In this study, a novel non-cooperative two-player game for minimizing static (Player 1) and dynamic (Player 2) compliances is introduced, implemented, and demonstrated using a multi-scale topology optimization framework for triply periodic minimal surface (TPMS)-based lattice structures. Player 1 determines the optimal macro-layout by minimizing the static compliance based on a micro-layout provided by Player 2. Conversely, player 2 identifies the optimal micro-layout (grading of the TPMS-based lattice structure) by minimizing the dynamic compliance given a macro-layout from Player 1. The multi-scale topology optimization formulations are derived using two density variables in each finite element. The first variable is the standard density, which dictates whether the finite element is void or contains the graded lattice structure and is governed by the rational approximation of material properties (RAMP) model. The second density variable represents the local relative density of the TPMS-based lattice structure, determining the effective orthotropic elastic properties of the finite element. The multi-scale game is implemented for three-dimensional problems, and solved using a Gauss-Seidel algorithm with sequential linear programming. It is numerically demonstrated for several benchmarks that the proposed multi-scale game generates equilibrium designs with strong performance for both static and harmonic load cases, effectively avoiding resonance at harmonic load frequencies. Validation is achieved through modal analyses of finite element models of the optimal designs.