The Algebraic Degree of Network Games via Tropical Geometry: A Geometric Perspective on Datta's Formula

TL;DR

This paper offers a tropical-geometric derivation of Datta's formula for the algebraic degree of network games, revealing structural properties and complexity measures of Nash equilibria through geometric and combinatorial insights.

Contribution

It introduces a tropical geometric perspective to derive Datta's formula, connecting algebraic degree with tropical intersection theory and network structure analysis.

Findings

Algebraic degree is multiplicative over strongly connected components.

Tensor-type couplings exhibit exponential growth in algebraic degree.

Numerical experiments support the tropical geometric interpretation.

Abstract

The algebraic degree of a network game measures the complexity of its totally mixed Nash equilibria. For sparse multilinear network games, Datta's formula expresses this degree combinatorially in terms of a permanent, but the geometric origin of this formula has remained unclear. In this paper, we provide a tropical-geometric derivation of Datta's formula by identifying totally mixed equilibria with stable intersection points of tropical hypersurfaces associated with the indifference equations. We show that the mixed cells arising from the multilinear Newton polytope structure induce the cycle-cover combinatorics of the polynomial graph, so that the permanent appears as a tropical intersection count. This interpretation yields several structural consequences. We prove that the algebraic degree is multiplicative over strongly connected components, and we establish a sharp contrast between two basic multilayer coupling mechanisms: Cartesian-type couplings remain bounded through a transfer-matrix trace formula, whereas tensor-type couplings exhibit exponential growth governed by the permanent of a local gadget. These results show that algebraic degree provides a structural complexity invariant for network architecture. We further illustrate the theory on coupled cyclic games and networked energy-market models, and we support the theoretical predictions with numerical experiments based on homotopy continuation.