On location problem of public facilities by collective choice model

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When agents are located on a network, the provision of public goods is considered. Each agent has a single-peaked preference. We showed that if there were at least four agents on a tree network, then no social choice rule existed that satisfied efficiency and replacement-domination. Efficiency is an axiom meaning Pareto optimality. Replacement-domination is an axiom in relation to the idea of solidarity among agents when changes in preferences occur. In fact, these properties were shown to be incompatible, even if agents' preferences were restricted to a smaller domain of symmetric single-peaked preferences.Moreover, it was shown that if agents' peak locations were restricted to either the nodes or the endpoints of trees, rules existed on a subclass of trees. We gave a characterization of a family of such rules for this tree subclass.