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Full characterization of core for nonlinear optimization games

Donglei Du, Qizhi Fang, Bin Liu, Tianhang Lu, Chenchen Wu

TL;DR

The paper develops a unifying framework for nonlinear cooperative games by exactly characterizing when the core is non-empty for the nonlinear optimization game $\nu_{X,f}$. Central to the approach is a basis-linear relaxation $F$ of the objective and an anchor relaxation on the cone $\text{cone}(X)$, yielding a sharp condition $\nu_{X,f}(\mathbf{1}_n)=\nu_{\mathbb{R}^m_{+},F}(\mathbf{1}_n)$ and a dual optimal set that coincides with the core when non-empty. The authors further present two equivalent characterizations via an upper and a lower game, extend the results to covering/partition games, finite-generator domains, and more general right-hand sides and objectives, and illustrate the applicability with nonlinear variants of classical linear games, including combinatorial quadratic and ratio problems. The work also clarifies the landscape of function classes (e.g., individually subadditive vs submodular) and provides a pathway to approximate cores, with several open questions about nonlinear duality and broader game classes. Overall, the results deliver a practical, polynomial-time check for core non-emptiness in a wide family of nonlinear optimization games and connect deep theoretical insights with classical cooperative-game theory constructs.

Abstract

We fully characterize the core of a broad class of nonlinear games by identifying a suitable relaxation for inherent nonlinearity, directly generalizing the linear frameworks in the literature. This characterization significantly expands the scope of cooperative games that can be analyzed and contributes to the literature on games induced from optimization models. We apply these insights to not only establish connections with and provide new insights on classical models but also solve new games untamed in the existing literature, including combinatorial quadratic and ratio games such as portfolio, maximum cut, matching, and assortment games. These results are further extended to more general models and also the approximate core.

Full characterization of core for nonlinear optimization games

TL;DR

The paper develops a unifying framework for nonlinear cooperative games by exactly characterizing when the core is non-empty for the nonlinear optimization game . Central to the approach is a basis-linear relaxation of the objective and an anchor relaxation on the cone , yielding a sharp condition and a dual optimal set that coincides with the core when non-empty. The authors further present two equivalent characterizations via an upper and a lower game, extend the results to covering/partition games, finite-generator domains, and more general right-hand sides and objectives, and illustrate the applicability with nonlinear variants of classical linear games, including combinatorial quadratic and ratio problems. The work also clarifies the landscape of function classes (e.g., individually subadditive vs submodular) and provides a pathway to approximate cores, with several open questions about nonlinear duality and broader game classes. Overall, the results deliver a practical, polynomial-time check for core non-emptiness in a wide family of nonlinear optimization games and connect deep theoretical insights with classical cooperative-game theory constructs.

Abstract

We fully characterize the core of a broad class of nonlinear games by identifying a suitable relaxation for inherent nonlinearity, directly generalizing the linear frameworks in the literature. This characterization significantly expands the scope of cooperative games that can be analyzed and contributes to the literature on games induced from optimization models. We apply these insights to not only establish connections with and provide new insights on classical models but also solve new games untamed in the existing literature, including combinatorial quadratic and ratio games such as portfolio, maximum cut, matching, and assortment games. These results are further extended to more general models and also the approximate core.
Paper Structure (34 sections, 20 theorems, 98 equations, 1 figure)

This paper contains 34 sections, 20 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Nonlinear games
  4. Relaxation vs extension
  5. Full characterization of core
  6. Immediate applications
  7. Equivalent characterizations
  8. Characterization via the upper game
  9. Characterization via the lower game
  10. Two concrete examples
  11. Further results
  12. Covering and partition games
  13. $X$ with a finite number of generators
  14. $X$ dependent on $w$
  15. More general right-hand side
  16. ...and 19 more sections

Key Result

Proposition 1

Given two subsets $X_1, X_2\subseteq \mathbb{R}^n$ such that $X_1\subseteq X_2$, let $F:X_2\mapsto \mathbb{R}$ be a relaxation of $f:X_1\mapsto \mathbb{R}$ such that $F|_{X_1}\ge f$. Consider the following two optimization problems, where the second is a relaxation of the first: Let $x^*\in \mathop{\mathrm{argmax}}\limits\limits_{x\in X_1} f(x)$ and $y^*\in\mathop{\mathrm{argmax}}\limits\limits_{

Figures (1)

  • Figure 1: Graph $G$.

Theorems & Definitions (46)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 2
  • ...and 36 more