The price of imperfect competition for a spanning network

We evaluate the price of imperfect competition (PIC), namely the ratio of the total price that could be charged to the buyer in some equilibrium, to the true minimal cost. If each seller can only bid for a single edge and costs satisfy the triangle inequality, we show that the PIC is at most 2 for an odd number of nodes, and at most 2n−1n−2 for an even number n of nodes. Surprisingly, this worst case ratio does not improve when the cost pattern is ultrametric (a much more demanding substitutability requirement), although the overhead is much lower on average under typical probabilistic assumptions. But the PIC increases swiftly when sellers can only provide a subset of all edges.