$\texttt{Flip-Team}$: Cooperative Takeover Games with Stochastic Human Override

Technical Summary: Flip-Team: Cooperative Takeover Games with Stochastic Human Override

Problem Statement
Shared autonomy systems, critical in domains like autonomous driving and assistive robotics, require principled mechanisms for transferring control between humans and autonomous agents. Existing approaches often rely on control blending or heuristic switching rules that lack theoretical guarantees and fail to account for the dynamics of authority transfer. Furthermore, many current frameworks assume symmetric roles or require full knowledge of human utility, which is often unrealistic. A specific gap exists in modeling authority switching where humans retain an asymmetric "override" capability (the ability to intervene even when the agent is in control) under stochastic conditions, while both agents pursue a common mission objective. This paper addresses the need for a cooperative framework that determines optimal switching policies without relying on ad hoc rules or continuous human engagement.

Methodology
The authors propose Flip-Team, a cooperative game-theoretic framework formulated as an identical-interest dynamic game. The core innovation is embedding the switching decision directly into the system dynamics rather than treating it as an external arbitration mechanism.

1. Game Formulation:

   • State Space: The system state $x_k$ evolves under either human control ($\alpha_k = H$) or autonomous control ($\alpha_k = A$).
   • FlipDyn State: A discrete state $\alpha_k$ tracks who holds authority. Transitions are governed by agent actions: "idle" (retain control) or "takeover/request."
   • Stochastic Override: A key feature is the stochastic human override. When the autonomous agent is in control ($\alpha_k = A$) and the human attempts to intervene ($\pi^H_k = 1$) while the agent does not request a handoff, the takeover succeeds with probability $p_k$. This captures interface latency and arbitration mechanisms where human authority is superior but not guaranteed.
   • Cost Function: The agents jointly minimize a total cost $J$ comprising state costs (tracking error, energy) and takeover costs (cognitive load, transition risk). The costs are asymmetric ($g^H \neq g^A$, $h \neq a$) to reflect different control effectiveness and switching burdens.

2. General System Analysis:

   • The problem is solved using dynamic programming. The authors define value functions and cost-to-go matrices for both FlipDyn states ($H$ and $A$).
   • Theorem 1 establishes the existence of team-optimal switching policies in pure strategies. It derives explicit threshold conditions for authority transfer based on the difference in future value functions ($\tilde{v}_{k+1} = v^A_{k+1} - v^H_{k+1}$) scaled by takeover costs and the override probability $p_k$.

3. Linear-Quadratic (LQ) Extension:

   • For linear systems with quadratic costs, the authors derive Corollary 1, providing closed-form recursions for the optimal switching policies and value function parameters ($P^H_k, P^A_k$).
   • Crucially, this formulation allows the switching thresholds to be computed offline and remain independent of the continuous state $x$ in the scalar case, or depend on the state only through a quadratic form in the multi-dimensional case. This ensures computational efficiency independent of the continuous state space cardinality.

Key Contributions

1. Cooperative Takeover Formulation: The paper formulates the human-autonomy takeover problem as an identical-interest dynamic game where switching decisions are embedded in system dynamics. This unified framework captures asymmetric authority, stochastic human override, and state-dependent costs.
2. Characterization of Optimal Switching: The authors establish the existence of team-optimal policies and characterize them via explicit threshold conditions. These conditions determine when authority transfers occur based on the trade-off between future cost differences and the probabilistic cost of switching.
3. Analytical Solutions for LQ Systems: For linear-quadratic systems, the paper derives closed-form recursions for optimal policies and value functions. This enables efficient computation of cooperative takeover strategies in continuous state spaces, with switching thresholds that are independent of the continuous state (in scalar cases) or defined by matrix inequalities (in multi-dimensional cases).

Results
The framework was validated on scalar and multi-dimensional linear systems:

• Scalar LTI System: Simulations over a finite horizon ($L=40$) demonstrated that switching behavior is highly sensitive to the override probability $p$. A critical threshold $p^* = H/(H+A)$ was identified; below this value, unilateral human override is never optimal. As $p$ increases, the frequency of optimal overrides increases, showing a non-monotonic policy evolution in intermediate regimes.
• Vehicle Lateral Regulation (2D): In a 2D vehicle lane-tracking task, the Flip-Team policy reduced the worst-case cost by 18.65% compared to an "always-autonomous" baseline and outperformed fixed-interval switching strategies.
• State Dimensionality: The results highlighted a fundamental structural difference between scalar and multi-dimensional systems. In scalar systems, switching is state-independent. In higher dimensions, the switching condition involves matrix inequalities ($\tilde{P}_{k+1} \succeq H/p$), meaning override may be optimal for states aligned with specific eigenvectors of the value difference matrix but not for others.

Significance and Claims
The paper claims that grounding shared autonomy in cooperative game theory provides a principled alternative to heuristic blending or arbitration. By treating human and autonomy as a unified team with a common objective but distinct roles, Flip-Team yields optimal switching policies that adapt to system dynamics, cost structures, and the reliability of human intervention. The framework explicitly addresses the trade-offs between human adaptability and autonomous efficiency.

The authors note limitations: the framework assumes identical interests (aligned objectives), treats override probability as known (requiring estimation in deployment), and restricts closed-form results to LQ systems. Future work is proposed to extend the framework to Bayesian games with private information, online learning of override probabilities, and physical validation with human-in-the-loop experiments.