When Does Selfishness Align with Team Goals? A Structural Analysis of Equilibrium and Optimality

This paper investigates the relationship between the team-optimal solution and the Nash equilibrium (NE) to assess the impact of self-interested decisions on team performance. In classical team decision problems, team members typically act cooperatively towards a common objective to achieve a team-optimal solution. However, in practice, members may behave selfishly by prioritizing their goals, resulting in an NE under a non-cooperative game. To study this misalignment, we develop a parameterized model for team and game problems, where game parameters represent each individual's deviation from the team objective. The study begins by exploring the consistency and deviation between the NE and the team-optimal solution under fixed game parameters. We provide a necessary and sufficient condition for any NE to be a team optimum, along with establishing an upper bound to measure their difference when this consistency fails. We then study how to steer the NE toward the team-optimal solution by adjusting game parameters in an incomplete-information leader--follower setting, where the leader observes only equilibrium responses rather than the exact lower-level game structure. To address this challenge, we develop a two-stage learned-response intervention framework: the leader first learns a behaviorally consistent lower-level model from observed equilibria and then computes the intervention over the learned response map using bilevel hypergradient optimization, followed by convergence analysis and simulation validation.