  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.
