[Paper Review] Lyapunov stochastic stability and control of robust dynamic coalitional games with transferable utilities

Dario Bauso, Puduru Viswanadha Reddy+1 more|arXiv (Cornell University)|Jun 9, 2011

Game Theory and Voting Systems19 references3 citations

TL;DR

This paper proposes robust dynamic allocation rules for coalitional games with transferable utilities under uncertainty, using Lyapunov stochastic stability to ensure convergence of average allocations to the core of the long-run average game. The method employs feedback control based on cumulative excess rewards—either fully or partially observed—guaranteeing almost sure convergence to the core and excesses to a predefined cone, even without knowledge of the underlying probability distribution of coalition values.

ABSTRACT

This paper considers a dynamic game with transferable utilities (TU), where the characteristic function is a continuous-time bounded mean ergodic process. A central planner interacts continuously over time with the players by choosing the instantaneous allocations subject to budget constraints. Before the game starts, the central planner knows the nature of the process (bounded mean ergodic), the bounded set from which the coalitions' values are sampled, and the long run average coalitions' values. On the other hand, he has no knowledge of the underlying probability function generating the coalitions' values. Our goal is to find allocation rules that use a measure of the extra reward that a coalition has received up to the current time by re-distributing the budget among the players. The objective is two-fold: i) guaranteeing convergence of the average allocations to the core (or a specific point in the core) of the average game, ii) driving the coalitions' excesses to an a priori given cone. The resulting allocation rules are robust as they guarantee the aforementioned convergence properties despite the uncertain and time-varying nature of the coaltions' values. We highlight three main contributions. First, we design an allocation rule based on full observation of the extra reward so that the average allocation approaches a specific point in the core of the average game, while the coalitions' excesses converge to an a priori given direction. Second, we design a new allocation rule based on partial observation on the extra reward so that the average allocation converges to the core of the average game, while the coalitions' excesses converge to an a priori given cone. And third, we establish connections to approachability theory and attainability theory.

Motivation & Objective

• To design allocation rules that guarantee convergence of average allocations to the core of the long-run average game despite unknown and time-varying coalition values.
• To drive coalitions' excesses to a pre-specified cone or direction, ensuring stability and fairness in dynamic settings.
• To develop robust control laws that function under partial or full observation of the cumulative excess reward, without requiring knowledge of the underlying probability distribution.
• To establish theoretical connections between Lyapunov stochastic stability and concepts in approachability and attainability theory.
• To ensure stability and fairness in dynamic coalitional games by modeling uncertainty via bounded mean ergodic processes and using feedback mechanisms based on surplus accumulation.

Proposed method

• The central planner uses a dynamic feedback control law that adjusts allocations based on the cumulative excess reward a coalition has received up to time t.
• For full observation, the control law uses the exact cumulative excess vector to steer the system toward a specific point in the core via a Lyapunov function-based design.
• For partial observation, the control law uses a sign-based approximation of the excess vector, with a gain parameter δ to ensure feasibility and stability.
• The control laws are derived using Lyapunov stochastic stability theory, with convergence proven via almost sure convergence of the normalized excess vector to zero.
• The method relies on the bounded mean ergodicity of the characteristic function, ensuring long-term average values are well-defined and known a priori.
• The framework incorporates budget constraints and saturation functions to maintain allocations within feasible bounds, using the nominal allocation and surplus as reference points.

Experimental results

Research questions

• RQ1 How can allocation rules be designed to ensure convergence of average allocations to the core of the average game when coalition values are uncertain and time-varying?
• RQ2 What control strategy guarantees that coalitions’ excesses converge to a pre-specified cone or direction under partial or full observation of the excess reward?
• RQ3 How can Lyapunov stochastic stability be applied to ensure almost sure convergence in dynamic coalitional games with unknown probability distributions?
• RQ4 What is the role of the cumulative excess reward in stabilizing the system and guiding allocations toward the core?
• RQ5 How do the proposed control laws relate to the theories of approachability and attainability in stochastic control and game theory?

Key findings

• The control law based on full observation of the excess reward ensures almost sure convergence of the normalized excess vector to zero, implying convergence of average allocations to a specific point in the core.
• The average allocations converge to the nominal allocation vector in the long run, as confirmed by simulation results showing convergence of the time-averaged allocation to the nominal value.
• The partial observation control law ensures that the average allocations converge to the core in probability, with the instantaneous allocations lying in a neighborhood of the nominal allocations.
• The simulation results demonstrate that the normalized excess for coalition {1,2} converges to zero over time, confirming the theoretical convergence under both full and partial observation.
• The control law with δ = 1 ensures feasibility by keeping allocations within the bounds defined by the minimum and maximum allowable values, as verified through conservative estimation of the gain parameter.
• The theoretical framework establishes a formal connection between Lyapunov stochastic stability and approachability/attainability theories, enriching the analytical foundation of dynamic coalitional games.