The fixed-point bundle method over product-of-simplex domains arising from game equilibria

This paper extends the fiber-bundle path-following framework of arXiv:2606.00778 to product-of-simplex domains, so that the algorithm solves arbitrary m-player n-action general-sum games to arbitrary precision, inheriting global linear convergence: exact for nonsingular equilibria, and up to any prescribed precision for singular ones.
Rather than re-proving convergence, we construct an equivalent variational inequality on a single simplex and establish a fiber bundle isomorphism between the two frameworks, complemented by a slice-extension argument that transfers all guarantees from the original paper. The resulting composite bundle also permits jumps along fibers that avoid singular points, where existing path-following guarantees would otherwise break down.
Experiments on 5600 random instances with dimensions ranging from 2-player 128-action to 128-player 2-action reduce the gap below 1e-5 in every case; code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.