Recently, the authors introduced a game invariant of graphs, called a game connectivity. In this paper, we investigate the edge version of the invariant, called a game edge-connectivity, by introducing a new combinatorial game on a graph, called a graph edge-cutting game. The game is one of generalizations of a classical combinatorial game, named the Shannon switching game. As an analog of the study of the Shannon switching game, we have a complete characterization of graphs with game edge-connectivity infinity in terms of the number of edge-disjoint spanning trees. As a corollary of the above, any graph with edge-connectivity at least 4 has game edge-connectivity infinity. Thus, since determining the game edge-connectivity of a given cubic graph is the most interesting problem, we give a useful tool to estimate the game edge-connectivity of cubic graphs. Other than those above, we study many fundamental topics of the game edge-connectivity of graphs, and propose several open problems.