This paper studies the welfare implications of using market mechanisms to allocate transmission capacity in recently liberalized electricity markets. It questions whether access to this essential facility should be traded on a market, or whether the incumbent should retain exclusive usage rights. We show that granting exclusive use to the incumbent might be optimal, if the capacity of the essential facility is small and the incumbent can reduce production costs by taking advantage of interregional production-cost differences. This result counters the intuition that arbitrage will improve the social surplus when there is no output contraction. The reason is that when competition is imperfect, arbitrage might reduce production efficiency. We advise policymakers to introduce market mechanisms for the allocation of transmission capacity only if sufficient investment in the network is ensured or if the market power of the incumbent is broken in at least one of the markets in which it is active.