This paper is devoted to the two-opposite-facility location games with a penalty whose amount depends on the distance between the two facilities to be opened by an authority. The two facilities are opposite in that one is popular and the other is obnoxious. Every selfish agent in the game wishes to stay close to the popular facility and stay away from the obnoxious one; its utility is measured by the difference between its distances to the obnoxious facility and the popular one. The authority determines the locations of the two facilities on a line segment where all agents are located. Each agent has its location information as private, and is required to report its location to the authority. Using the reported agent locations as input, an algorithmic mechanism run by the authority outputs the locations of the two facilities with an aim to maximize certain social welfare. The sum-type social welfare concerns with the penalized total utility of all agents, for which we design both randomized and deterministic group strategy-proof mechanisms with provable approximation ratios, and establish a lower bound on the approximation ratio of any deterministic strategy-proof mechanism. The bottleneck-type social welfare concerns with the penalized minimum utility among all agents, for which we propose a deterministic group strategy-proof mechanism that ensures optimality.
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