Strategy-proof location of public bads in a two-country model

We consider the joint decision of placing public bads in each of two neighboring countries, modelled by two adjacent line segments. Residents of the two countries have single-dipped preferences, determined by the location of the nearest public bad to their dips. A social choice function or rule takes a profile of reported preferences as input and assigns the location of the public bad in each country. All rules satisfying strategy-proofness, country-specific Pareto optimality, non-corruptibility, and the far away condition are characterized. These rules pick only boundary locations