We consider electricity pool markets in radial transmission networks in which the lines have capacities. At each node there is a strategic generator injecting generation quantities into the pool. Prices are determined by a linear competitive fringe at each node (or equivalently a linear demand function) through a convex dispatch optimization. We derive a set of linear inequalities satisfied by the line capacities that gives necessary and sufficient conditions for the unconstrained one-shot

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... ot equilibrium to remain an equilibrium in the constrained network. We discuss the extension of this model to general networks and to lines with transmission losses, and conclude by discussing the application of this methodology to the New Zealand electricity transmission network.