Prices and locations in a spatial duopoly under uniform delivered pricing

I analyse a two-stage location-price duopoly game under uniform delivered pricing when firms produce homogenous goods and are unable to ration the supply. Two tie-breaking rules (TBR) are studied: consumers either buy from the nearest firm or buy from either firms with equal probabilities. Under the first TBR, I find multiple single-price equilibria. Equilibrium locations are shown to be symmetric and to be such that the distance between firms increases (decreases) with the transportation cost (c) when c is high (low). Under the second TBR, firms cluster to the centre of the market line and choose the price that gives them zero profits. Surprisingly, when c is low, consumers are better off when they randomly select from which firm to buy.