We introduce Robust Multi-Objective Decoding (RMOD), a novel inference-time algorithm that robustly aligns Large Language Models (LLMs) to multiple human objectives (e.g., instruction-following, helpfulness, safety) by maximizing the worst-case rewards. RMOD formulates the robust decoding problem as a maximin two-player game between adversarially computed reward weights and the sampling policy, solvable through a Nash equilibrium. We demonstrate that this game reduces to a convex optimization problem to identify the worst-case reward weights, with the optimal sampling policy analytically derived. For practical applications, we propose an efficient algorithm of RMOD tailored for contemporary LLMs, introducing minimal computational overhead compared to standard non-robust Controlled Decoding methods. Experimental results across the range of popular alignment datasets with up to 10 objectives show the effectiveness of RMOD and its distilled version, consistently outperforming baselines in worst-case rewards and win rates.