Robust Chance Constrained Complex Zero-Sum Games

TL;DR

This paper introduces a comprehensive framework for complex-valued zero-sum games, extending classical results to the complex domain and incorporating uncertainty via chance constraints with convex reformulations.

Contribution

It develops a unified complex game theory framework, including chance constraints and uncertainty modeling, with explicit saddle-point characterizations and practical numerical validation.

Findings

Established minimax theorem and saddle-point structure for complex games.

Derived convex second-order cone representations for probabilistic constraints.

Validated the framework through numerical experiments on waveform interaction models.

Abstract

This paper develops a unified framework for zero-sum games in which both the pure strategies and the payoff matrices contain complex-valued entries. By leveraging a linear isomorphism between complex and real vector spaces, we extend key results from real-valued convex analysis to the complex domain, establishing the validity of the minimax theorem and the preservation of saddle-point structure. Building on this foundation, we formulate a complex zero-sum game model that enables mixed strategies to interact with the real and imaginary components of the payoff matrix, and we characterize its saddle-point equilibrium through associated primal and dual problems. To incorporate uncertainty, we introduce a complex chance-constrained zero-sum game model (3CP) that handles individual probabilistic constraints defined by complex linear functionals. We first study the 3CP formulation under known exact distributions, focusing on Complex Elliptically Symmetric random variables, which generalize the complex Gaussian family. The framework is then extended to moments-based ambiguity sets, including: (i) distributions with known first two moments, (ii) distributions with unknown second-order moments, and (iii) fully distributed with unknown moments. In all cases, the probabilistic constraints admit deterministic second-order cone representations, ensuring convex feasible strategy sets and enabling explicit characterization of the complex game value. Numerical experiments, including a transmitter--jammer waveform interaction model, show how the proposed framework captures the behavior of complex mixed strategies. Additionally, we evaluate out-of-sample rates and confirm that practical behavior closely aligns with the theoretical guarantees.