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In order to mitigate the effects of the contraction in demand during economic crises, firms face the need to reduce the number of facilities in their networks. This reduction must be conducted taking into consideration both the possible actions of their rival firms and the reaction of the affected customers, so that the loss of market share is minimised. In this article, we analyse the facility closing problem of two firms operating in a duopolistic market. The problem is modelled as a non-cooperative game over a binary integer programming formulation of the firms' delocation problem. The possible outcome of the game is analysed for three different competitive scenarios: Myopic behaviour, Cournot conjectures, and Stackelberg strategies. These scenarios are analysed under the assumption that customers show certain level of loyalty to the firm that they initially resorted to. This assumption entitles us to establish the existence of Nash equilibria in the delocation game bymeans of the introduction of a social planner. Moreover, this social planner provides a mechanism for computing the Nash equilibrium in the Cournot delocation game. Additionally, we develop an algorithmic approach that provides a simple mechanism for finding equilibria under Stackelberg strategies. Experimental results indicate that for small reductions in the network size the solution under myopic behaviour is often an equilibrium. For large reductions, evidence has been found that there is a first mover advantage in the Stackelberg delocation game.