The fixed-point bundle method over product-of-simplex domains arising from game equilibria
Abstract
Developing general-purpose algorithms with global asymptotic convergence guarantees for Nash equilibria of general-sum games remains a longstanding challenge. This paper extends the fixed-point bundle framework for finite-dimensional variational inequalities (VIs) from simplex domains to product-of-simplex domains, enabling its direct application to Nash equilibrium computation. The resulting fixed-point bundle exhibits a composite fiber bundle structure. The key innovation is to construct an equivalent VI on a simplex and establish the equivalence between the two fixed-point bundle frameworks via a fiber bundle isomorphism. Exploiting this geometric equivalence, we show that the path-following algorithm for VIs on product-of-simplex domains inherits the global linear convergence guarantees of the simplex-domain framework. In numerical experiments, the algorithm attained a precision of 10−5 on all 5600 randomly generated game instances, with dimension ranging from 2-player 128-action to 128-player 2-action. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.
Related
- Seifert type magnetohydrostatic equilibria in rotationally symmetric toroidal domains
- Selfless Reduced Crossed Product C*-Algebras Arising from Almost Periodic Actions
- Intermediate subalgebras of cross-product von Neumann algebras arising from weakly compact actions
- Finite-Time Singularities of the Kähler--Ricci Flow on a ℂPm-Bundle over a Product of Kähler--Einstein Manifolds
- Boundary Rigidity of Line-Bundle Shilov--Poisson Transforms on Bounded Symmetric Domains
BibTeX
@misc{sun2026fixedpoint,
title = {{The fixed-point bundle method over product-of-simplex domains arising from game equilibria}},
author = {Hongbo Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.16158},
eprint = {2609.16158},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16158},
}
RIS
TY - UNPB
TI - The fixed-point bundle method over product-of-simplex domains arising from game equilibria
AU - Hongbo Sun
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.16158
UR - https://arxiv.org/abs/2609.16158
AB - Developing general-purpose algorithms with global asymptotic convergence guarantees for Nash equilibria of general-sum games remains a longstanding challenge. This paper extends the fixed-point bundle framework for finite-dimensional variational inequalities (VIs) from simplex domains to product-of-simplex domains, enabling its direct application to Nash equilibrium computation. The resulting fixed-point bundle exhibits a composite fiber bundle structure. The key innovation is to construct an equivalent VI on a simplex and establish the equivalence between the two fixed-point bundle frameworks via a fiber bundle isomorphism. Exploiting this geometric equivalence, we show that the path-following algorithm for VIs on product-of-simplex domains inherits the global linear convergence guarantees of the simplex-domain framework. In numerical experiments, the algorithm attained a precision of 10^−5 on all 5600 randomly generated game instances, with dimension ranging from 2-player 128-action to 128-player 2-action. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.
ER -