surely degenerates to a singleton. We further derive an asymptotic expression for the minimal number of balancing coalitions, explaining this phenomenon. Both theoretical analysis and experiments consistently show that the probability of the core being a singleton converges to one as the number of players increases.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Cooperative games with nonempty core are called balanced, and the set of balanced games is a polyhedron. Given a game with empty core, we look for the closest balanced game, in the sense of the (weighted) Euclidean distance, i.e., the orthogonal projection of the game on the set of balanced games. Besides an analytical approach which becomes rapidly intractable, we propose a fast algorithm to find the closest balanced game, avoiding exponential complexity for the optimization problem, and being able to run up to 20 players. We show experimentally that the probability that the closest game has a core reduced to a singleton tends to 1 when the number of players grow. We provide a mathematical proof that the proportion of facets whose games have a non-singleton core tends to 0 when the number of players grow, by finding an expression of the aymptotic growth of the number of minimal balanced collections. This permits to prove mathematically the experimental result. Consequently, taking the core of the projected game defines a new solution concept, which we call least square core due to its analogy with the least core, and our result shows that the probability that this is a point solution tends to 1 when the number of players grow.

Problem

• cooperative games
• balanced games
• empty core
• closest balanced game
• core

Innovation

• closest balanced game
• least square core
• orthogonal projection
• minimal balanced collections
• cooperative game theory