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Monte Carlo Techniques for Approximating the Myerson Value -- Theoretical and Empirical Analysis

Myerson first introduced graph-restricted games in order to model the interaction of cooperative players with an underlying communication network. A dedicated solution concept -- the Myerson value -- is perhaps the most important normative solution concept for cooperative games on graphs. Unfortunately, its computation is computationally challenging. In particular, although exact algorithms have been proposed, they must traverse all connected coalitions of the graph of which there may be exponentially many. In this paper, we consider the issue of approximating the Myerson value for arbitrary graphs and characteristic functions. While Monte Carlo approximations have been proposed for the related concept of the Shapley value, their suitability for the Myerson value has not been studied. Given this, we evaluate and compare (both theoretically and empiraclly) three Monte Carlo sampling methods for the Myerson value: conventional method of sampling permutations; a new, hybrid algorithm that combines exact computations and sampling; and sampling of connected coalitions. We find that our hybrid algorithm performs very well and also significantly improves on the conventional methods.