Measuring contributions is a classical problem in cooperative game theory where the Shapley value is the most well-known solution concept. In this paper, we establish the convergence property of the Shapley value in parametric Bayesian learning games where players perform a Bayesian inference using their combined data, and the posterior-prior KL divergence is used as the characteristic function. We show that for any two players, under some regularity conditions, their difference in Shapley value converges in probability to the difference in Shapley value of a limiting game whose characteristic function is proportional to the log-determinant of the joint Fisher information. As an application, we present an online collaborative learning framework that is asymptotically Shapley-fair. Our result enables this to be achieved without any costly computations of posterior-prior KL divergences. Only a consistent estimator of the Fisher information is needed. The effectiveness of our framework is demonstrated with experiments using real-world data.

We address the problem of mitigating congestion and preventing hotspots in busy water areas such as Singapore Straits and port waters. Increasing maritime traffic coupled with narrow waterways makes vessel schedule coordination for just-in-time arrival critical for navigational safety. Our contributions are: 1) We formulate the maritime traffic management problem based on the real case study of Singapore waters; 2) We model the problem as a variant of the resource-constrained project scheduling problem (RCPSP), and formulate mixed-integer and constraint programming (MIP/CP) formulations; 3) To improve the scalability, we develop a combinatorial Benders (CB) approach that is significantly more effective than standard MIP and CP formulations. We also develop symmetry breaking constraints and optimality cuts that further enhance the CB approach's effectiveness; 4) We develop a realistic maritime traffic simulator using electronic navigation charts of Singapore Straits. Our scheduling approach on synthetic problems and a real 55-day AIS dataset results in significant reduction of the traffic density while incurring minimal delays.

With the advent of e-commerce, logistics providers are faced with the challenge of handling fluctuating and sparsely distributed demand, which raises their operational costs significantly. As a result, horizontal cooperation are gaining momentum around the world. One of the major impediments, however, is the lack of stable and fair profit sharing mechanism. In this paper, we address this problem using the framework of computational cooperative games. We first present cooperative vehicle routing game as a model for collaborative logistics operations. Using the axioms of Shapley value as the conditions for fairness, we show that a stable, fair and budget balanced allocation does not exist in many instances of the game. By relaxing budget balance, we then propose an allocation scheme based on the normalized Shapley value. We show that this scheme maintains stability and fairness while requiring minimum subsidy. Finally, using numerical experiments we demonstrate the feasibility of the scheme under various settings.

Game theory has gained popularity as an approach to analysing and understanding distributed systems with self-interested agents. Central to game theory is the concept of Nash equilibrium as a stable state (solution) of the system, which comes with a