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 the consistency condition fails. The exploration then focuses on aligning NE strategies towards the team-optimal solution through the adjustment of game parameters, resulting in a non-convex and non-smooth bi-level optimization problem. We propose a hypergradient-based algorithm for this problem, and establish its convergence to the critical points. Finally, we validate our theoretical findings through extensive simulation studies.