Robust n-Agent Heterogeneous Investment-Consumption Game Under $$\alpha $$ α -Maxmin Mean-Variance-Utility Criterion

This paper investigates a robust heterogeneous n-agent stochastic differential game under a mean-variance-utility criterion, where agents compete based on relative performance in the presence of model uncertainty. Model uncertainty is represented by a set of equivalent probability measures, with Novikov’s condition imposed to guarantee their mutual equivalence. Ambiguity attitudes are characterized by the $$\alpha $$ α -maxmin model. Agents invest and consume in a financial market exposed to both common and idiosyncratic risks, aiming to maximize relative terminal wealth with the mean-variance criterion and the expected utility of relative consumption under the $$\alpha $$ α -maxmin model. We formulate a heterogeneous game that emphasizes outperforming a specific group of competitors. Agents focus on a weighted average of their peers’ wealth and consumption. This optimization problem is inherently time-inconsistent, and we derive the associated extended Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations within a game-theoretic framework. The robust best response strategies are composed of a myopic component and another component that reacts to the actions of other agents. We obtain closed-form solutions for the robust Nash equilibrium investment-consumption strategies through a system of linear equations. This paper explores how levels of ambiguity, ambiguity aversion, risk aversion, and competition affect Nash equilibrium strategies, uncovering the herd effect that competition has on agents’ strategies.