This paper studies a spatial duopoly under uniform delivered pricing when firms do not ration the supply of the good, thus extending to a spatial context the analysis of oligopolistic markets with no rationing. The paper shows the existence of the equilibrium in prices under different tie-breaking rules (TBR) and compare the features of the equilibria found under these rules, thereby allowing to highlight the importance of the choice of the TBR in studying these models. When consumers buy from the nearest firm in case of equal prices (efficient TBR), any symmetric price pair within a given range is a Nash equilibrium, with each firm serving exactly half of the market line. If demand in each local market is equally split between the firms charging the same price (random TBR), the only equilibrium price is the one that gives zero profits to each firm. The degree of competitiveness of the market crucially depends on the TBR. Under the efficient TBR, all (but one) price equilibria deliver positive profits to both firms. Under the random TBR, the market outcome is very competitive in that firms make zero profits. None of the equilibria found under any tie-breaking rule are allocatively efficient.