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On the closest balanced game

Pedro García-Segador, Michel Grabisch, Dylan Laplace Mermoud, Pedro Miranda

TL;DR

The paper tackles the problem of sharing the value when the core is empty by projecting a TU-game onto the set of balanced games to obtain the closest balanced game $v^*$, and then taking its core as a solution. It introduces a fast algorithm, CLOBIS, that computes $v^*$ efficiently up to at least 20 players and proves that the probability that $C(v^*)$ is a singleton tends to 1 as the number of players grows, supporting the usefulness of the least square core concept. The authors also analyze the combinatorial structure of minimal balanced collections and show that almost all facets (and, asymptotically, faces) correspond to singleton-core games, which under a mild distributional hypothesis implies a high likelihood of single-point imputations. Taken together, the results provide both a practical computational tool and a rigorous asymptotic foundation for the proposed least square core as a robust solution concept in large coalitional settings.

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.

On the closest balanced game

TL;DR

The paper tackles the problem of sharing the value when the core is empty by projecting a TU-game onto the set of balanced games to obtain the closest balanced game , and then taking its core as a solution. It introduces a fast algorithm, CLOBIS, that computes efficiently up to at least 20 players and proves that the probability that is a singleton tends to 1 as the number of players grows, supporting the usefulness of the least square core concept. The authors also analyze the combinatorial structure of minimal balanced collections and show that almost all facets (and, asymptotically, faces) correspond to singleton-core games, which under a mild distributional hypothesis implies a high likelihood of single-point imputations. Taken together, the results provide both a practical computational tool and a rigorous asymptotic foundation for the proposed least square core as a robust solution concept in large coalitional settings.

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.
Paper Structure (15 sections, 22 theorems, 142 equations, 2 figures, 7 tables, 1 algorithm)

This paper contains 15 sections, 22 theorems, 142 equations, 2 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Finding the closest balanced game
  4. The analytical way
  5. The optimization way
  6. CLOBIS algorithm
  7. Simulation study
  8. Probability to have a singleton core
  9. Probability of facets and faces
  10. On the probability that $v^*$ has a singleton core
  11. Proportion of facets whose games have a singleton core
  12. Proportion of faces whose games have a singleton core
  13. Probability that $v^*$ has a singleton core
  14. Concluding remarks
  15. Projections on $\mathscr{BG}_\alpha(3)$

Key Result

Theorem 1

Consider a game $(N, v)$. Then, $C(v)\ne \varnothing$ if and only if $v$ is balanced.

Figures (2)

  • Figure 1: Projecting on a triangle. Each region shows the set of points projecting on each face (edges and vertices) of the triangle.
  • Figure 2: Comparison between estimated probability (red) and $1/2^{n-1}$ (blue) in logarithmic scale.

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 31 more