On the Set of Balanced Games

TL;DR

The paper provides a complete polyhedral treatment of the set of balanced cooperative TU-games, characterizing BG(n) as a nonpointed cone with explicit lineality, extremal rays, and facets, and BG_α(n) as an affine analog. It also fully analyzes BG_+(n), a bounded polytope, giving its vertices, a uniform random vertex generation algorithm, adjacency structure, and the fact that its vertex graph is Hamilton-connected. A central theme is when the core reduces to a singleton, with complete results for BG(n) and BG_α(n) and partial results for BG_+(n). The findings have implications for projection problems onto balanced sets, optimization over balanced allocations, and connections to Boolean-function combinatorics, with potential applications across operations research and decision theory.

Abstract

We study the geometric structure of the set of cooperative transferable utility games having a nonempty core, characterized by Bondareva and Shapley as balanced games. We show that this set is a nonpointed polyhedral cone, and we find the set of its extremal rays and facets. This study is also done for the set of balanced games whose value for the grand coalition is fixed, which yields an affine nonpointed polyhedral cone. Finally, the case of nonnegative balanced games with fixed value for the grand coalition is tackled. This set is a convex polytope, with remarkable properties. We characterize its vertices and facets, study the adjacency structure of vertices, develop an algorithm for generating vertices in a random uniform way, and show that this polytope is combinatorial and its adjacency graph is Hamiltonian. Last, we give a characterization of the set of games having a core reduced to a singleton. Funding: This work was supported by the Spanish Government [Grant PID2021-124933NB-I00].

Figures (3)

• Figure 1: Non-adjacency of $v_1,v_2$, with associated collections $\mathscr{D}_1,\mathscr{D}_2$. Case (a): $\mathscr{D}_3=\mathscr{B}_1\cup(\mathscr{D}_1\cap \mathscr{D}_2)\cup\mathscr{B}_3$, $\mathscr{D}_4=\mathscr{B}_2\cup(\mathscr{D}_1\cap \mathscr{D}_2)\cup\mathscr{B}_4$; Case (b): $\mathscr{D}_3=\mathscr{D}_1\cup\mathscr{B}_3$, $\mathscr{D}_4=(\mathscr{D}_1\cap\mathscr{D}_2)\cup\mathscr{B}_4$ (similar when $\mathscr{D}_1,\mathscr{D}_2$ exchanged); Case (c): $\mathscr{D}_3=\mathscr{D}_1\cup\mathscr{D}_2$, $\mathscr{D}_4=\mathscr{D}_1\cap\mathscr{D}_2$.
• Figure 2: Adjacency graph of $\mathscr{BG}_+(n)$ for $n=3.$ Vertices are defined in Table \ref{['vert_n=3']}
• Figure 3: Adjacency graph of $\mathscr{BG}_+(n)$ for $n=3.$ The Hamiltonian path joining $u_{123}$ and $d_{2,23}$ is indicated in thick red line.

Theorems & Definitions (70)

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