Comparative statics of minimum-cost-spanning-tree games

We conduct a comparative static analysis for the cores of minimum-cost-spanning-tree games. We introduce a new category of cover, termed as the matrix-exact cover. Our investigation underscores the pivotal role played by the matrix-exact cover, in conjunction with the classical irreducible cost matrix. (i) When edge costs experience a decrease, the core remains invariant as long as the costs of all edges are weakly above their respective matrix-exact covers. (ii) When the costs of certain edges do fall below the matrix-exact cover, and simultaneously all edge costs remain weakly above the irreducible costs, the core contracts, forming a proper subset of the original core. (iii) Furthermore, should the costs of some edges continue to decrease and fall below the irreducible costs, the core shifts away from the original core.